Modal Representations for the High-Frequency Response of Elastic Plates Citation Randles, Philip Wayne (1969) Modal Representations for the High-Frequency Response of Elastic Plates. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/F4BM-J980. https://resolver.caltech.edu/CaltechTHESIS:06082017-101810031 Abstract Representations for the high-frequency response of a suddenly loaded infinite plate are obtained from the modal form of the exact solution. The method of approach is presented by treating a linearly elastic, homogeneous, isotropic plate subjected to a normal impulsive line load on one face. An investigation of the branches of the governing Rayleigh-Lamb frequency equation is given. These branches are closely related to the modes of propagation, the sum of which is the modal solution. The relationship between the high-frequency portions of the underlying frequency spectra and the high-frequency response is brought out. Series representations for the branches are used to facilitate a summation over the branch (or mode) numbers. This results in convenient high-frequency representations, which exhibit all of the expected singular wave fronts in the plate. The method appears to be applicable to a broader class of problems than other methods which have been used for the high-frequency response of a plate. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Applied Mechanics and Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Mechanics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Miklowitz, Julius Thesis Committee: Unknown, Unknown Defense Date: 13 May 1969 Funders: Funding Agency Grant Number NSF UNSPECIFIED Caltech UNSPECIFIED Record Number: CaltechTHESIS:06082017-101810031 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06082017-101810031 DOI: 10.7907/F4BM-J980 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10315 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 23 Jun 2017 16:22 Last Modified: 06 May 2024 19:21 Thesis Files Preview PDF - Final Version See Usage Policy. 63MB Repository Staff Only: item control page

MODAL REPRESENTATIONS FOR THE HIGH-FREQUENCY RESPONSE OF ELASTIC PLATES Thesis by Philip Wayne Randles In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1969 (Submitted May 13, 1969) -11- ACKNOWLEDGMENTS The author wishes to thank his thesis advisor, Professor J. Miklowitz, for his guidance and many helpful discussions throughout this work. The author would also like to thank Professor T. K. Caughey for his interest and help in this work. The financial support of a National Science Foundation Graduate Fellowship, a National Science Foundation Traineeship and a teaching assistantship from the California Institute of Technology are greatly appreciated. -iii- ABSTRACT Representations for the high-frequency response of a suddenly loaded infinite plate are obtained from the modal form of the exact solution. The method of approach is pre- sented by treating a linearly elastic, homogeneous, isotropic plate subjected to a normal impulsive line load on one face. An investigation of the branches of the governing Rayleigh-Lamb frequency equation is given. These branches are closely related to the modes of propagation, the sum of which is the modal solution. The relationship between the hi g h-frequency portions of the underlying frequency spectra and the high-frequency response is brought out. Series representations for the branches are used to facilitate a surrunation over the branch (or mode) numbers. This results in convenient high-frequency representations, which exhibit all of the expected singular wave fronts in the plate. The method appears to be applicable to a broader class of problems than other methods which have been used for the high-frequency response of a plate. -iv- TABLE OF CONTENTS CHAPTER TITLE PAGE 1 INTRODUCTION 1 2 STATEMENT OF THE PROBLEM AND THE MODAL SOLUTION . . . • . 7 2.1. STATEMENT OF THE PROBLEM 7 2.2. THE MODAL SOLUTION . . 10 2.3. CHANGE OF VARIABLES IN THE MODAL SOLUTION . . . . . 18 3 THE BRANCHES n=nn(x) 3.1. ANALYTICITY OF THE BRANCHES 3.1.1. 3.1.2. 3.1.3. 3.2. 36 Proof of Analyticity and the Criteria for Singular Points of the Branches 37 The Nature of the Singular Points 44 The Number of Branch Points and Their Location 47 THE BRANCHES ON THE REAL x-AXIS AND ON THE UNIT CIRCLE Ix I=1 . . 3.2.1. 3.2.2. 3.2.3. 3.3. 35 ... . 54 . . . 55 . . . . .. 60 The Real Branches on the Real x-Axis . . .. The Branches on the Unit Circle lxJ=l The Complex Branches on the Real x-Axis SERIES REPRESENTATIONS OF THE BRANCHES n ABOUT THE ·POINTS . x=l AND x=-1 . . . . . . 0 . 0 ... 72 . . . . . 76 -v- CHAPTER TITLE 3.4. ANALYTIC CONTINUATIONS OF THE BRANCHES . . o 3.4.1. 3.4 . 2. 4 PAGE • 101 • Analytic Continuations About the Branch Points 102 Analytic Continuations on a Closed Path . . • . • 111 THE APPROXIMATE REPRESENTATION OF THE HIGH-FREQUENCY RESPONSE . .. . . . . . 4 .1. THE MODAL SOLUTION ON THE EQUIVOLUMINAL AND DILATATIONAL BRANCHES . . . . . . . . . . . . THE HIGH-FREQUENCY APPROXIMATION . . 4. 2. 4.2.1. 4.2.2. 4.2.3. . . The High-F~equency Expansion of the Modal Solution . . . The Geometry of the Wave Fronts . . . . . 116 117 124 124 . Discussion of Specific Terms Yielding Wave Fronts. An Example of a Solution Approximated Near a Wave Front . . . . . . . . . . 145 157 BIBLIOGRAPHY . 169 FIGURES 171 APPENDIX A THE FUNCTION R(X) 193 APPENDIX B ADDITIONAL INFORMATION ABOUT THE BRANCHES n=n n (x) . . . . 201 B.l. B.2. THE BRANCHES AT SPECIAL POINTS . . . 201 THE BEHAVIOR OF THE BRANCHES UNDER THE MAPPING x+l x . . . 212 -vi- CHAPTER B.3. APPENDIX C PAGE TITLE THE COMPLEX BRANCHES ON THE REAL x -AXIS . . . . ADDITIONAL MATERIAL FOR ANALYTIC CONTINUATIONS c .1. C.2. C.3. 226 AN EXPANSION OF THE FREQUENCY EQUATIONS FOR 1-v SMALL . . . . 226 CONTINUATIONS OF THE EQUIVOLUMINAL AND DILATATIONAL BRANCHES ABOUT THEIR SINGULAR POINTS x=-1 and x=l . 228 THE NATURE OF "THE SET OF POSITIVE BRANCH POINTS" 1 FOR V=~ . . C.4. 215 . . . . . . • • . 230 EXAMPLES OF ANALYTIC CONTINUATIONS ON CLOSED PATHS . . . . . . . . . . • . . . 231 -vii- LIST OF FIGURES FIGURE TITLE PAGE 1 The Line Load Problem . 2 Mapping of the Real K-Axis, 0++ onto the Integration Path C in the 4 171 00 , x-Plane 3 . . Cl • • • • • • • • • • o • Mapping of the Real x-Axis, 0++ onto the Integration Path c S for the 0 Symmetrical Rayleigh Branchr and the Integration Path c A for the Asymmetrical 0 Rayleigh Branch in the x-Plane . . . . . 00 , and x , where R(x)=O, 01 02 and of x R' where R( x )= in the x-Plane , as Poisson's Ratio v Varies . The Variation of xR with v . • • . 9 174 1 2 . . . . . . . . . . . . 175 1 e for v=0,0.3, 2 i8 ), . . . . . . . . . . . 176 Branch Points of the Branches n on a-1 02x~a+l (v=0.3). The Triplets (±,j,k) Identify the Branch Points as Zeros of the Functions 6~k( x ) 8 . . y( 8 ), the Argument of R(e versus 7 . • . R( x ) and [R(X) ]-l versus x, -12 x21, for v=0,0.3, 6 173 The Loci of x 00 5 172 • . . . . . . . The Real Branches on the Real x -Axis for -12x21 (v=0 . 3) . . 177 . . .. . . 178 The Equivoluminal Branches on the i8 Unit Circle x=e for 0<8<11' (v=0 . 3) . . .. .. 179 -viii- TITLE FIGURE 10 11 The Dilatational Branches on the · · 1 e x=e ie f or o< e< TI (v= o . 3) Unit Circ 13 xR<x~O • ... 181 The Complex Branches on the Real a-1 x-Axis for -1,::x,::a+l (v=O. 3) ... 182 The Real Branches on 02x21 for the 1 Special Cases v= and v=l 183 A Path in the x-Plane for a Continuation Over Symmetrical Branches . . . . . . . . . . . 184 A Continuation Over Symmetrical Branches Corresponding to the Path in Fig. 14 .. . ..... . 185 A Path in the x-Plane for a Continuation Over Asymmetrical Branches . . . . . . . . . . . 186 A Dilatational Continuation Over Asymmetrical Branches Corresponding to the Path in Fig. 16 . . . . . . • 187 A Path in the x-Plane for a Continuation Over Symmetrical Branches . . . . . . . . . . . 188 An Equivoluminal Continuation Over Symmetrical Branches Corresponding to the Path in Fig. 18 . . • . . • 189 . 2 14 15 16 17 18 19 180 The Bounds on Imn (v=0.3). The Lower Bound is a Portion of the Symmetrical Rayleigh Branch on -12x<xR and i t is the Asymmetrical Rayleigh Branch on 12 PAGE -ix- FIGURE 20 21 TITLE PAGE A Schematic Representation of the Branch Points Common to Pairs of a-1 Real Branches on a+l~X~l (v=0 . 3) • • • • • • The Integration Paths CE and CD. A Deformation of CE. A Deformation . . 22 190 191 The Wave Fronts in the Plate at T=6.3 (V=0.3) ••• 192 -x- NOMENCLATURE A ref er$ to asynunetrical part when used as a subscript A pq see equations A pq see equation (4.2-6) Arg(-) argument of a complex quantity (4.2-4) and (4.2-5) ratio of wave velocities, see equation (2.1-2) B see equation (4.2-10) c integration path in the x-plane, see equation (2.3-lSa) and Fig. 2 path in x -plane on whichjRj=l, see Appendix A integration paths in x-plane, see equation (4.1-2) and Fig. 21 integration paths in x-plane, see equation (2.3-lSa) and Fig. 3 c refers to critical point or branch point when used as subscript Cd' c s wave velocities, see equation (2.1-1) cg group velocity, see equation (2.3-1) c phase velocity, see equation (2 .3-1) p CR Rayleigh surface wave velocity D(x) see equation d refers to dilatational when used as subscript or superscript (2.3~12b) -xi- refers to equivoluminal when used as subscript or superscript e see equation (3.3-7b) see equations (2.2-8) and (2.3-12) see equation (4.1-la) see equation (4.1-lb) see equations (3.3-26) and (3.3-27) see equations (3.3-9) and (3.3-12) see equation (3.3-5) half plate thickness H Heaviside step function see equation (2.1-7) I see equation (4.2-26a) imaginary part of complex quantity i=r-T L=~ 'I normalized propagation coordinate 0 order symbol , see following equation (2.3-3) p used as Laplace transform variable only in Chapter 2 see equation (4.2-20) see equation (2.3-12a) and Appendix A -xii- Re(,...,) real part of complex quantity s refers to symmetrical part when used as a subscript ski see equation (4.2-2a) t time ust;, UAt; I us i'.; I see equation (2.3-16) UAi'.; ~= (ut;, 0 ,ui'.;) u displacement vector see equation (2.1-5) x' u z Ut; I components of displacement vector, see equation (2.1-5) UC'; us t; , UAt; I Us i'.; I see equation (2.2-9) UAi'.; x propagation coordinate y see equation (3.2-6) ze pq' zd normalized thickness coordinates, see equation (4.2-20) pq z thickness coordinate 1 a~(:2 w2-K2r dilatational thickness wave number 1 f3 = (w 2 -K r (X) 2 2 ) equivoluminal thickness wave number see equation (3.1~7a) -xiii- y(6)=ArgR(e ie ) see equation (A-6) in Appendix A Y1(x)=ArgR(X) for x E:: cl see Appendix A see equation (3.1-9) delta functions, see equation (2.1-7) used as a small quantity see equation (3.3-7a) z l;=- dimensionless thickness coordinate n=S+a new variable replacing the dimensionless frequency H n ~o) (x) am see equation (3.3-24) n~k) (x) see equations (3.3-23) and (3.3-25) ( 0) n en ( X) see equation (3.3-3) n (j) (x) e see equations (3.3-14) and (3.3-15) n 1 =Imn nR=Ren nn' nm' np' nq branches of the frequency equation 6=Arg( x ) forJxJ=l eo see equation (2.3-8) and Appendix A e-1 see equation (A-la) in Appendix A -xiv- dimensionless wave number K Lame constants v Poisson's ratio x ~=­ dimensionless propagation coordinate p' material density H stress components, see equation (2.1-6) c t s dimensionless time i:=-- H dilatational phase function, see equation (4. 2-1) see equation (4.2-7) dilatational potential, see equation (2.2-1) S- a x=s+a new variable replacing the dimensionless wave number Xo see equation (2.3-8) and Appendix A see equation (2.3-9) and Appendix A see Appendix A equivoluminal phase function, see equation (4.2-1) 1¥ e ,Q, pq I '¥ d,Q,pq see equation (4.2-7) equivoluminal potential, see equation (2 . 2-1) dimensionless frequency w W n ( K) branches of the frequency equation -1- CHAPTER 1 INTRODUCTION This work is concerned with representations for the high-frequency response of a suddenly loaded infinite plate governed by the equations of motion from linear elasticity theory. The problem has been given attention in the liter- ature only recently. The method of attack has almost ex- elusively exploited Cagniard's method , geometric ray theory, and wave front approximations, as the most recent of these works, that by Rosenfeld and Miklowitz [l] 1 , exhibits. On the other hand, the present work is an investigation which logically asks whether high-frequency representations can be extracted from the modal form of the solution to the problem, which is based on the underlying frequency spectra. Indeed, it is shown here that such representations can be written, and the method appears to be applicable to a broader class of problems than those that Cagniard's method can handle. The method, and representations, are brought forth here by treating the problem of an infinite plate, subjected to an impulsive line load applied normal to one of 1 Numbers in brackets designate references in the bibliography. -2- the faces, which are otherwise free. The plate is assumed to be homogeneous and isotropic. The modal form of the exact solution is a sum over an infinite number of integrals, each of which represents a mode of propagation. The modes are directly related to the branches, or roots, of the Rayleigh-Lamb frequency equation giving a functional relationship between the frequency and wave number of straight-crested waves propagating in the plate. The frequency spectrum of an infinite plate is then the plot of frequency versus wave number. The branches of the Rayleigh-Larr~ frequency equation are now quite well understood, chiefly through the recent efforts of Mindlin and his coworkers (see [2]). Most of the previous work on evaluating the modal solution has been limited to relatively low-frequency, longwavelength calculations so as to limit consideration to a few of the lowest modes or just to the lowest mode of propagation. For example, Miklowitz [3] considered a plate, t h e faces of which were subjected to symmetrical normal point loads having a step dependence on time. Calculations were t h en based on the low-frequency portion of the lowest mode of p ropagation. Such approaches are very useful when only the lowfreque n c y response of the plate is of interest , however, -3- they cannot possibly predict the multitude of wave fronts, which are expected to be present in a · plate subjected to an impulsive loading. The reason for this is that discon- tinuities identified as wave fronts must necessarily be composed of a superposition of high-frequency, short-wavelength waves. To obtain the high frequency response of the plate, Rosenfeld and Miklowitz [l] used Cagniard's method which deals initially with a double-integral representation rather than with the modal solution. This representation results from use of integral transforms. The double- integral representation is expanded in a series, each term of which is finally warped into a single Laplace transform on time, then inverted by inspection to obtain wave front expansions. The wave front expansions are very accurate near the fronts, but they diverge quite rapidly away from them. Earlier uses of Cagniard's method of inversion for plate problems were given by Mencher [4], Broberg [5] and Davids 16]. Further studies on the high- and low-frequency response of plates are pointed out by Miklowitz [7]. Implicit in the problem of finding the high-frequency response of a plate is the need of a method of solution which is valid both at the wave fronts and for some distance behind the fronts. Also, there are two other very -4- important aspects of this problem. First, a clear indica- tion of the relationship between the high-frequency portions of the frequency spectrum and the high-frequency response of the plate including possible singular wave fronts is needed. This will open the way for the investi- . gation of the high-frequency response of other wave propa. gation problems whose solution can be formulated as a sum over modes of propagation. Such problems include other loading configurations on plates as well as wave propagation in anisotropic plates, circular rods, and layered media. Second, a method of solution for the high-frequency response which does not depend on Cagniard's method of inversion is needed since some of the problems just mentioned are apparently not solvable by his method. In Chapter 2 the modal solution for the response of the plate is derived. The frequency and wave number are then replaced by new variables which allow a much more direct investigation of the high-frequency response of the plate. An extension of Mindlin's [2] investigation of the branches of the Rayleigh-Lamb frequency equation is contained in Chapter 3. The most important new result found here, which relates to the high-frequency response, is the existence of analytic continuations of the branches which -5- are closely associated with the dilatational waves in the plate. These continuations are called the dilatational branches and they are found in a region where both the frequency and wave number are pure imaginary. The impor- tance of these branches cannot be overemphasized in the context of this work. Series representations of the dila- tational branches and of the usual branches of the RayleighLamb frequency equations, which are called the equivoluminal branches here, are given in Chapter 3. These series repre- sentations are extremely accurate for high frequency, maki n g them the device which can be used to obtain representations of the high-frequency response of the plate. These series representations for the branches are quite unique since the individual terms in the series · are found, in Chapter 4, to be closely related to the individual wave fronts in the plate. Also contained in Chapter 3 are de- scriptions of analytic continuations of the branches of the Rayleigh-Lamb frequency equations. These continuations are essential for manipulating the integrals involved in the modal solution. In Chapter 4 , the modal solution is shown to be equivalent to sums of integrals over the dilatational branches and over the equivolurninal branches. This equiva- lence is the key point in the solution method used in this -6- work. This is true since the series representations of Chapter 3 can then be used to derive representations of the response of the plate which are valid for high frequency and which exhibit all of the expected wave fronts. The series representations of Chapter 3 are used as if they were exact representations of the branches, and this makes t h e method approximate. However, the series representa- tions for the branches are very good approximations and the validity of the final representations for the response of the plate does not diminish rapidly away from the wave fronts, as is the case for wave front expansions. -7- CHAPTER 2 STATEMENT OF THE PROBLEM AND THE MODAL SOLUTION 2.1. STATEMENT OF THE PROBLEM An infinite plate subjected to an impulsive line load is considered in this work . The loading configuration, which makes this a plane strain problem, is shown in Fig. 1 with the space coordinates x and z indicated . The load is applied in the form of a stress normal to the upper face z=H (the uniform plate thickness being 2H) and it is concentrated on the line x=O. of the plate are traction-free. Otherwise the faces The plate material is linearly elastic, homogeneous and isotropic with no body forces acting. The object is to obtain a representation which is capable of exhibiting the high-frequency response of the plate in terms of the displacement components. It is convenient to define the material constants and wave velocities at this point . cs = (2 . 1-1) are the dilatational and equivoluminal body wave velocities respectively where A andµ are the Lame constants and p' is -8- the material density. a = c = s The ratio of the velocities is (l-\!) ~= ~2 1-2\) (2.1-2) where \! is Poisson's ratio. The notation in the following work is simplified by writing this problem in terms of the dimensionless space and time coordinates ~ x = H l; = Hz T = cs t (2.1-3) H This problem is governed by the equation of motion (2.1-4) for -oo<~<oo,-l<l;<l,-r>O where ~=(u~,O,uz;) is the displace- ment vector and V is the del-operator with respect to the rectangular Cartesian coordinates ~ and z;. (2.1-4) is homogeneous with respect to ~ and hence the dimensions are arbitrary. However, it is advantageous to take (2.1-5) where u x and u z are the original displacement components in the x,z coordinate system. components, u~ Thus, the displacement and uz;, have length squared dimensions. -9- The pertinent stress-strain relations are ass = au au~ (A+2µ) ass + A---aT (2.1-6) a~s = ]J ('u<a s 5-) -- + a~ The boundary conditions on the faces ~=±1 now take the form ass(~ ,-1,T) where I = o is a positive constant, (2.1-7) o (~) is the Dirac delta function and o+(T) is the one-sided delta function. For example , 2 . . (N)~ -7T e -N~ o (U = 11m N-++oo 0,T <O O+(T) - 2 { limN Te -NT , T~O N-++oo (N real and positive) are possible definitions of these ge neralized functions. Hence, from (2.1-7) and these -10- definitions, a quantity measuring the impulse is unit length along the line of loading. The initial condition of quiescence at T=O and the radiation condition (quantities vanish as J~J+ 00 ) are imposed on all independent variables and their derivatives. An important consideration is that (2.1-4) and (2.1-6) are physically accurate only if the strains are infinitesimal, which cannot be expected to be the case for the severe boundary conditions (2.1-7). This problem is justified on the grounds that any solution can be used in conjunction with convolution integrals to solve other more realistic problems in which only infinitesimal strains are expected. 2.2. THE MODAL SOLUTION This problem is easily solved, formally at least, by first representing the displacement vector in terms of displacement potentials and then solving the resulting differential equations by means of integral transforms. This results in a double-integral representation or an infinite sum of single integrals the displacement components. (the modal solution) for -11- The representation u \7¢ + Vx (O,\j!,O) (2.2-1) where ¢ is the scalar displacement potential associated with dilatational motion and (O,\j!,O) is the vector displacement potential associated with equivoluminal motion (l/J is a scalar), reduces the equation of motion (2.1-4) to a2 <D ---'- + as 2 a2¢ 1 a2¢ = 2 --2 ac;; 2 a OT (2.2-2) a2 + a21/J = a21/J -2 2 ac;;2 OT as __ \j!r (2.2-2) applies on the same domain as was indicated follow- ing (2.1-4). The boundary conditions on¢ and \j! are given by (2.1-7) through (2.1-6) and (2.2-1) . The required quiescence and radiation conditions are (2 . 2-3) and lirn[ ¢ (s,1;;,T),l/J(s,c:::,T), ... J = o lsl+ (2 . 2-4) 00 where (2.2-4) contains derivatives of¢ including the second order. and~ up to and -12- A Fourier transform with respect to the propagation coordinate ~ and a Laplace transform with respect to T are applied to ¢ and ~ to reduce (2.2-2) to a pair of ordinary differential equations with s being the independent variable. Solution of these transformed equations subject to the boundary, initial and radiation conditions and also to the inversion integrals given by the pairs 00 F(p) = ~ f (T)e-pTdT f (T) = 2;i~F(p)eP"dp 0 Brl and (2.2-5) 00 F( K) =~~ ( < )eiK<d< 00 f (~) = _!_~F(K)e-iK<dK 2TI · -oo gives double-integral representations of the displacement components u~ and us . Br 1 is the Bromwich contour in the ri ght half of the p-plane. Th e following derivation of the modal solution was done for the symmetrical case b y Miklowitz [3] and the double-integral representation appears in the work of Rosenfeld and Miklowitz [l]. Some of the derivation and the results are repeated here so that they can be used later in this work. -13- Without indicating the intervening algebraic calculations, the double-integral representations for the displacement components take the form (2.2-6) w=-ip is a dimensionless frequency, which is obviously real when p is pure imaginary, and K is a dimensionless wave number for propogation in the ~ direction. The functions (2.2-7) are the dilatational and equivoluminal thickness wave numbers respectively and the functions (2.2-8) -14- will be called the symmetrical and asymmetrical Lamb functions respectively. Rayle~gh­ The functions US~ ( W 1 K1 (;) 2 2 = (w -2K )cosaz;;sinS - uA~(w,K,1;;) 2 2 = (w -2K )sinaz;;cosS 2aSsinacosSz;; - 2aScosasinSz;; (2.2-9) 2 2 2 US(;(w,K,i;;) = (w -2K )sinaz;;sinS + 2K sinasinSz;; 2 2 2 U.Az;;(w,K,(;) = (w -2K )cosa1;;cosS + 2K cosacosSi;; are proportional to components of the transformed displacement. They are written so that the first term in each results from the dilatational potential~ through (2.2-1) and the second term in each results from the equivoluminal potential ~· The decomposition into symmetrical and asymmetrical parts, denoted by the S and A subscripts respectively in (2.2-8) and (2.2-9), results from a convenient separation into two problems both with loadings on the upper and lower faces of the plate. In one problem the load is symmetrical with respect to z;;=O ( oz;;z;;(~,±1,l) -~Io(~)o+(L) = from the loading (2.1-7)) and in the other the load is asymmetrical with respect to i;;=O +~I o (~)o+(L)) . ( oz;;z;; (~,±1 ,l) = Superposition of the two problems gives the problem posed in this work. Actually the purpose of this -15- decomposition is to separate the Rayleigh-Lamb functions (2.2-8) as they appear in (2.2-6). The integrands in (2.2-6) are even with respect to both a and B; hence the radicals in (2.2-7) do not give rise to branch points. It is well known that each of the functions F (w,K) 8 and FA(w,K) in (2.2-8) have doubly infinite sets of simple zeros and no others for 0<K< 00 • Even though the symmetrical and asymmetrical sets of zeros are distinct, the zeros of both cases will be denoted by w=±w (K) with n=0,1,2, ... n and it will be clear from the context whether they are symmetrical or asymmetrical zeros. are ordered as O<w <w <w ... 0 1 2 For 0<K< 00 the zeros The zeros were shown to be simple by Scott and Miklowitz [8] for the symmetrical case and it will become evident for both cases in this work. The zeros B=O for F (w,K) and a=O for FA(w,K) are 8 superficial since they are seen to be removable in (2.2-6~ Thus, the integrands in (2.2-6) have doubly infinite sets of simple poles located on the imaginary p-axis for 0 <K<oo and residue theory is used to carry out the integration over p. A detailed analysis of the representation (2.2-6) or the simple observation that (2.2-2) are hyperbolic differential equations leads to the conclusion that -u=o- for 2 T<!'1~ +(1-s) 2 . a calculation results in l~ 2 . For T2a ~ +(1-s) 2 the residue -16I U.;(.;,s,T) = Trµ f: us.; i= sinK[,sinwT 3FS n=O 0 3W I + -Trµ f: n=O f dK F s =0 w=w (K) n sinK[,sinwT 0 UA.; 3FA 3W dK F =O A w=w (K) n I Us(.;,s,T) = Trµ t (2.2-10) 00 USs JacosK[,sinwT 3FS n=O 0 aw dK F s =0 w=wn (K) t UAs 1:cosK[,sinwT Trµ 3FA n=O 0 I 3W ~- dK FA=O w=w (K) n The modal representation of the solution is precisely (2.2-10) with each of the terms n=0,1,2, . .. called a mode of propagation. The integrands represent residues and are evaluated at the simple poles resulting from F =0 or FA=O 8 or equivalently at w=w (K) where w (K) are the zeros of n the appropriate function F n 8 or FA. Thus, the residue calculation has g{ven rise to the Rayleigh-Lamb frequency equations . F =0 or from (2.2-8) 8 -17tanB + tano: 2 4K 0'.(3 ( W 2 -2K 2) 2 = 0 (2.2-11) is the symmetrical Rayleigh-Lamb frequency equation for a plate which admits branches wn(K), n=0,1,2, . . . . FA=O or from (2.2-8) tan(3 + tano: (2.2-12) is the asymmetrical Rayleigh-Lamb frequency equation for a plate which admits branches w (K), n=0,1,2, . . . . n An important point is that the integration over K and the summation over n have been interchanged in the representations (2.2-10) without mathematical justification. The singular loading (2.1-7) is expected to produces~gular wave fronts; hence either the integration or the summation in (2.2-10) must diverge at these points. If theorems requiring uniform convergence with respect to the parameters are to be used to justify this interchange, either the loading must be made more smooth or the generalized functions in (2.1-7) must be replaced by their defining sequences with the limit to the generalized function taken after these summations and integrations are carried out. The latter is the most direct way to justify (2.2-10) as the response to the loading (2.1-7) and this can be -18- done in a manner similar to that which Lighthill [9] used to treat generalized functions. 2.3. CHANGE OF VARIABLES IN THE MODAL SOLUTION Much of the high-frequency response of a plate, which must be contained in the representation (2.2-10), has apparently been inaccessible through direct integration and summation even when approximations such as asymptotic expansions are employed. The independent variable K in (2.2-10) will be replaced by a new variable in an effort to make the modal solution and especially the frequency spectrum (made up of the branches w (K) as functions of n K) more easily handled with tools such as analytic function theory. Various properties of the branches w (K) of the symn metrical and asymmetrical Rayleigh-Lamb frequency equations (2.2-11) and (2.2-12) respectively, which were given by Mindlin [2] or are implicit in his work, are useful to recount at this point. is real and nonnegative. It is assumed throughout that K The properties are applicable to both the symmetrical and asymmetrical branches unless specified otherwise. (1) The branches are ordered O<w <w <w < ... for 0 1 2 0<K< 00 with wn(O)>O for n=l,2,3, ... and w (0)=0 0 (w (K) will 0 be called the Rayleigh branch in this work; however, i t is -19- usually called the lowest branch of the Rayleigh-Lamb frequency equation). n~l certain integers in (2.1-2) Also,wn_ 1 (0)<wn(O)=wn+l(O)<wn+ 2 (0) for when Poisson's ratio v is such that a is a rational number. Otherwise, w (0), n n=l,2,3, ... , are ordered just as for O<K<oo. W (K) ( 2) n The ratio is a strictly decreasing K W (K) function of K> O as K increases from O++oo with lim K+OO for n=l,2,3, ... Wo (K) . Als~ n K =l for the Rayleigh branches n=O, is a strictly decreasing function of K>O for the K synunetrical case and a strictly increasing function of K>O for the asynunetrical case as K increases from 0++00 with w (K) lim K+oo 0 K = CR c < 1 for both cases (cR being the Rayleigh s surface wave velocity for a half-space). The preceding remarks depend on the well-known fact that c >c in the p g intervals mentioned (except for the asymmetrical Rayleigh branch n=O on which c <c p c g = p for K>O) where wn (K) K c S (2.3-1) are the phase and group velocities respectively for waves propagating in the s direction. These inequalities on cp -20- and c g will be seen to be related to the fact that the branches in the new variables have no singular points on the path of integration except possibly at the endpoints. It is also useful to state the results (2.3-2) for the symmetrical Rayleigh branch and (2.3-3) for the asymmetrical Rayleigh branch as K~O in each case. The order symbol O, as in (2.3-2) and (2.3-3), will be understood to mean the following in this work: (3) if f (K)= For any change of variables, wn(K) and K range through the values 0 < w (K) < oo for 0 < K < oo and 0 0 < w ( K) < oo for 0 < K < oo (n=l, 2, 3, .•. ) • n The new variables are defined as x = [3-a [3+o:. n = S+a t) ( 2. 3-4 I -21where x will replace K as the independent variable in the integrals (2.2-10) and n will play the role of w (K) in n that it is a function of x through the Rayleigh-Lamb fre quency equations. a and S are the thickness wave numbers defined by (2.2-7). Holden [10] used a and S as variables to replace w and K in the Rayleigh-Lamb frequency equations, thus making the frequency spectrum simpler in some respects. The choice of the variables not unlike using a and x and n to replace w and K is S and can be considered as a map- ping of the frequency spectrum into yet another set of variables . However, the x,n variables differ from the w,K and a,S variables in several respects one of which is that all infinities (cases where K-ro:> or w+ 00 ) map into the finite x-plane. The thickness wave numbers have the inverses \ (2.3-5) in terms of x and n. cals defining a and The particular branches of the radi- S in (2.2-7) were unimportant in the preceding integrals, however, now a definite branch must be assigned to each so that x and n will be uniquely defined . This is arbitrary and the result of other choices -22- will be clear later. The branches of a and S are com- pletely defined by requiring that a > 0 for 0 < K < lw (K) - a n a = -iJaJ for lw (K) < K a n (2.3-6) S > 0 for 0 < < W (K) K n S = -i J S J for w ( K) < K n The inverses of (2.3-4) giving w and K as functions of x and n are 1 w = a -Va 2 -l nx2 (2.3-7) 1 K )]2 G( X- a+ a-l)(a+l = 21 n C 1 a -1 -x where the branches of the radicals are chosen so that w and K are always real and nonnegative in the integrals (2. 2-10) (unless the integration path is deformed). The following remarks about the . definitions x result from (2.3-4) and (2.2-7) and (2.3-6) of a and s. W ( 1) x is a real positive function of W n n=l,2,3,. • • / on oo> - n ( K) K ( K) K ->a which is strictly increasing -23a-1 wn(K) from x=a+l -++l as K decreases from +co-+a. In this case and in the cases to follow, the strictly increasing or decreasing property of the functions does not necessarily include the endpoints of the intervals. The point = +oo(K=O) maps into x = a-1 a+l and the point = a (K=!w a n (K)) . maps into x = +l. ( 2) x=ei 8 with 8=2tanW 1 (1%1), 0~8~TI, where 8 ( K) n is a real positive function of ~~~, n=l,2,3, •.. , on K wn(K) a2 K 21, which is strictly increasing from 8=0-+n as decreases from a+l. . t Th e poin ( K) nK =a maps into W 8=0 or x=+l as given above and the point = 1 (K=+oo) maps into x = -1. (3) For the symmetrical Rayleigh branch n=O, x=ei 8 with 8=2tan- 1 (1%1), 80~8~TI, where 8 is a real posi- w ( K) tive function of 0 K -24WQ (K) which is strictly increasing from e=e +n as 0 creases from ~ + 1 . e K de- is defined by 0 = 1 - 2v 2 + 2iv..J1-v 2 so that O::_e < 0 1 7T 2 for O::_v~ 2 . (2.3-8) The point = ~ (K=O) maps into x = x 0 and the point = 1 maps into x = - 1. (4) For the symmetrical Rayleigh branch n=O, X W Q ( K) is a real negative function of K W Q ( K) on l> K CR > - C s ' w (K) 0 which is strictly increasing from x=-l+xR as K decR creases from l+C I where s 2 J-(1- ::: ) ( 1cR2 XR = - Cd (1- :::)~ (1-::22) + (2.3-9) -25O~v~ with all of the radicals positive and O>xR>-1 for 1 2. WO (K) The point K =l maps into x=-1 as given above and the point CR = c (K=+oo) maps into x = XR· s (5) For the asymmetrical Rayleigh branch n=O, x is a real negative function of w0 (K) w0 (K) CR on O<--<- , K K which is strictly decreasing from x=O+xR as cR creases from 0+-. cs - C s w (K) 0 K in- The point = O(K=O) maps into x = 0 and the point = The preceding remarks and the remarks at the first of ( K) W this section about the variation of n K as K increases on the various branches specify a continuous one-to-one mapping of the real positive K-axis, which is the integration path in the integrals (2.2-10), into the x-plane for -26- each branch w (K). n In the following work 0 < - K - l:w a (2.3-10) < will be called "the dilatational sector" of the w,K-plane and 1 -uJ<K<W a - (2.3-11) will be called "the equivoluminal sector" of the w,Kplane . This terminology is not physically precise and is only used for identification purposes. The mapping of the integration path on the real K-axis 0<K< 00 in the integrals (2.2-10) onto the integration path C in the x-plane for the branches n=l,2,3, ... is shown in Fig . 2. In that figure it is seen that the dilatational a-1 sector maps onto the segment a+l~X~l of the real x-axis and that the equivoluminal sector maps onto the upper half unit · 1 e x=e ie I ClrC o2e2TI• The direction on C in the x-plane is a-l · · · a+l +1+- 1 correspon d'ing to 0 ++oo on t h e positive rea 1 K-axis with :~i+l on the positive real x-axis and l+-1 on the unit circle x=e i8 I 02 e2TI. The numbers 1 , 2 I 3 in Fig. 2 denote speci fie points on a given branch w (K), with 3 meant to indicate n the limiting point K+ 00 , and they also indicate the mapping of these points into the x-plane . The integration path C is valid for all branches n=l,2,3, . . . . The corresponding integration paths for the Rayleigh branches n=O are shown in Fig. 3 . For the symmetrical -27- Rayleigh branch the real K-axis with the direction O++oo maps onto the integration path direction x +-l+xR· 0 (2.3-9). x 0 c 0 s in the x-plane with the is given by (2.3-8) and xR by The portion x +-l of COS is on the upper half 0 unit circle and the portion -l+xR is on the negative real axis. The numbers 1, 2, 3 in Fig. 3 denote points on the symmetrical Rayleigh branch and their mapping into the x-plane with 3 indicating the limiting point K+ 00 • For the asymmetrical Rayleigh branch the real K-axis with the direction 0++00 maps onto the integration path c 0 A in the x-plane with the direction O+xR on the negative real x-axi& The numbers I, II in Fig. 3 denote points on the asymmetrical Rayleigh branch and their mapping into the x-plane. The w and K scales in Fig. 3 are expanded with respect to those in Fig. 2 so that the details of the Rayleigh branches can be seen more clearly. If branches other than those defined by chosen for a and also the paths lows: (2.3-6) are B, the integration path C in Fig. c 0 S and c 0 A in Fig. 2 and 3 are changed as fol- the paths of integration are reflected about the real x-axis if the sign of a (or B) is changed only when it is pure imaginary; the paths of integration are reflected about the unit circle JxJ=l by the mapping x+l if x the sign of a (or B) is changed throughout. In the latter case all points interior to JxJ=l map to the exterior like -28- a-1 a+l 1 a+l + a-l' xR+xR' etc., and the points on the unit circle are just reflected about the real x-axis. Any of these three other choices represents the mapping of the positive real K-axis into the x-plane equally well. It is significant that the mapping determined by (2.3-4) and (2.3-6) for this work has mapped all of the infinity points in the original w,K-plane, that is, the points where K+oo or w (K)+oo for any branch n=0,1,2, •.. , n into the finite x-plane with !xl~l. Also, from the fore- going, it appears that all of the complex K-plane, taken as a projection from the w,K-space with w and K related by one of the Rayleigh-Lamb frequency equations, has been mapped onto the unit disk lxl~l. Likewise, the mapping can be onto lxl~l by choosing other branches of a or$. Expressions required to change the integrals (2.2-10) from the K independent variable to the x independent variable are listed below in terms of Lamb functions x and n. The Rayleigh- (2.2-8) become (2.3-12) where -29- = 1 (l+x) X ( 1 +x) 3 3 - a (1-v) (l+vx)x a (1-v) ( v+x) x (2.3-12a) and D (x) = (l+x) 3 - a (1-v) (v+x) x (2.3-12b) Hence, the symmetrical Rayleigh-Lamb frequency equation (2.2-11) becomes (2.3-13) sinn = -R(x)sinxn and the asymmetrical Rayleigh-Lamb frequency equation (2.2-12) becomes sinn = R(x)sinxn These equations give the branches n=n which are identified with w (K) n (2.3-14) n (x), n=0,1,2, ... , through (2.3-4). Hence- forth (2.3-13) and (2.3-14) will just be called the symmetrical and asymetrical frequency equations respectively. In the preceding forms of the frequency equations, use has been made of the fact that -30- is precisely the equation whose zero gives the Rayleigh surface wave velocity. S defined by It is easy to show that, for a and (2.2-7) and (2 . 3-6), this equation has only w CR the zero -= - , that is, the asymptote of the Rayleigh K branches, on CS 0<K~ 00 • Hence, all of the branches of the Rayleigh-Lamb frequency equations (2.2-11) are contained in (2.3-13) and (2.2-12) and (2.3-14) respectively as branches defined in the x-plane. It should be noted that x =xR from (2.3-9) is a zero of D(X) and that, D(X) being a cubic, the other two zeros are usually discarded as being identified with physically meaningless Rayleigh velocities. There are other expressions such as ~~(;~ F=O)-l• which are required as functions of x and n for the integrals; however, they will not be listed here. This par- ticular expression requires the chain rule of differentiaCJ F tion to calculate 3W from (2.3-12) and (2.3-7) and requires the derivative ~~ from (2.3-13) or (2.3-14) to calculate dK dx from (2 . 3-7). In terms of x and becomes n, the modal solution (2.2-10) -31US (i; , I'; 1 T ) = aI 2 ~ (usi; n +uAi;n) 7TJJ..J"Ti n-0 4 0 ~o (2.3-15) ()() Ul';(/; 1 1';,T) = 4TIµ n- (us1;n -uA1;n) where usi;n =I (l-x 2 )sin(Ki;)sin(w-r)u 8 ~ 3 c x 2 x D <x) (cosn+xRcosxn) 1 (l-x 2 )sin(K~)sin(w-r)UA~ UAl;n = c i x 3 2 X D(X) (cosn-xRcosxn) (l-x) 2 (l+x)cos(K~)sin(w-r)U 8 s ussn = c -=-3~~~~~~~~~~~1~~~~~~~-ax - [ UAl';n - 2 x D <x) [ ( ' x-:~i)( :~i-x)J2 (cosn+xRcosxn) . (l-x) 3 , 2 (1+x)cos(Ki;)sin(w-r)UAI'; 1 c /o (x) [ (x-::i) (:~ i-x)]2~ (cosn-xRcosxn) X (2.3-15a) In (2.3-lSa) w and K are given by (2.3-7) with n=n (X), n n=0,1,2, . .. , understood to be branches of the symmetrical frequency equation (2.3-13) for integrals with an S subscript and branches of the asymmetrical frequency equation -32- (2.3-14) for integrals with an A subscript. C is the inte- gration path shown in Fig . 2 for n=l,2,3, ... and it is c 0 S and c 0 A shown in Fig. replaced by 3 for the symmetrical and asymmetrical Rayleigh branches n=O respectively. The radicals in (2.3-15a) are defined just as in (2.3-7) so that w and K are real and nonnegative. The two factors cosn±xRcosxn in (2.3-15a) will be found to play a very important role in determining the properties of the branches n=n n (X). They result from the term ~~ which is required for calculating ~~ in the change of variables. These factors replace the rather complicated functions of w and K , (2.2-10). The functions oF s oFA 8w- and --aw' in the representation -33- ) . n (1-x) . n (l+x) c:.; + ( x- a-l)(a+l a+l a-l -x sin 2 2 sin a-l)(a+l ) n (1-x) n (l+x) r; + ( x-a+l a-1-x cos 2 cos 2 are proportional to the functions (2.3-16) u in (2.2-9) and are also written with the first term in each deriving from the dilatational potential ¢ through (2.2-1) and the second term from the equivoluminal potential~· The integrals (2.3-lSa) actually offer no computational advantages over those in (2.2-10). Neither form can be integrated explicitly and they exhibit no characteristics which make numerical computation seem useful. These remarks should be qualified for the Rayleigh modes of propagation n=O where a wave singularity arises from the single branches and no summation is involved if only this contribution to the response is sought. (K) W equivoluminal asymptotes, Also, the n K +las K++oo for n=l,2,3, .. • , may lend themselves to computation which would yield some wave fronts traveling with the equivoluminal body wave velocity. The frequency spectrum is well understood due to Mindlin [2] and others; however, the complexity of much -34of the high-frequency portion of the spectrum is not at all conducive to integration . Hence, any advantage which can be derived from the form (2.3-lSa) must be obtained by manipulation of the integrals such as deforming the contours of integration, expanding functions, etc. As will be shown later, it is by these methods that the integrals in (2.3-lSa) can be evaluated for the high-frequency response for the present problem. -35- CHAPTER 3 THE BRANCHES n=n n (x) A great deal must be known about the branches n=n n (X) satisfying the symmetrical frequency equation (2.3-13) and the asymmetrical frequency equation (2.3-14) before the integrals (2.3-lSa) can be evaluated even in an approximate sense. In this chapter, existence of the branches on the real x-axis and on the unit circle JxJ=l is shown and properties of these brances which may be useful in evaluating the integrals (2.3-lSa) are developed. The branches which map into the real w, real K- or the real w, imaginary K-plane by (2.3-7) were completely explained by Mindlin [2] and that work is repeated here only to show how these branches map into the x,n-space. The branches on which w is real and K is complex (K being .neither pure real nor pure imaginary) map into the x,nspace with both x and n complex and are not convenient to examine in these variables. complex K Other than for these real w, branches, this chapter is a limited extension of Mindlin's work in that branches are considered on which both w and K are complex. The function R(x) defined by (2.3-12a) is obviously important in determining the properties of the branches n. R(x) is treated in Appendix A and the results contained -36- there will be referred to throughout this chapter. Figs. 4, 5 and 6 refer to R(x). Also, Fig. 4 shows some special 1 points where R=O or [R]- =0, Fig. 5 shows a sketch of R and of [R]-l on -l~x~l and Fig. 6 shows a sketch of y(8)=ArgR(e ie ) on x=e ie I 0~8~TI. Appendix B supplements Section 3.2 and Appendix C pertains to the analytic continuations of the branches, which are discussed in Section 3.4. 3.1. ANALYTICITY OF THE BRANCHES Perhaps the most important property of the branches is that they are analytic functions of the complex variable x except at certain singular points in the x-plane. Thus, the powerful tools of analytic continuation and the Cauchy-Goursat theorem for integration in the complex xplane are available for use on the modal solution integrals (2.3-lSa). If a branch can be identified and under- stood on a subregion such as a segment of the real x-axis or of the unit circle lxl=l, then the branch can be continued into any part of the x-plane where it is known to be analytic. Analyticity of the branches is shown and an investigation of the singular points is given in this section. -37- 3.1.1. Proof of Analyticity and the Criteria for Singular Points of the Branches. The frequency equations, (2.3-13) and (2.3-14), can be written in the form (3.1-1) where F is analytic with respect to both x and n except at a few isolated subregions of the x,n-space. F(x,n) For example, can be taken as one of the Rayleigh-Lamb functions defined in (2.3-12), which are analytic functions of x and n except at n=oo and at x= 00 • Hence, the derivative of n with respect to x exists and is given by 8F = _ 8x (3.1-2) 8F 3r1 and, by definition, n is an analytic function of x in any domain where F(x,n) is analytic and where derivative ~~~O. The (3.1-2) is written explicitly as dn = _nRcosxn + R'sinxn dX cosn + xRcosxn (3.1-3) for the branches satisfying the symmetrical frequency equation (2.3-13) and as dn = nRcosxn + R'sinxn dx cosn - xRcosxn (3.1-4) -38- for the branches satisfying the asyrrunetrical frequency equation (2.3-14) where R'=~~ in each case. Singular points where the branches are not analytic because the derivatives (3.1-3) or (3.1-4) fail to exist may occur at any point where (1) R + 00 (2) n + 00 ( 3) cosn ± xRcosxn = 0 ~ (3.1-5) The first case only occurs at x=O and x=xR in Ix Isl as can be seen from Appendix A and the second case will be found to occur only at a few isolated points. In fact, it can be shown directly from the frequency equations and the properties of R(x) given in Appendix A that the branches n are always bounded on lxl~l except possibly on the real x-axis. This is done by assuming that n+ 00 and showing that both sides of the frequency equations cannot be balanced regardless of the manner in which n becomes unbounded as long as x is not real. The cases on the real x-axis where n+ 00 will be considered in a later section. The third case in (3.1-5) occurs at many isolated points in the x-plane which will be shown to be branch points common to pairs of branches. A useful property of the branches is that they are also analytic with respect to Poisson's ratio v as can be -39- seen by extending the foregoing argument. (2.3-12a) R(x)=R(x,v) in is analytic with respect to both x and v except at isolated poles. Hence, the frequency equations can be written as in (3.1-1) with F also analytic with respect to v. This property makes it possible to observe the branches as v varies and to reach conclusions about n=nn(X) for v fixed. The third case of (3.1-5) is now considered in more detail. If this case is satisfied for the symmetrical branches, then some pairs x,n (these are not yet known to be isolated points in the x,n-space) must simultaneously satisfy sinn = -Rsinxn } (3.1-61 cosn = -xRcosxn the first of these relations being the symmetrical frequency equation (2.3-13). Simultaneous satisfaction of these two equations is equivalent to requiring n to satisfy (3.1-7) n = .12£. x simultaneously where -401 r = r(x) = (1-x 2~2)2 (3.l-7a) i-x j and k are arbitrary integers except that j+k must be odd. The choice of the ± signs must be consistent for both equations . The pair (3.1-7) is derived by squaring both equa- tions in (3 . 1-6) and adding to give the second equation in 2 (3.1-7) and by squaring both, multiplying the first by x , and adding to give the first of (3.1-7) . Requiring that j+k be odd insures that (3.1-7) are equivalent to (3.1-6). This is seen by taking n from the first of (3 . 1-7) for the left side of each of (3.1-6) and xn from the second of (3 . 1-7) for the right side of each of (3.1-6). Then it is '+k seen that (-l)J =-1 is required to make (3.1-6) and (3.1-7) equivalent. r is a single-valued, analytic function of x in a cut x -plane (meaning that the branch points resulting from the radical in (3.l-7a) are located and connected by branch cuts in such a way that r is single-valued on all of the x -plane e x cepting the branch points and branch cuts). only has branch points at x =0, 00 ,x 0 r and x * (x * is the com0 0 plex conjugate of x ) as can be deduced from Appendix A. 0 A branch cut can be taken on the negative imaginary x-axis connecting x =O and x =-i 00 and another branch cut can connect x =x 0 and x =x * so that it is not interior to the unit 0 -41- circle lxl=l. For definiteness, r is taken to be real and positive when l-x2R2 is real and positive. 2 1-x Hence, the two possible branches resulting from the radical are denoted by ±f which accounts for the ± signs in (3.1-7). The inverse sine functions in (3.1-7) are also defined as single-valued, analytic functions of x in a cut x-plane and they approach zero as x approaches zero. The functions n defined by (3.1-7) have all of the branch points and branch cuts of r plus branch points at a-1 a+l x=a+l' a-l where R=l and f=l, which produces square root branch points in the inverse sine functions. Also, n given by the first equation of (3.1-7) has logarithmic branch points at x=xR' ~1 -, Xo1 1 Xo2 where R=oo and f =00 and n given by the second equation of (3.1-7) has logarithmic branch points at x=x 01 , x 02 , 1 X-' 00 1 where R=O and Rr= 00 • R Appendix A defines these points. The multitude of combinations of ±,j,k in (3.1-7) then represents the various branches n (not to be confused with the branches n=nn(x) satisfying the frequency equation). A point x where any branch n given by the first expression in (3.1-7) is common with any branch n given by the second expression is a possible singular point of some branch n=n n (x) of the symmetrical frequency equation. point x=-1 is an exception for the symmetrical branches The -42- since (3.1-7) are satisfied simultaneously at this point for some combinations of ±,j,k, but it will be shown not to be a singularity of the branches. This occurs only at x=-1 where the pair (3.1-6) reduces to the identities sinn=sinn and cosn=cosn. These equations appear to be satisfied by any value n(-1) and certainly (3.1-7) are satisfied simultaneously at x=-1 with the proper choice of ±,j,k. This is a special case on the symmetrical branches which is treated in B.l of Appendix B where it is shown that the symmetrical branches are analytic at x=-1. For the asymmetrical branches, simultaneous satisfaction of sinn = Rsinxn } (3.1-B) cosn = xRcosxn the first being the asymmetrical frequency equation (2.3-14) and the second being the third case of (3.1-5), is analogous to the pair (3.1-6) for the symmetrical branches. This is equivalent to requiring n to satisfy (3.1-7) simultaneously with j+k even. Thus, simultaneous satisfaction of (3.1-7) locates singularities of both the symmetrical and asymmetrical branches with j+k determining the particular case. Th e point x =l is an exception for the asymmetrical branches for the same reason that x=-1 was for the -43- symmetrical branches. For x=l, (3.1-8) reduces to sinn=sinn and cosn=cosn allowing any value n(l) as a solution and (3.1-7) are satisfied for certain choices of ±,j,k. However, in B . l of Appendix Bit is shown that the asymmetrical branches are analytic at x=l. In discussing these singular points, it is more natural to consider the zeros of the difference between the functions n defined by (3.1-7). This being done, the singular points of the branches arising from the third case of (3.1-5) are identified as zeros of the function (3.1-9) with j+k odd for the symmetrical branches and j+k even for the asymmetrical branches. For each fixed combination of j,k and either the plus or minus signs in (3.1-9) + (again + these distinguish a given branch 6jk), 6jk is a singlevalued, analytic function of x in a cut x-plane. The func- + tion 6jk has all of the branch points and branch cuts of the two functions n defined in (3.1-7). The singular points of the branches n=n n (x) , which + are identified as zeros of 6jk' are certainly isolated because the zeros of an analytic function are isolated (see Alfors (11], p. 102). Furthermore, it is obvious that if each of these zeros is identified by a pair, say -44+ Xc'nc where Xe is a zero of 6jk for a fixed combination ±,j,k and nc corresponds to Xe through either of (3.1-7) then this pair determines a point in the four-dimensional x,n-space which is distinct from any other point corresponding to another pair x ,n c c identified with a different combination ±,j,k. 3.1.2. The Nature of the Singular Points. Knowing that the singular points are isolated, it is advantageous to investigate the nature of these points before proceeding to determine their number and location. For this purpose the symmetrical branches are considered, and it is assumed that a singular point is given by the pair Xc'nc' which is identified with a zero of 6~k' as in the foregoing, for a fixed combinatiqn ±,j,k with j+k odd. For the moment it is assumed that n c R(x c )cosx c n c + R' (x c )sinx c n c so that the derivative t- o (3.1-10) ~~ in (3.1-3) truly does not exist. Based on an argument analogous to that given for the branches n=nn(x)' an inverse function x=x(n) with xc=x(nc) exists, which is analytic in a tieighborhood of and including the point nc in the n-plane. The derivative of this ax(n > function is given by the inverse of (3.1-3) with dn c =O -45- by the definition of the pair xc,nc and the second derivative is = nRcosxn + R'sinxn (3.1-11) x=xc n=n c 2 d X (nc) Hence, dn 2 10 unless xc=±l or sinnc=O. The points x=±l have already been seen to be exceptions and from (3.1-7) sinnc=O implies that r=O which only occurs at x=O,~,x 0 ,x * 0 so that all of these are special points, which can be examined separately. Other than at these exceptional points, the local Taylor's series development of x(n) about nc is (3.1-12) which can be inverted locally to give nc + (3.1-13) This behavior of n(x) near x=xc shows clearly the nature of the singularity at this point, and it is a square root branch point with the function n(x) being -46- double-valued. Thus, x c is a branch point common to two branches, say n n (x) and n n , (x), which are defined locally by (3.1-13) if they are identified with the two branches of the radical defined to be single-valued in the x-plane with a branch cut emanating from Xe· The case where (3.1-10) does not hold and dn in dx 0 (3.1-3) takes the 0 indeterminant form at x=x c is now considered. This gives another relation, in addition to the two in (3.1-6), since the numerator of dn also vandx ishes and these three conditions can be shown to cause + d ll jk to vanish at x=xc. lljk and dx + Hence, if dn takes the -0 0 dx + form, Xe is a zero of order higher than one of lljk· This can also be interpreted as a point of tangency between the functions n defined by (3.1-7). Such a case is like the + end result of a merging of two or more zeros of lljk to the single point Xe· For all of the points satisfying this situation, other than some of the exceptional points already mentioned, this merging can be made to occur by varying Poisson's ratio \). If this involves the merging of only two branch points, as will be shown later to be the case, and if they are common to the same pair of branches, say n (x) and n , (x ), then the branch points n n merge to cancel each other and the branches become analytic at the point x . c A simple example illustrating this local -47- behavior of two merging branch points is given by the 2 2 ~ double-valued function (x -s ) as s+O. This can be shown to be precisely the behavior of the branches n=n n (X) as v varies when the case under discussion occurs. It will soon become apparent that this is a very restricted case. In fact, it is precisely the case which was investigated by Onoe, McNiven and Mindlin [12] for the similar circular rod frequency equation. In that work this critical point was called a saddle point in the w,K,vspace. The asynunetrical branches have branch points which are exactly analogous to those just discussed for the symmetrical branches. 3.1.3. The Number of Branch Points and Their Location. With the exception of the points x=O and x=x , it can 0 be shown that there are no branch points resulting from the third case of (3.1-5) on the segments -l~x~O or on the unit circle x=e i6 ~1 a+l<x~l for 0<6<TI. and This is done by using the inequalities (A-5) and the form (A-6) from Appendix A and just showing that the frequency equations and the equations in the third case of (3.1-5) cannot be satisfied simultaneously on these segments. The excep- tions inciude the w=K=O points of the Rayleigh branches -48ie where x=x =e 0 0 and n=O, with x 0 given by (2.3-8), for the symmetrical Rayleigh branch and x=n=O for the asymmetrical Rayleigh branch. a-1 On the segment O<x<-- , there are zeros of the funca+ 1 + tions 6jk in (3.1-9) and, hence, there are branch points common to pairs of branches n=n n (x). This is shown in Fig. 7 by indicating the intersection points of the functions n given in (3.1-7). The functions n are sketched, rather + than the functions 6jk' so that the branch points occur in the Re x ,Ren-plane and they are actually points common to pairs of branches n=n n (x) which will be considered in the next section (Re and Im are used to denote the real and imaginary parts respectively of the quantities in question~ + Each function 6 jk can be deduced from Fi.g. 7 as the difference between two of the functions n, the two being given by (3.1-7) for a fixed choice of ±,j,k. Each branch point in Fig. 7 is designated by the + triplet (±,j,k) indicating that it is a zero of 6jk given by (3.1-9). + A given function 6jk has either one zero or a-1 a-1 no zeros at all on O<x<a+l except possibly near x=a+l where some functions two zeros. 6~k with ±(-l)k=+l and j,k>l may have Two such zeros of 6 3111 corresponding to branch points of asymmetrical branches are shown in Fig. 7 between a-1 n=llrr and n=l2n and near x=a+l· + If a function 6jk does have two distinct zeros, then a slight increase in -49- Poisson's ratio v causes these to merge into a double zero and then vanish completely from the real x-axis. This is the case of branch points merging to cancel each other that Alternatively, decreasing v causes was spoken of earlier. branch points to migrate onto the real x-axis for O<x<:~i and they apparently come from the complex x-plane. The point x=:~i is a singular point of the function + 6jk· Specifically, it is a branch point through the in- verse sine functions since R=l and f=l at this point; however, it is possible for this point to also be a branch point of the branches n=n n (x) for some integers n. This will be the case if (3.1-6) or (3.1-8) are simultaneously · f'ie d at x =a+l a-l or i' f satis k kTI - J.Tia+l + (-1) TI = Q a-1 a-1 where ~ has been taken for the value of the inverse sine functions at this point. This equation can be solved for a to give the rational number a = k+j±(-l)k k-j which indicates that these branch points are precisely the mapping into the x-plane of the points on K=O where wn(O)=wn+l(O) is possible as was discussed in Chapter 2. -50- These points are not singular in the w,K-space and they have become branch points in the x,n-space as a result of the particular mapping (2.3-4) introduced in this work. The question of branch points interior to the upper half unit disk (lxl<l and Imx>O) is now investigated. This is accomplished by use of the argument principle (see Alfors [11], p. 123). . The argument principle is applicable + since each branch, 6jk' is analytic and single-valued interior to the upper half unit disk except for a logarithmic branch point at x=x 01 due to the fact that R(x 01 )=0 (XOl is interior only if vc<v~~ with vc defined following (A-3) in Appendix A). This logarithmic branch point can be excluded a-1 by a branch cut from x 01 to the boundary a+l<x<l, which + avoids any of the zeros of 6jk· The point x 01 is not a branch point of the branches n=nn(X) since neither (3.1-6) nor (3.1-8) can be satisfied simultaneously at this point. + Actually, knowledge of 6jk interior to the upper half unit disk, other than the property of analyticity, is not required for application of the argument principle; how+ ever 6jk must be known on the boundary of this region. As mentioned before, the functions 6~k are implicit in Fig. 7 for a-1 a+ 1 O<x<~- . + The form of 6jk is not complicated on the remainder of the real x-axis. On the unit circle, takes a simple form due to (A-6) in Appendix A. + ~jk With the form of 6~k known, the real and imaginary parts of ~jk and, -51+ hence, the change in the argument of 6jk can be deduced for one traversal of the boundary in the positive direction. Once this is done for a fixed combination ±,j,k, it is a simple matter to deduce the results for other combinations due to the simple dependence of 6jk on ±,j,k in (3.1-9). + The number of zeros of the functions 6jk<x) in (3.1-9) or, equivalently, the number of branch points of the branches n=nn(x), as calculated by the argument principle, is given below. Case I: k ± (-1) =+l. (a) The function 6jk (X) =kTI-?+sin- (r)-~ sin- (~r) 1 1 has one simple zero interior to the upper half unit disk for each pair j,k with j=l,2,3, .•• and k=l,2,3, ..• provided that 1 -1(1 (j-2)7T < k1T < min[? - sin -1 (f) + Xsin Rf)] a-1 O<x<--1 a+ (these will be called "the set of positive branch points" henceforth) . The upper bound on k involves a minimum over a-1 O<x<a+l of a quantity in brackets which is actually just + kTI-6jk· This bound is explained by saying that when it is . + a-1 violated the function 6jk has one or two zeros on 0 <x<a+l as shown in Fig. 7. Hence, this bound on k defines the transition between branch points on the real x-axis and -52- branch points in the complex x-plane off the real axis. The bound j-2<k has no obvious explanation, but it can be shown to distinguish between branch points exterior to and those interior to the unit circle lxl=l. At these branch points the bounds kTI < Ren < (k+l)TI on Ren result from (3.1-7) and knowledge of the range of 1 sin- (r) on the upper half unit disk. (b) + 6jk has one simple zero interior to the upper half unit disk for each pair j,k with k=l,2,3, ••• and j=-1,-2,-3, ... provided that k > -(j+l) (these will be called "the set of negative branch points" henceforth). Again, the bound on k has no obvious expla- nation, but it can be shown to distinguish between branch points exterior to and those interior to the unit circle. The value of Ren at these branch points satisfies kTI < Ren < (k+l)TI with the bounds explained just as in (a). This includes all of the zeros of Case I for k=0,1,2, ... and j=0,±1,±2, . . . . -53Case II: ± (-1) k =-1. + -i TI -1 (f)+Xsin 1 -1( Rr 1 ) The function lljk(x)=kn-7-sin has no zeros interior to the upper half unit disk for any pair j,k with k=0,1,2, . .. and j=0,±1,±2, • . . . A natural analytic continuation of this function into the lower half unit disk does have zeros which are complex conjugates of the zeros found in Case I . Negative values of the integer k have not been considered since they only lead to branch points for branches with Ren<O which are simple reflections of the branches whose branch points were just considered . In sununary, all of the branch points or singularities due to the three cases in (3.1-5) have been identified for those branches with on the real x-axis . the branches n=n n Ren~O at the point ~ith some exceptions These exceptions will be examined when (x) are understood in more detail. It is interesting to examine the mapping of these branch points back into the w,K-space by (2.3-7). With the exception of the branch points which may occur at a-1 map into K=O with wlO, all branch points with x=-a+l and the local behavior (3.1-13) give the local behavior in the w,K-space with wc,Kc (both may be complex) being -54- t n e mapping of the branch point xc,nc. This is equivalent to requiring that K=K c which means that the phase and group velocities defined in (2.3-1) are equal at K=Kc in the special case where both wc and Kc are real . branches n =n n Thus, absence of branch points of the (X) on the integration paths c 0 S a nd c 0A in Fig. c in Fig. 2 and 3, as has been shown in all but a few e x ceptional cases, guarantees that the phase and group velocities are not equal on the real positive K-axis for any of the branches w=w (K), n=0,1,2, . . • . n 3.2. THE BRANCHES ON THE REAL x-AXIS AND ON THE UNIT CIRCLE Jx J=l One result of the change to the variables x ,n in Chapter 2 is that the frequency equations (2.3-13) and ( 2. 3-1 4 ) are satisfied by simple forms of the branches n=nn (X) on the real x -axis and on the unit circle JxJ=l. Th is sectio n contains verification of the existence and form of n=n n (x ) on which both w and K are real from (2 . 3-7), plus other branches on these subregions . These oth er branches will be shown in a later section to be -55- analytic continuations of n=n n (x); however the notation n (x) is reserved for the branches on which w and K are n real with n ,n , etc., denoting other branches. m q The discussion is restricted to values of n with Ren~O since it is obvious that if n satisfies either of the frequency equations then -n does also. 3.2.1. The Real Branches on the Real x-Axis. In this subsection the branches n which are real on the real x-axis are considered and most of the results are Only the segment -l~x~l of the real shown in Fig. 8. x-axis is considered since B.2 in Appendix B gives the 1 mapping of the branches under x+-. x A convenient form of the symmetrical frequency equaa-1 tion (2.3-13) on a+l<x<l is (3.2-1) n=l,2,3, . . . . If n is assumed to be real, the inverse sine function is defined by requir~ng that . -1( ( ) . TI R x sinxn ) < 2 2 < sin TI as can be done on this interval since -l<R(x)<l by (A-5) in Appendix A. The form (3.2-1) does not yield explicit solutions n; however it immediately shows the existence -56- of the branches n=n n (x) and also it provides a method of finding solution points if they are required. This is seer by considering the simultaneous solution of the two parametric equations 1 n = nn + (-l)n+lsin- IR(x)sinPn] } Pn n = x where P is a positive, constant parameter. (3.2-2) This pair is equivalent to (3.2-1) as can be seen by setting xn=Pn. As P is varied continuously, the solution points of (3.2-2) form a branch n=n n (x), which is a real, analytic function of x in this interval for each n=l,2f3, • . . • The bounds (3.2-3) result from (3.2-1) and (3.2-2) by setting sinxn=±l to make I (-l)n+lsin- 1 [R(x)sinxn] J a maximum. Thus, one of the equalities in (3.2-3) is taken by a branch nn(x) at a (2m+l)TI 2x m=0,1,2, ... , so that sinxn =(-l)m (or equivalently, if n P= 2m;l in (3.2-2)) with the upper equality taken in the case where (-1) m+n R(x)~O case where (-l)m+nR(x)~O. and the lower equality in the These requirements on the sign -57- of (-l)m+nR(x) merely indicate whether the plus or minus sign precedes sin- 1 jR(x) I in (3.2-3). All of the results to this point apply equally well to the asymmetrical frequency equation (2.3-14) and to its branches n=n n (x), n=l,2,3, ... , if R(X) is replaced by -R(x) throughout. a-1 Analogous to (3.2-1), on the interval -l<x<a+l' a convenient form of the symmetrical frequency equation is n = mrr ( -1) m+l . -1 [ 1 . ] X+ ~x~ sin R(x)sinn (3.2-4) m=0,±1,±2, ... , where, if n is assumed to be real, . -1[ 1 . ] - 2TI < sin R(x)sinn rr · < 2 1 defines the inverse sine function since -l<R(x)<l from (A-5) in Appendix A. and (3.2-4) It should be stressed that (3.2-1) are equivalent and, in turn, are both equiva- lent to the symmetrical frequency equation (2.3-13). The new forms are introduced only because they are open to each interpretation in the respective intervals mentioned. The form (3.2-4) lends itself to a parametric solution just as did (3.2-2). However, the interpretation of con- tinuing branches of the frequency equations is not so a-1 obvious because of the branch points on O<x<a+l' which were found in Section 3.1 and sketched in Fig. 7. These branch -58- points show up as vertical tangents of the functions n in Fig. 8. This only means that, if a continuous function is to be traced through solution points from the parametric solution, multiple-valued functions must be accepted. This is certainly not a difficulty and the presence of branch points does not hinder the parametric method of finding solution points which form continuous functions be they multiple- or single-valued. Further analysis will be done later to identify analytic continuations of single branche~ but that is not important here. The bounds I-x mn I - 11Xsin . -1 ( R(x) 1 )I (3.2-5) result from (3.2-4) and are valid on the whole range a -1 a+1· -l <x <~- 0 These bounds are derived J ust as (3.2-3) was derived from (3 . 2-1). One of the equalities in (3.2-5) is . . f'ies n (2n+l)TI , t a k e n at a point x on 1 y i' f n a 1 so satis 2 n=O, 1, 2, . . . , with the upper equality being taken if (-l)m+n (-l ) m+n <0 and the lower equality being taken if XR(x) ~0. x R( x ) For the case m=O, can be used as an O~n(x)~J~sin-l(Rfx))I improved bound to replace (3.2-5). a-1 Again, all of the results for -l<x<a+l apply to the asymmetrical branches if R(x) is replaced by -R(x). -59- The hyperbolas n= 1~~' p=l,2,3, ... , and n=r+~' q=l,2,3, ... , are curves of constant a and 8 by (2.3-5) and they form a grid in the x 1 n-plane, which can be used as an aid for sketching the real branches. This grid is pre- cisely the mapping of Mindlin's [2] "grid of bounds," which also consists of hyperbolas, but in the w,K-plane. The real n branches are sketched on the real x-axis in Fig. 8 by using the bounds (3.2-3) and (3.2-5) and the special solution pairs from B.l in Appendix B. The symmet- rical branches are shown in heavy solid lines and the asymmetrical branches are shown in dashed lines in Fig. 8. . gri'd consisting . . ~ _-9..:!!_ . A1 so, t h e h yper b o 1 ic o f n=l-x an d n1 +x is shown in Fig. 8. a-1 The branches, shown in Fi.·g. 8 on ~- <x<l, are denoted a+ 1 - by n , n=l,2,3, ... , for both the symmetrical and asymmetrin cal branches and they are one-to-one mappings of the real branches w (K), n=l,2,3, ... , from the dilatational sector n (2.3-10) of the Rew,ReK-plane into the x,n-plane by (2.3-4). The branches on a-1 O<x<~- -M 1 are one-to-one mappings of the branches in the Rew,ImK-plane, which were discussed by Mindlin [2]. The branches on -l~x<O give both w and K pure imaginary by (2.3-7) and apparently these branches of the Rayleigh-Lamb frequency equation have not been considered before. All of these branches will play a role in the analytic -60- continuation of the branches n=n n Those with w and (X). K pure imaginary seem to have no special physical significance except that they add to the overall understanding of the branches of the frequency equations. 1 . d b e f ore, th e b ranc h poin · t s on O<x<aAs men t ione -a+ 1 appear in Fig. 8 as vertical tangents on the branches and they can be identified with the branch points shown in Fig. 7 by comparing the two figures. The scale of the real x-axis in Fig.7 is expanded five times that of Fig. 8, however, the vertical scale for real n is the same in both figures. 3.2.2. The Branches on the Unit Circle Jxl=l. In Chapter 2 it was found that the portions of the branches w (K), n=l,2,3, ... , and the portion of the symmetn rical Rayleigh branch w (K) which occur in the equivolumi0 nal sector (2.3-11) map onto the upper half unit circle x=e ie I o~e~n. The definition (2.3-6) of a and the particular mapping given by (2.3-4). B determines It is sufficient to consider the upper half of the circle since simple relations in B.2 of Appendix B give the continuations onto the lower half of the circle. In this subsection these branches n=n gated as functions of e through x=e ie o n (x) are investi- Also, a second set of branches are found on the upper half unit circle which -61map into the w,K-space with both w and K pure imaginary. The former set of branches, which map into the usual real branches w (K), is shown in Fig. 9 as continuations of the n real branches n=nn(X) on the real x-axis (shown in Fig. 8 and discussed in Subsection 3.2.1) through the point x=l or e=O. The latter set of branches, which map into branches with w and K imaginary, is shown in Fig. 10 as continuations of the real branches on the real x-axis through the point X=-1 or 8 =TI • In B.l of Appendix B, it is shown that all of the branches of the frequency equations which are bounded at x=l and x=-1 are analytic at these points. The function n satisfying either of the frequency equations, (2.3-13) or (2.3-14), is just an unknown complex · f unction of eon x=e i 8 h · · · ; .ence, it is JUSt as we 11 to ta k e (3.2-6) where y is another unknown complex function of e. n is taken in the form (3.2-6) to simplify the frequency equations on x=e ie (3.2-6) and the form (A-6) of R(ei 8 ) from Appendix A are used to write the synunetrical frequency equation (2.3-13) in the form -62- sinn + R (x) sinxn = e x=e iy_[ 2 _iy_ 2 . ( -iQ) 2 e sin ye ie ei):sin 0e i ~)] = 2e i): [cos):s in (ycos~) co sh (ys in~) + - sin):cos (ycos~) sin+s in~)] = 0 on x=e ie Thus, the symmetrical frequency equation can be written as ( 3. 2-7) on the unit circle x=e ie . If (3.2-7) is written as tanY 2 the right side of this equation is real since y in (A-6a) in Appendix A is real and it is a simple matter to show that the left side is real only if y is either pure real or pure imaginary. Thus, using (3.2-6), n must take one of the forms .e -1- ( 1) n(eie) = JnJe 2 (3.2-8) ( 2) n(eie) = iJnJe .e -12 -63- These £orms are chosen so that Ren~O. As mentioned previ- ously, -n is also a solution if n given by either of (3.2-8) satisfies the symmetrical frequency equation. The form (1) of n in (3.2-8) maps into w and K real 1 and nonnegative by .e 2 12 (2 . 3-7) by taking x =e , etc., and branches of this description map into the branches wn(K) in the equivoluminal sector (2.3-11) which were discussed in Chapter 2. into w and Likewise, the form (2) of n in (3.2-8) maps K both pure imaginary by (2.3-7). For reasons wh ich will become apparent shortly, branches n(ei 8 ) of the form (1) in (3.2-8) are called "the equivolurninal branches" and t h ose of the form (2) are called "the dilatational branches." The symmetrical frequency equation , {2.3-13) or (3.2-7), takes the form (3.2-9) for the symmetrical equi volurninal branches nn (e 1. 8 .e ) =Inn Ie -i2 n=0,1,2, . . . , and the form (3.2-10) -64- for the symmetrical di1atationa1 branches nm(e ·e 1 >=i In mIe .e -12 , m=0,1,2, . . . . Just as the real branches in the preceding subsection satisfied the form (3.2-1) or (3.2-4) of the symmetrical frequency equation, the symmetrical equivoluminal branches nn(e ·e 1 .e >=lnnle -1 2 , n=0,1,2, ••. , satisfy the form (3.2-11) of (3.2-9) and the symmetrical dilatational branches .e -1- 2 nm (ei8)=il nml e of (3.2-10). by (A-6a) From knowledge of the function y=y(8) defined in Appendix A and sketched in Fig. 6, the ranges 0 apply on , m=0,1,2, .•. , satisfy the form ~ tan~coth (I nm Icos~) :S + o~e~n; 00 thus (3.2-11) and (3.2-12) are completely defined by requiring that -65- O < tan -l [tan}tanh(I nnl sin~)] ~ i O < tan- 1 [tan~oth(1nm1 cos~)] '.: i The forms (3.2-11) and (3.2-12) lend themselves to easy solution for In! by considering simultaneous solutions of equivalent parametric equations as was done in (3.2-2) for (3.2-1). It should be indicated that the branches n=O in (3.2-11) and m=O in (3.2-12) can be shown, by solving for In! using parametric equations or otherwise, to terminate i8 in a branch point at the point xo=e 0 given by (2.3-8). The behavior of these branches near the branch point x is 0 given by (B.1-1) in Appendix B, and this can also be derived from either (3.2-11) or (3.2-12) . The branch n=O defined by (3.2-11) is the synunetrical Rayleigh branch and it only exists on the portion circle. 8 0 ~8~TI of the upper half unit Likewise, the branch m=O, defined by (3.2-12), is a continuation of the synunetrical Rayleigh branch about x0 =e i8 0 and it exists only on the portion upper half unit circle. 0<8~8 0 of the The termination of these branches is seen in Figs. 9 and 10. 0 From (3.2-11) the moduli of the synunetrical equivo- in a branch point at 8=8 . i8 luminal branches n (e )=jn n .8 n je -i2 , n=0,1,2, ••• , have the -66- bounds (3.2-13) 8 2 cos- with the upper equality occurring only at 8=8_ 1 where y( 8 _ )=rr and the lower equality occurring only at 8=0 (the 1 branch n=O does not extend to 8=0) . The bounds (3.2-13) result from the allowable range of the inverse tangent function in (3.2-11) Likewise, from (3.2-12) the moduli of the dilatational branches n m (e i8 .8 . -i2 )=iln le I m=0,1,2, ••• , have the bounds m (3.2-14) . 8 2 sin- with t h e lower equality occurring only at 8_ 1 for m=l,2,3, ... and the upper equality never occurrin9. These branch es satisfy (B.1-10) in Appendix B at 8=n and they approach the upper equality in (3.2-14) form large as can b e seen from the asymptotic expansion for n (-1) following m (B. 1-10). The solution points at x=l or 8=0, x=x_ 1 or 8=8_ , 1 and x=-1 or e=n are listed in B.l in Appendix B where the -67- subscripts n and m refer to equivoluminal and dilatational branches respectively. deduced from (3.2-11) These solution points can also be and (3.2-12) the points where the bounds taken. and they make up some of (3.2-13) and (3.2-14) are These points are evident in Figs. 9 and 10. It is apparent from the bounds (3.2-13) that the syrn- metrical equivoluminal branches n=l,2,3, ... become unbounded as 8+TI or x+-1 on the unit circle (the symmetrical Rayleigh branch n=O is an exception since it is bounded at x =-1, yet i t is natural to call i t an equivolurninal branch in the present context). Also, from (3.2-14) the syrnrnetri- cal dilatational branches are unbounded as 8+0 or x+l on the unit circle. velocity c p As was mentioned in Chapter 2, the phase in (2.3-1) takes the value c , the equivolumis nal body wave velocity, at x=-1 and the value cd' the dilatational body wave velocity, at x=l. Hence, the names equivoluminal and dilatational branches have been used since these branches give the equivoluminal and dilatational phase velocities at their singular points ~=-1 and x=l respectively. The equivoluminal branches are well known and are just the mapping of the usual real w, real the equivoluminal sector (2.3-11) K branches from into the x,n-space. The dilatational branches apparently have not been investigated because they map into the w,K-space with both -68- w and K pure imaginary. However, these branches are highly significant since they have an asymptote (point where lnl~ 00 ) where the phase velocity takes the value of the dilatational body wave velocity in the limit. Also, they are smooth monotonic functions of e as contrasted with the terraced portion of the frequency spectrum in the dilatational sector (2.3-10) or on a-l<x<l in Fig. 8, which is a+l- the only place that the dilatational body wave velocity is approached in the Rew,ReK-plane. Thus, if analytic contin- uations can be found onto these branches, an improved representation for the dilatational waves in the plate may be possible by manipulating the integrals (2.3-lSa). The branches · satisfying the asymmetrical frequency (2 . 3 - 14) on th e upper h a lf unit · circ · 1 e x=e ie , . equa t ion O~ e~n , are now considered. Just as in the symmetrical case, the asymmetrical branches must take one of the forms (3.2-8) with those of the first form called the equivoluminal branches and those of the second form called the dilatational branches. In the same manner as (3.2-9) and (3 .2 -10) were derived, the asymmetrical frequency equation (2.3-14) takes the form cot (1n Icos~) = -tan~oth (1 n Isin!) (3.2-15) -69· far the asymmetrical equivoluminal branches nn (e ·e 1 ) =I nn·I e .e -12 , n=l,2,3, ... , and the form tan (1 n Isin~) = -tan}tanh (1 n Icos~) fortheasymmetrical dilatational branches n (e m m=l,2,3, . . . . (3.2-16) 1·e )=iln le Analogous to (3.2-11) and (3.2-12), m .e -1 2 ' (3.2-15) and (3.2-16) can be written as n=l, 2, 3 , ... , and m=l,2,3, ... , respectively. The inverse tangent functions in (3.2-17) and (3.2-18) have the same ranges as those given following (3.2-12), which results in the bounds (n-!)TI cos£ 2 nTI e cos2 (3.2-19) -70n= l ,2,3, ... , for the moduli of the asymmetrical equivoluminal branGhes nn(e ·e 1 .e )=lnnle -i2 and the bounds (3.2-20) . e sin2 m=l,2,3, ... , for the moduli of the asymmetrical dilata- .e -i- 2. m m The upper equality in (3 . 2-19) is taken only at the tional branches n (eie)=iln le point e_ taken. 1 where y(e_ )=n and the lower equality is not 1 At 8=0 or x=l, (3.2-17) takes a limiting form which is equivalent to (B.1-4) in Appendix B; hence In n I in (3.2-19) approaches the lower equality at 8=0 for n very large. This can be seen from the asymptotic expansion for n n(l) following (B . 1-4). The lower equality in (3.2-20) is taken only at the point e _ 1 and the upper equality is taken only at the point e =n or x =-1 . These as y mmetrical equivoluminal and asymmetrical dil a tational branches are also unbounded at x=-1 and x=l respectively and the description of the symmetrical branches also applies here. . ie_ 1 The point x _ =e defined by (A-la) in Appendix A 1 maps into the Lame point w=l2K and the special solutions -71- (B.1-12) and (B.1-14) in Appendix B taken by the equivo- luminal branches at this point are identified with the Lame modes of harmonic wave propagation. These consist of equivoluminal waves meeting and reflecting from a boundary at an angle of 45° with no mode conversion to dilatational waves. This is expected since R(x_ )=-l and R is the re1 flection coefficient as stated in Appendix A. Both the symmetrical (in heavy solid lines) and the asymmetrical (in dashed lines) equivoluminal branches are sketched in Fig. 9 with the continuations through the point 8=0 or x=l onto the real branches of Fig. 8 shown at the left side. Likewise, the symmetrical (in heavy solid line~ and the asymmetrical (in dashed lines) dilatational branches are sketched in Fig. 10 with the continuations through the point 8=TI or x=-1 onto the real branches of Fig. 8 shown at the right side. The hyperbolas In I- qn e 2cos 2 1 or n= ~~, q=l,2,3, ... , are also shown in Fig. 9 and the . hyperbolas In I PTI or n_.E!!_ , p= 1 , 2 , 3 , •.. , a r e s h own in · 1 -x . e · 2 sin 2 Fig. 10. These branches can be calculated approximately by a few simple iterations on equations ( 3 . 2-1 7) and ( 3 . 2-18) . This is done (3.2-11) / (3.2-12), by taking Inn I= n7T e cos 2 as a first try in the expression of the right side of (3.2-11) to give an improved value of In n I, etc. -72- 3.2.3. The Complex Branches on the Real X-Axis. These complex branches serve to complete the picture of the branches of the Rayleigh-Lamb frequency equations on the real x-axis and on the unit circle lxl=l. In prin- ciple at least, analytic continuations of the branches can then be made to the interior of the unit circle and the mappings to the exterior are given by (B.2-2) and (B.2-4) in Appendix B. Actually, a detailed description of these branches serves no other purpose than to complete the understanding of the branches so that such continuations can be made with more confidence. The method to be employed in this work for evaluating the high-frequency response of the plate does not depend on the complex branches. Hence, the com- plex branches are treated in B.3 of Appendix B and only a description of the branches is given here by referring to B.3 and to Figs . 11 and 12. The complex loops in the w,K-space with w real and K complex as given by Mindlin [2] are somewhat like the branches which are described here, however, they are not the same since both w and K will be complex by (2.3-7) for branches with n complex (not pure imaginary) and x real. Such complex branches must surely be analytic continuations of the real w, complex K branches. -73- n =Imn, the imaginary part of n, is determined by 1 (B.3-3) and nR=Ren, the real part of n, is determined by (B.3-4) by the method described in B.3 of Appendix B. Fig. 11 shows the upper and lower bounds of n =Imn 1 . given by (B.3-7) where the lower bound is the asymmetrical Rayleigh branch on xR<x~O and it is a portion of the symmetrical Rayleigh branch on -l~X<XR· The remaining portion of the symmetrical Rayleigh branch .is on x=e ie and it is shown in Fig. 9. A sketch of some of the complex branches is shown in Fig. 12 as a perspective in the Rex,Ren,Imn-space along with the projections onto the Rex,Ren- and Rex,rmn-planes. Only branches with Imn~O are shown, but the complex conju- gates, n*, are also branches satisfying the frequency equations. Portions of the real branches on x>O from Fig. 8 are shown at the right of Fig. 12 to illustrate that the branch points on a-1 O~x~a+l complex branches. are common points of the real and Loops between branch points occur above the level Ren=5TI in this figure and they become very numerous for Ren large. All of the projections onto the Re x ,Ren-plane up to the level included in Fig . 12 are sketched, but only a few of the projections onto the Rex 1 Imn-plane are shown since these fall on top of each other and just become confusing. Only the lowest asymmet- rical complex branch (the branch with the smallest value -74- of Ren=nR) is sketched as x+-1 where nR+ 00 and Imn=n 1 oscillates more and more rapidly as shown in the projection onto the Rex 1 Imn-plane. All other complex branches behave in the same manner and their general form can be deduced from their projections onto the Rex,Ren-plane in Fig. 12 and by the limited variation of Imn=n bounds. 1 allowed by its The real part of n jumps by an amount ± : as 1 XR the complex branches are continued about the logarithmic branch point at x=xR· This can be seen in Fig. 12 or from the asymptotic approximation (B.3-9) in Appendix B. In general, the imaginary part of n consists of loops between the branch points on a-1 O<x~a+l where Imn=O, and the loops are tangent to the upper bound given by the second equation in (B.3-7) with the equality. ~he branch points on this interval are shown as points in the Rex,Ren-plane in Fig. 7. A final loop from a branch point near x=O fails to return to the Rex 1 Ren-plane with Imn=O and, instead, it crosses x=O with a value of n satisfying (B.1-7) of Appendix B if it is a symmetrical complex branch or (B.1-8) if it is an asymmetrical complex branch. On -l~x<O, n 1 oscillates between the upper and lower bounds given by (B.3-7) and shown in Fig. 11 and becomes unbounded as x+xR· The approximation to the complex branches near x=xR is given by (B.3-9). Since nI takes -75- 2 2 the lower bound when sin nR=sin (l+x)nR=O and the upper 2 2 bound when sin nR=l and sin Cl+x)nR=O, i t oscillates more and more rapidly as x=-1 is approached with the frequency of oscillation going to infinity at x=-1. As is mentioned following (B.3-8) of B.3 in Appendix B, the separation of the bounds remains finite as x=-1 is approached. oscillation of n 1 The is a result of an essential singularity at x=-1 on all of the complex branches except the symmetrical Rayleigh branch, which is analytic at x=-1. The unexpected feature of all complex branches having a singularity at x=xR seems to have little physical significance. The Rayleigh branches are unique in this respect simply because they are pure imaginary and, of course, w and K given by (2.3-7) are real. All other complex branches have a finite real part, nR' as x+xR with the imaginary part, n , unbounded. 1 Then by (2.3-7), w and K have real parts which become unbounded just as do the Rayleigh branches, but they also have finite imaginary parts which are usually identified with a disturbance that decays in time and space and does not give rise to wave fronts. More important is the fact that analytic continua- tions of the Rayleigh branches are now known about the infinity point (w,K+ 00 ) and the continuation goes onto other neighboring complex branches all of which are interconnected by a common logarithmic branch point at x=xR. -76- This gives further information needed to make analytic continuations in the complex x-plane. With regard to analytic continuations, there is a striking similarity between the singularity of the complex branches at x=-1 and the singularity of the equivolurninal branches on JxJ=l at the same point x=-1 or 8=TI. This is seen by solving (B.3-4) in Appendix B for nR so that it has the leading term r:x' q=l,2,3, •.. , and comparing this with (3.2-11) and (3.2-17). in each case is Then the dominate term of n ;+; or r+; and this makes it seem likely that the complex branches continue about x=-1 onto the equivoluminal branches. A comparison of Figs. 9 and 12 also indicates this possibility. The fact that there are no complex branches on the real x-axis near x=l which can be considered as analytic continuations of the dilatational branches onJxJ=l and shown in Fig. 10 poses a difficult problem which can be answered only by investigating the analytic continuations about the branch points near x=l. 3.3. SERIES REPRESENTATIONS OF THE BRANCHES n ABOUT THE POINTS x=l AND x=~l All of the preceding work in Section 3.2 concerning the existence and form of the branches on certain subregions in the x-plane is not sufficient to allow even a -77- good approximate evaluation of the integrals (2.3-15). The infinite sums of integrals and the complicated relationship between n and x present the same difficulties that were present in the integrals (2.2~10) with w and K as the variables. In this section representations of the branches n as explicit functions of x are derived in the form of a series of functions of X· These series are derived by expanding the frequency equations about the points x=l and x=-1, which are singular points of the dilatational and equivoluminal branches respectively of Subsection 3.2.2. The fact that n+oo on these branches as x+l,-1 or as 8+0,TI on x=eie means that w+oo and K+oo from (2.3-7). These singular points are expected to be very significant with regard to their contribution to the high-frequency wave fronts in the plate. The reason for this is that ~ and dw both approach 1 as x+-1 or as 8+TI on the K dK equivoluminal branches, other than on the synunetrical Cd c as x+l or as s e+O on the dilatational branches. ~c and dwc are the K S dK S phase and group velocities respectively as defined by Rayleigh branch, and they both approach . a=~ (2.3-1) when both w and K are real, which is the case on the equivoluminal branches. tion 3.2.2 that both w and It is recalled from SubsecK are pure imaginary on the -78- dilatational branches and it remains to be seen if these branches contribute to the response of the plate. For much the same reason as was given in Subsection 3.2 . 2 for using the names equivoluminal and dilatational branches, the points x=-1 and x=l are assumed to be identified with equivoluminal and dilatational wave fronts respective . Apparently, the only convenient series representations, such as was a function of K, have been about points where both w and Taylor's series . K are finite. These usually consist of Obviously such representations are of very little use for high-frequency calculations where w may become unbounded. It is true that a few terms are apparent in an asymptotic approximation as K+ 00 on the Rayleigh branches or on any of the other branches. For example, CR w ( K )"'cK on the Rayleigh branches as 0 s and K+co (3 :: 1 ( wn 1 2 (K)-K 2)2"'~TI as K7 00 on all other branches with m=2n and n=l,2,3, .. . on the symmetrical branches and w°ith m=2n-l and n=l,2,3, ... on the asymmetrical branches. Attempts to find more than a few succeeding terms or to find an expression for a general term in these expansions appear to be doomed to failure . The reason for this is the complicated functional relationship between w and the frequency equations (2 . 2-11) and (2.2-12). K It seems in -79- to be practical to seek asymptotic expansions only if the variables w and K are grouped into convenient functional forms such as x and n. The symmetrical branches nn' which are identical to the equivoluminal branches of the form (1) of (3.2-8) when x =eie, will be considered first. x will not necessarily be restricted to the unit circle x=eie, however, the branches considered are analytic continuation~ of the equivoluminal branches. As has been the usual practice, the form (2 . 3-13) of the symmetrical frequency equation is altered to obtain the more convenient form t an (l+x )n 2 R-1 = R+ltan (1-x)n 2 (3.3-1) This can be recognized as the original form (2.2-11) of the symmetrical frequency equation by using (2.3-5) and (2.3-12a) to return to the w,K variables. Series representations are now sought for the syrnmetrical branches n which satisfy (3.3-1) and are singular as x+-1 in some sector about that point. elude the path x=e ie , 0~8<TI , branches are on this circle. The sector must in- as 8+TI since the equivoluminal Also this sector may include the real x -axis as x+-1 from the right where the complex branches on the real x-axis of Subsection 3.2.3 were found -80- to be singular. It was speculated there that these branches might be analytic continuations of the equivoluminal branches. The singular nature of the equivoluminal branches is shown in Fig. 9 and that of the complex branches in Fig. 12 as x+-1 in each case. It is convenient to examine the case x=ei 6 first, realizing that by means of analytic continuations most of the results will extend to neighboring portions of the x-plane. n takes the form (1) of (3.2-8) and the fre- quency equation (3.3-1) becomes (3.2-9) on x=e R(ei 6 )=eiy(S) by (A-6) in Appendix A. i8 Further, where (3.2-9) can be written as (3.2-11) which has the obvious approxima ti on (3.3-2) as 8+n where mr + y(6) cos 2cos --e 2 (3.3-2a) e 2 The subscript e refers to equivoluminal branches. is valid for n=l,2,3, ..• since lnnl+oo and (3.3-2) tanh(innlsin~)+l as 8+n. This result is extended to the general x-plane by replacing e i6 with x and y(8) with -iLogR(x) and recalling -81- 2nrr - - i.LogR(x) l+x i+x (3.3-3) is taken for the first term in the series expansion for nn(x) as x+-1. iy(8) When x=e The term is well-defined by taking LogR(x)= ie I . . . ( ) 2nrr . Th e approximation nn x - 1 +x gives 0~8~7fo 1 s=2(l+x)nn-nrr, which was mentioned as an asymptotic approx- imation to the symmetrical branches relating wn(K) to K as K+oo. The second term in (3.3-3) is finite as x+-1 with . LogR(x) 1 im l+ x+-1 x by (A-4) in Appendix A. (3.3-4) = The single term In~~) (ei 8 ) I in (3.3-2a) gives an excellent approximation to the equivolu. 1 b ranc h es on x=e ie b eing · mina exact at 8= o,e_ e_ 1 is defined by (A-la) in Appendix A. 1 an d as 8 +rr. The approximation is so accurate that, on the scale shown in Fig. 9, no 8 apparent difference exists between lnn(ei ) I and In~~) (ei 8 ) I for the symmetrical branches. By comparing Figs . 6 and 9, it is seen that the fact that y(e_ )=rr gives 1 the exact solution at 8=8_ = 1 since -82- in agreement with (B.l-12) in Appendix B. Another result of this good approximation is that succeeding terms in the series representation must be very small on x=e The one-term approximation n n ie . (x)~n en (O) (x) given by (3.3-3) also has many of the characteristics of the complex branches on the real x-axis of Subsection 3.2.3. This is true both as x+-1 and near the point x=xR as can be seen by comparing (3.3-3) with (B.3-9) in Appendix B. The question of a region or sector of convergence of the series representation is discussed at the end of this section and it is only assumed here that such a region will include the Circular path x=e ie I o~e~TII aS 8+TI. For the purpose of deriving a series representation, it is convenient to define another variable, gn(X), replacing nn(X) by setting = n (O) (x) + 2gn (x) en l+x where n (O) (x) is given by (3.3-3). en (3.3-5) There is no reason to exclude the Rayleigh branch n=O, and (3.3-5) will be taken for the definition of gn(X) for n=0,1,2, ••• with gn(X) expected to vanish as x+-1. The form (3.3-1) of the symmetrical frequency equation then becomes -83- e - i ( 1-x h1 ( o) (x ) en = 2ii~~gn -2Rsingn 2 2 (R +l)sing - i(R -l)cosg n · n e (3.3-6) by using (3.3-5). The left side of (3.3-6) approaches zero exponentially fast as x+-1 in O<Arg(l+x)<TI and, since 2 gn R -l=O(jl+xJ) as x+-1 from (A-1), l+x must also approach zero exponentially fast as x+-1 in this sector. Rayleigh branch n=O is an exception with e The -i Cl-x)n~~) <x> + 4 e 1-v as x+-1 regardless of Arg(l+x), but this exception is of very little concern in this work. The definitions = e -i (1-x>n (O) Cx) en (3.3-7a) (3.3-7b) with n (O) (x) given by (3.3-3), reduce (3.3-6) to en (3.3-8) For the present En' gn and x are thought of as three completely independent complex variables except that they are -84- related by the functional relationship (3.3-8). the fact that £ n That is, is a function of x by (3.3-7a) is not critical in this derivation. F(gn 1 x), defined by (3.3-7b), is an analytic function of both gn and x with three obvious . excep t ions: g =oo, x=- 1 / an d t ang n 2 .R -l wit ' h sing ' ~o in ' / n R2 +l n =i~~ the last two cases. The inverse of (3.3-8) must then exist in any domain of the x s n -space in which ~Fgn ;fO and Fis an analytic func1 tion of g n and X· Furthermore, the representation of the analytic function . g n can be taken as a Taylor's series in sn about sn=O with the coefficients being functions of X· Thus, a representation of the form (3.3-9) which satisfies (3.3-8) will be sought. The question of the existence of an explicit inverse function has been answered by the analytic properties of the functional relationship (3.3-8) an~ it is left to find the coefficients g~j) (X) in (3.3-9). It simplifies the algebra of finding these coef ficients to consider the special case x=e return to the x-plane in general. ie , o:::e:::1T, and then The form of En(e not important since it is still considered to be an i8 ) is -85- independent complex variable, however, it is listed here along with F(gn,eie) and from (3.3-7a) and (3.3-7b) they take the forms = e F(g ,e ie n R(e ·e 1 · (e) 1 )=e Y - ( 2mr+y) tan e 2 (3.3-lOa) . 2 g tane sing n n 2 ) = -sin(y+g ) 0 (3.3-lOb) n from (A-6) has been used again. The symmet- rical frequency equation (3.3-8) then becomes e singn 2gntan2 -----"""' sin(y+g ) I n which can also be written as = -g n cotg n siny - g n cosy (3.3-11) (3.3-11) is expanded in powers of gn and (3.3-9) is substituted for g n to give relationships for the coeffi- cients g(j) by equating the coefficients of Ej. n n The first few of these coefficients are easy to find , but i t becomes increasingly more difficult as more coefficients are sought . A general recurrence relation can be -86- determined which gives gn(j+l) in terms of g(j) g(j-l) n ' n ' (1) ... , g for j=l,2,3, ... , however, the form of the general n term as a function only of 8 has not been found. work on the integrals Further (2.3-15a) will show that only a finite number of the terms in this series are required to represent a finite numbe.r of wave fronts in the plate; hence it is not so important to find all of the coefficients g(j) n . An interesting feature of these coefficients is that they are independent of the branch number n. Thus, the coefficients are denoted by g(j)~g(j) with the e n subscript e added to distinguish these as equivoluminal coefficients as opposed to the dilatational case, which will be considered. A few of these coefficients, derived by the procedure just described, are -siny 1 . 2 2sin y - 2 sin . 2y t an e 2 . . . 3 t 2 e (3) i . e ge = - 3 sin 3y + 3 sinysin 2ytan - 6 sin y an 2 2 (4) ge = i . 4sin 4y - 3 3 3) ( . i . . 4 siny\sin y+ sin y tan e 2 . 2 . 2 t 2e 64 . 4 t 3 e + 16 sin ysin y an - --r-sin y an 2 2 (3.3-12) -87- A pattern does emerge in the form of . g(j) e and it is clear that g ( j) = O (n-e) as e+n for fixed j since y(6)= e 2 2 6 -iLogR(ei )=~ (n-e)+O(n-e) from (A-4). Also, the first 1 -v and last terms in g ( j) can be written in general as e (2')j-l · ·-1e ( -1) j --.- sinjy and -~ sinJytanJ 2 respectively, however, J the other terms become more numerous and complicated as j increases. It is true that the last term in g(j) domie nates for j very large and it becomes unbounded as j+00 for a general point 6; however this is of very little consequence in the power series (3.3-9) since the magnitude of En greatly influences convergence. The symmetrical branches on x=e ie , 0S6STI, were de- scribed in Subsection 3. 2. 2 by the modulus Inn ( eie) I where .e nn(e i e )= j nn(e ie ) j e -i2 from (1) of (3.2-8) luminal branches. Thus, from (3.3-2a), for the equivo- (3.3-5), (3.3-9) and (3.3-lOa), the series representation for the symmetrical equivoluminal branches is oo (.) + I::gJe . 1 e J= -j (2mr+y (6)) tan~] . (3.3-13) n=0,1,2, ... , where the first four coefficients g(j) are e given in (3.3-12). Before the series representations are derived for the -88- asymmetrical equivoluminal, symmetrical dilatational and asymmetrical dilatational branches; it is useful to write the series representation for the symmetrical equivoluminal branches nn(x) for a point x not necessarily on the unit circle x=e ie . From (3.3-5) and (3.3-9), this series is written as (3.3-14) n=0,1,2, . . . , where n(O) (x) is given by (3.3-3) and n(j) (x)= en · e 2 ( j) l+x ge (x). The form (3.3-7a) of £n(X) has been used. The first four coefficients result directly from {3.3-12) by replacing eie with x and eiy with R(x) to give g(j) (x) and e 2 then n ( j) (X) = - -g ( j) ( x) These coefficients are e l+x e • ne(2) (x) = _ _i_R2-l[!(R2+1) R2 l+x 2 + _l ( 2 1- x )2(Rl+x 2-1 )2 J (l-x)R2-1] i+x -89- n ( 4 ) (x) e 8 3 R2 -l 3] - -(1-x) (l+x -) 3 (3.3-15) 2 R -l The term l+x is bounded, except at points where R is un- R2-l 4 bounded, and lim 1 + -l-v from (A-3) and (A-4) in Appendix A . x+-1 x The series representation for the asymmetrical equivoluminal branches n (X), n=l,2,3, . . • , is obtained by simply n replacing R(x) with -R(X) in all of the foregoing expressions. This is valid because the sign of R(x) is the only difference between the frequency equations (2.2-13) and (2.2-14). In this case the Rayleigh branch n=O is not expected to occur in the domain of interest which includes x=e ie , o,:::e_:::TI. In order to have Ren n (x)>O on all branches - n=l,2,3, ... , it is necessary to take Log(-R)=LogR-iTI which is equivalent to replacing n with n-! in the definition (3.3-3) of n(O)(x). Therefore, y(8) is replaced with en i8 y (8)-TI on x =e . In summary, the series representation for the asymmetrical equivoluminal branches n Cx) v n=l,2 , n 3t • o • I is -90-i j ( 1-x) n ( 0 > (x) 1 oo en-2 = n ( o) 1 (x) + (-l)jn (j) (x)e (3.3 - 16) . 1 e enJ= L 2 where n (O) (x)-( 2 n-l) en-1 l+x -i Lol~~x) results directly from the 2 definition (3.3-3) for n (O) <x>, and the first four coeffien ( .) cients n J e (x) are given by (3.3-15) . + oo • (.) 2: (-1) Jg e J e . 1 On x=e .e 1 -j ( (2n-l) TI+y (8)) I o::e<7r I tan~] ( 3. 3-17) J= .e with nn(e i8 )= I nn(e i8 ) I e -i.2 from (1) of (3.2-8), and the first four coefficients g(j) are given by (3.3-12) . e The symmetrical dilatational branches are now considered. The series representation of these branches can be derived in much the same way as was done for the equivoluminal branches. However, due to the unusual symmetry of t h e frequency equations, these representations for both the symmetrical and asymmetrical cases can be derived directly from the equivoluminal representations by simple changes of variables. -91- This comes about as follows for the symmetrical case: The representation (3.3-14) is an explicit function of x and R which satisfies the symmetrical frequency equation (2.3-13). This, of course, assumes that all of the co- efficients n(j) can be found and that the series converges e to n n in some domain. Hence, that function can be written as n = f(x,R) (3.3-18) and it is incidental that R is a function of X· It will be assumed for the time being that x and R are independent complex variables and then n given by (3.3-18) satisfies the equation (3.3-19) sinn = -Rsinxn on some domain D in the x,R-space. (3.3-19) includes the symmetrical frequency equation as the special case where R=R(x) given by (2.3-12a). 1 R~-R . Now, i' f t h e mappings . are carried out, sin[f(x,R)) = -Rsin[xf(x,R)] becomes [-lf -l)] !sin R X (-! X' R 1 x an d x~-- -92- which can be written as Thus, the function n = _l:_f X (-! _!) (3.3-20) X' R satisfies the same equation (3.3-19) as does f(x,R). In this case, n given by (3.3-20) satisfies (3.J-19) on the -1 which is the mapping of D under x+-1 and R+-1 X R domain D in the x ,R-space. Setting R=R(x), this means that n given by (3.3-20) is an equally good solution of the symmetrical frequency equation. For the special case when x=e ie , O~e~rr, (3.3-18) for the symmetrical equivoluminal branches becomes .e n(eie) = f(eie ,eiy) = e-i2jf(ei8 ,eiy) where form (1) of (3.2-8) has been used. I The indicated by (3.3-20) are applied to (3.3-21) (3.3-21) cha~ges for f (ei e ,eiy) , with x+-l and R+-l: being equivalent to 8+rr-e X R . and y+rr-y respectively , to give new solutions -93- . TI-8 -i8 -i~2~lf( = -e e e i(TI-8) ,e i(TI-y)) I = (3.3-22) on x=e i8 Hence, the new set of branches obtained in this manner must be the symmetrical dilatational branches since (3.3-22) is identical with form (2) of (3.2-8). The changes indicated by (3.3-20) are carried out on (3.3-14) where the right side of that equation is taken to be f (x ,R) . Replacing n with m and using the subscript d to distinguish these as dilatational branches, this operation gives the series representation (3 . 3-23) .for the symmetrical dilatational branches n m (X), m=0,1,2, ... ,where n~~)(x)=-~ln~~)(x)J L _ x+-1 x with the indicated R+-1 R n+m changes applying only to n{U) (X) given by (3.3-3). en taking By Log(-~)=iTI-LogR , this gives = (2m+l)TI + iLogR(x) 1-x 1-x (3 . 3-24) -94- The coefficients ndk) (x) are obtained by performing the same operations indicated by (3.3-20) on the coefficients n~j) (x) given by (3.3-15). Thus, the first four coeffi- cients ndk) (x) to be used in (3.3-23) are + l (1 + ) 2 (1- R 2 )2 2 x 1-x 2 3] x i-x + ~(l+ )3(1-R) 3 J . 8 The series for Jnm(ei ) (3.3-25) I, the modulus of the symmetri- cal dilatational branches on x=eie, can be obtained from -95- (3.3-13) by simply making the changes 8+rr-e and y+rr-y in that expression. This is true since the term -!=-e-ie x multiplying f in (3.3-20) has unity modulus. Thus, since 8 8 t 8 ,an d y+rr-y · 1 ies · th at co~+sin · 8 an d t an~co e+n-e imp 2 2 implies that sinjy+(-l)j+lsinjy, the series representation for the symmetrical dilatational branches, nm(ei 8 )= .e i nm e -i2 / m=0,1,2, ... , On x=e ie is = - .1-[(m+!) e 2 . TI - ~ 2 (8) sin' ·2 (3.3-26) as obtained from (3.3-13). The first four coefficients g~k) in (3.3-26) result from (3.3-12) in the same manner .a nd they are ( 1) gd = -siny g~ 3 ) = -~sin3y - 3sinysin2ycot~ - 6sin 3 ycot2 ~ -96- . 4y - 4 siny . ( sin . 3 y+ 1 sin . 3 y ) co t8 gd( 4) = - 1 sin 3 4 . 2 . 28 16 sin y sin 2 ycot 2 64 . 4 38 2 - ~sin ycot 2 (3 . 3-27) The series representation for the asymmetrical dilatational branches, n (x ), m=l,2,3, . . • , is obtained by replacm ing R( x ) with -R( x ) in all of the expressions for the symmetrical dilatational branches. In this case Log(-R)= LogR+iTI is taken so that all branches with Ren · 1ud e d • m= 1 , 2 , 3 , . . . are inc y (8 )+TI. (x)>O for m on x=e i 8 , y (8) is · rep 1 ace d wi·th Thus, the series representation for the asymmetri- cal dilatational branches is ik(l+x)n(O) 00 L: (--1) kn~k) (x) e 1 dm-- <x> 2 k=l (3.3-28) m=l, 2 , 3 , .. . , with n(o) ( x ) given by (3.3-24) and the first dm four coefficients ndk) (x) given by (3.3-25). x=e i8 Likewise , on ' o:::.8:5.TI , l j nm(ei8) j = _ . 8 sin2 [m~ _2Y1 (8) (3.3-29) -97- .e where nm(e . e )=i nm(e e ) e -12 by form (2) 1 J 1. first four coefficients J of (3.2-8) and the g~k) are given by (3.3-27). The forms of the representations (3.3-23) and (3.3-28) for the symmetrical and asymmetrical dilatational branches respectively show that there is no chance of convergence on the real x-axis. In fact, these series are complex for a-1 real and only real branches are possible on ~-<x<l. a+ 1 - - x Also, the individual terms in the series seem to increase in magnitude as k increases. This is contrasted with the repre- sentations for the equivoluminal branches which seem to converge to the complex branches on some of the interval -lsxsO. This difference can be shown to be closely related to the distribution of branch points near the points x=l,-L The derivation of these representations has revealed a very significant symmetry of the branches of the RayleighLamb frequency equations. This symmetry is evident in the close relationship between the equivoluminal and dilatational branches as demonstrated by (3.3-18) and (1.3-20), and it is not necessary to derive the series representations to find this relationship . Symmetries have long been known to exist between the symmetrical and asyrr~etrical branches, shown in this work by replacing R(x) with -R(X) , but the lack of any equivoluminal-dilatational symmetry has been a troublesome aspect .in modal solutions . The -98- only part of the frequency spectrum with both w and K real which seems to be related to dilatational waves is a portion in the dilatational sector (2.3-10) where the branches are terraced and very dissimilar to the branches in the shear sector (2.3-11). However, this symmetry between branches with an equivoluminal singularity and those with a dilatational singularity is not really surprising. Once the dilatational branches have been identified and sketched, it is almost obvious in Figs. 9 and 10 that the equivoluminal and dilatational branches are closely related. Also, on physical grounds, there is a great similarity between the geometry and form of the dilatational and equivoluminal wave fronts, which will be found to be closely identified with the dilatational and equivoluminal branches. Convergence of the series representations that were derived in this section can be shown on certain regions in the x-plane by a lengthy method, which is not included here. Instead, the method is described briefly and the results are listed. Uniform convergence of the series representations can be shown on certain regions by taking advantage of the similarity between series such as (3.3-9) and a Taylor's series in powers of (3.3-9) is not a £ n • Actually, Taylor's series since both £n and the coefficients are functions of X· g~j) (X) -99- As might be expected, the region of convergence of the series is determined by the location .of the branch points n. and other singular points of the branches The region of convergence is then determined by locating the singularity "nearest" to E n =O. However, since both E n and g(j) (X) in n (3.3-9) are functions of x 1 the term "nearest" must be interpreted very carefully. This fact makes the problem quite difficult and too lengthy to include here. Convergence of the series representations is only considered on x=e ie 1 0~8~TI, and on the path C , which is de1 scribed at the end of Appendix A. These paths are good choices for possible alternate integration paths for the modal solution (2.3-15), and they are convenient to investigate due to the simple forms of R(x) given by {A-6) and (A-8) in Appendix A. The results for convergence are listed below. The series (3.3-14) for the symmetrical equivoluminal branch n=O (the symmetrical Rayleigh branch) converges · h respect to x=e ie on any c 1 ose d segmen t in· uni· f orm 1 y wit terior to 8 0 <8~TI. e0 is defined by (2.3-8). The series (3.3-23) for the symmetrical dilatational branch m=O converges uniformly with respect to x=e any closed segment interior to o~e<e 0. ie on In B.l of Appendix B it is shown that the symmetrical Rayleigh branch n=O and the symmetrical dilatational branch m=O share a branch -100i60 point at x =e 0 6=6 0 This branch point determines the limit of these regions of uniform convergence. The series (3.3-14) for the symmetrical equivoluminal branches n=l,2,3, ... and the series (3.3-16) for the asym- metrical equivoluminal branches m=l,2,3, ... converge uniformly with respect to x=e i6 on any closed segment interior to 0<8 <TI. The series (3.3-23) for the symmetrical dilatational branches m=l,2,3, ... and the series (3.3-28) for the asym- metrical dilatational branches -m=l,2,3, .•. converge uniformly with respect to x=e i8 on any closed segment interior to 0<8<TI. None of the series representations converge on all of the path c 1 , which has the endpoints x=:~~ and x_ 1 given by (A-la) in Appendix A. However, as the branch number, n for the equivoluminal branches and m for the dilatational branches, increases the series representations converge uniformly with respect to x (: c closed segment of 1 on a larger and larger c1 which includes the endpoint x_ 1 and · a-l approac h es th e en d point x=a+l" · A cone 1 usion, wh'ic.h can be a-1 verified by investigating the branch points near x=a+l' is that the extent of the region of uniform convergence is determined by the location of branch points on or near c which are common to the branch n in question. 1 -101This result for uniform convergence on c 1 is precisely that which is required since this work involves the highfrequency response of a plate. As such, calculations are necessary which require an approximation that improves in accuracy as the branch number and, hence, the frequency increases. 3.4. ANALYTIC CONTINUATIONS OF THE BRANCHES The purpose of this section is to incor~orate the in- formation already derived about the branches in this chapter. Specifically, analytic continuations will be used to continue the branches n on a closed curve consisting of the boundary of the upper half unit disk (ixl<l, Imx>O) with the singularities on the boundary being excluded. Also, by excluding the branch points which are interior to the half disk, the branches become single-valued, analytic functions of x on the closed curve. The branches as a function of are described in Section 3.2 for all of this boundary and the branch points are described in Section 3.1. Analytic continuations about each of the particular classes of branch points and continuations about the aforementioned closed curve are described in this section. Reference is made to Appendix C where supplementary material is given for the continuations and continuations for specific branches are described. x -102- 3.4.1 . Analytic Continuations About the Branch Points. In this subsection the nature of the analytic continuations of the branches of the frequency equations about their branch points is investigated by varying Poisson's ratio v. This causes many of the branch points to migrate onto the real x-axis where they are much easier to understand. In this context, the nature of the analytic contin- uations about the branch points merely means the knowledge of which branches are common to the particular branch points. It is recalled from Subsection 3.1.2 that each branch point is common to only two branches. An argument was given in Subsection 3.1 . 1 showing that the branches n are analytic functions of Poisson's ratio v as well as being analytic functions of X· Hence, v is varied and the fact that the position of all the branch points, except the branch point of the asynunetrical Rayleigh branch at x=O, are functions of n makes it possible to bring many of these points into view on the real x-axis. The analyticity of the branches n with respect to v insures that the nature of the branches remains generally the same as v is varied. Specifically, given branch points continue to be shared by the same two branches. From Subsection 3.1.3 and from Figs. 7 and 8 1 branch points are plainly visible and easily understood on the -103- a-l o f th e real x-axis · . segmen t 0 ~X~a+l Also, from (2.1-2) . giving a as a function of v, the point a-1 a+l = 12(1-v) - II=2V = 3 - (3.4-1) 4v - 212(1-v)(l-2v) /2(1-v) + ./l-2v . is an increasing function of v for O~v~ 1 2. Hence, increas- 2 causes :~i to increase to 1 and the interval 1 ing n to O~x~l. containing branch points on the real x-axis becomes The branches are sketched on the interval 1 v= 2 in Fig. 13. O~x~l for The general behavior of the branches re- mains essentially the same on -l~x~O and on x=e ie , 0~6~TI, O~v~~ with no change in the number of for all of the range branch points. The transition of the branches which occurs as v increases toward 1 2 is caused by the merging of the branch points from "the set of positive branch points 12 of Subsection 3.1.3 with their complex conjugates. As v continues to increase, the complex conjugate pairs then separate and remain on a-1 O<x~a+l· This can be visualized in Fig. 8 for various pairs of branches. n 4 and n5 approach in a maximum on n4 and a mini- branches mum on n For example, the symmetrical 5 a-1 near x=a+l" A slight increase in v over the value v=0.3 in Fig. 8 will cause branch points common to a-1 n 4 and n 5 to merge on the real x-axis near x=a+l; hence, -104- n 4 and n5 also merge at this point and, as described in Subsection 3.1 . 2, the branch points cancel and leave the branches analytic at this point. A further increase in v will cause these branch points to separate and remain on a-1 O< x :'.:a+l on the real x -axis. n 4 and n 5 will then be left connected by one of the branch points, as are the symmetria-1 cal branches n 9 and n 10 near x =a+l in Fig. 8 . By this method, branch points common to n and n are identified as 4 5 (+,1,4) from "the set of positive branch points 11 and as (+,2,5) which is the complex conjugate of (+,1,4) for v=0.3 . The triplets (±,j,k) identify these branch points + as zeros of the function ~jk in (3.1-9). The transition just described has occurred for all of the branch points which are seen on O<x<l in Fig. 13 as v increased to v=1· These branch points are iden t ified as the vertical tangents where ~~-00 in Fig. 13 and they are circled and labeled with the triplets (±,j,k) to identify + them as zeros of the functions ~;k in (3.1-9). 1 For v=all of the branch points from "the set of 2 positive branch points" have migrated onto the segment O<x <l of the real x-axis with the exception of an infinite subset of branch points . Before discussing this subset , the nature of the branch points which do migrate onto the real x-axis can be described by comparing Figs. 8 and 13 . -105- All of the branch points (±,j,k) with ±(-l)k=+l, which . . . t o v= 1 , are migrate onto t h e rea 1 x-axis as v increases 2 common to the real branches nk and nk+l· Referring to Subsection 3.1.3, ±(-l)k=+l identifies these as branch points from "the set of positive branch points" and they are common to symmetrical branches if j+k is odd while they are common to asymmetrical branches if j+k is even. The previously mentioned subset of branch points which remains off the real x-axis is just "the set of positive branch points" for v=!· The identification of these branch points by the triplets (±,j,k) depends on the definition of + the functions 6jk from (3.1-9). This definition is made precise by specifying the branch cut to the logarithmic branch point at x + 6jk· x 01 01 which is common to all of the functions is located as shown in Fig. 4 for v=21 . In Fig. 13 for v=! the choice of this branch cut is indicated by a vertical line at x=0.6 which represents the intersection of the real x-axis and the branch cut to x 01 . This is chosen quite arbitrarily so as to avoid the lower branch points on O<x<l. This being the case, "the set of positive branch points" for v=!2 which remain off the real x-axis are identified by the triplets (+,3,4), (-,2,1), (+ 1 3,2), (-,2,3), (-,4,5), etc., for the symmetrical branches and by ( - 'l , 1) ' ( + I 2 I 2) I ( - ' 3 I 3) ' the asymmetrical branches. ( +, 4 I 4) y ( - y 3 , 5) , etc. , for -106"The set of negative branch points" of Subsection 3.1.3 does not appear on the real x-axis for v= 1 2. These branch points are investigated in the following by letting 1 . v exceed 2 There is no physical reason to consider values of Poisson's ratio v exceeding !, however, the branches are analytic functions of v for v>! and they can be studied for any range of v. For ~<v<l the branches remain essen- tially the same as they were for v=~ with no more branch points appearing on the real x-axis or on lxl=l. a-1 a+l ~-given by (3.4-1) does move onto x=e The point ie with 0<8<n caus- ing some change in the branches. The limiting case v=l causes many things to happen to the branches. . a-1 All of the points a+l' xR' x 01 , x_ 1 and x 0 and their inverses and complex conjugates have merged to x=-1 leaving R=~ from (2.3-12a). then become sinn=±isinxn. The frequency equations The first obvious result is that the branches n are now even functions of X· The branches are sketched in Fig. 13 for v=l on beside the case v=! for comparison. valid for -l~x~o O~x~l The sketch is also by just replacing x by -x because the branches are even functions of X· A result, which is of no importance here, is that the asymmetrical Rayleigh branch n <x> and the asymmetrical 0 branch n1 (x), which shared the branch point at x=O, have -107become identically zero for v=l. The reason for looking at the branches for v=l is apparent from Fig. 13. The branch points on 0SXS1 for 1 in pairs to form third order 2~v<l have merged together 2 3 d d d branch points with dX-~=O~~ at the points. n dn dn For example, the two synunetrical branch points (+,1,2) and (-,1,2) in Fig. 13 for v= 1 1 2 have merged at x= 2 and n=2TI for v=l, the two asynunetrical branch points (-,1,3.) and (+,1,3) have 1 merged at x=j and n=3TI, etc. tion is the same on -l~x~O More importantly, the situa- for v=l since the branches are even functions of x; therefore, "the set of negative branch points" have merged with their complex conjugates on that segment of the real x-axis. Now all of the branch points have been located since the branches have no branch points other than those on the real x-axis for v=l. This can be seen by setting R=! in + x ; 7T (3.l-7a) to give f=O and b. -:k=k - . ..L:.:.... from ( 3 .1-9) so that J + zeros of b.jk' which are also branch points of the branches, x occur only at x=~ with n=kTI resulting from (3.1-7). This value of n causes sinn to vanish in (3.1-11) to explain how 2 d ~-O comes about at the branch points. dn This discussion has not explained what has happened to "the set of positive branch points" for off the real x-axis as v~. v=~, which remained These branch points have _ -108- migrated toward x=-1 as v+l and at v=l they merge with their complex conjugates at x=-1 to cancel each other and leave the branches n analytic except for the branch points already discussed. The nature of the branch points belonging to "the set of negative branch points" is not apparent even though they have migrated onto the real x-axis for v=l. These branch points are investigated in C.l of Appendix C where the frequency equations are approximated under the assumption that Poisson's ratio v is slightly less than V=l. The result of great importance which is found in C.l is that only the complex branches with Imn>O have branch points belonging to "the set of negative branch points" and these branch points are shared with real branches on -l<x<O. The complex branches with Imn<O have no branch points from ''the set of negative branch points" and, in fact, these complex branches share branch points with the real branches on -l<x<O which are the complex conjugates of branch points from this set . It is interesting that the form of the branch points, which occur on the real x-axis or in complex conjugate pairs adjacent to the real x-axis, is related to the bounds on R(x) and xR(x) given by (A-5) in Appendix A. That is, a-1 on ~a+_1 <x<l where -l<R<l and -l<xR<l, the branch points are -109- in complex conjugate pairs which are common to neighboring real branches. <a-1 where R>l and -l<xR21, the On O< X-a+l branch points are in pairs on the real x-axis and they are common to complex loops and real branches. On -l<x<O where 1 -l<fRJ- <1 and -l<[xRJ- 1 <1, the branch points are in complex conjugate pairs with the upper branch points (Imx>O) common to the real branches and only the complex branches with Imn>O. Continuations about the singular points x=-1 and x=l of the equivoluminal and dilatational branches respectively are described in C.2 of Appendix C. There it is found that · i umina · 1 b ranc h es on x=e i 8 , th e equivo o2e2n, continue · ab out x=-1 onto the complex branches with Imn<O on -l<x<O. The complex branches with Imn>O continue about x=-1 onto the dilatational branches on x=eie ~ 0~82TI· This difference is a result of the nature of the branch points from "the set of negative branch points," which was just discussed. Not surprisingly, the dilatational branches continue a-1 about x=l onto the real branches on a+l<x<l since these are the only branches on this segment of the real x-axis. These continuations of the equivoluminal and dilatational branches provide information about the branch points near x=-1 and x=l. Additional information about the branch points can be . gained by approximating the function ~jk in (3.1-9) in a -110- Taylor's series about x=-1 and x=l. The zeros of the + approximated function 6jk then gives the approximate location of branch points near these points. Sets of branch points corresponding to j fixed and k large and varying are found corresponding to the triplets (±,j,k). Obviously, these sets become infinite as k++ 00 and it is found that they have limit points at x=l. For ±(-l)k=+l and the appropriate requirements on j and k, these branch points belong to "the set of positive branch points," which were found to be common to neighboring real branches, nk and nk+l' fork fixed and j varying. These branch points are also common to the dilatational branch n. for j fixed and J k varying. Likewise, "the set of negative branch points," represented by (±,j,k) with ±(-l)k=+l and j=-1,-2,-3, ... , was found to be common to real and complex branches as described. For j fixed and k varying, the approximate loca- tion of these branch points shows that the infinite sets as k+oo have limit points at x=-1. The abundance of these branch points near the real x-axis as x+-1 is reflected by the rapid oscillation of the imaginary part of n on the complex branches. The details of the local analytic continuation about a2x 1 a branch point where dx_o d and ----'27'0, which includes all but n an -111- a few branch points, is given by (3.1-13); thus, the nature of any branch point can now be understood so that analytic continuations can proceed. 3.4.2. Analytic Continuations on a Closed Path. A discussion of analytic continuations of a general branch of either frequency equation is given in this subsection. In C.3 of Appendix C, "the set of positive branch points" for v=!, which were not satisfactorily explained in Subsection 3.4.1, are examined by using these analytic continuations on a closed path. The conclusions of the present subsection are based on the ahalytic continuations of three specific branches of the frequency equations corresponding to a closed path in the x-plane, which are described in C.4 of Appendix C. The continuations of the branches are conformal mappings since the branches n are analytic functions of x and they are single-valued mappings if no singular points of n are interior to or on the closed path in the x-plane. The paths for the analytic continuations described here and in C.4 of Appendix C consist of the boundary of the upper half unit disk ( j x I <l , Imx>O) taken in a counterclockwise direction with all of the branch points of the branch in question excluded by making indentions about the -112- points and their branch cuts . The branch cuts can be taken perpendicular to the real x-axis for convenience. The continuation illustrated by Figs. 14 and 15 invalves the symmetrical branch n dilatational branch no on x=e cal equivoluminal branch n branch) on x=e ie , 8 0 <8~TI. 0 i8 a-1 at x=a+l' the symmetrical 1 , 0<8<80, and the symmetri- (the symmetrical Rayleigh This analytic continuation is explained in C.4 of Appendix C. The point x =e 0 i80 is de- fined by (2.3-8) and it is a branch point common to these two branches on x=e i8 as is seen from (B.l-1) in Appendix B. Similarly, Figs. 16 and 17 illustrate a continuation a-1 which involves the asymmetrical branch nl at x=a+l and the asymmetrical dilatational branch n 1 on x=e i8 , 0<8<n; and Figs. 18 and 19 illustrate a continuation which involves a-1 ns at x=a+l and the symmetrical the symmetrical branch equivoluminal branch n 2 on x=e i8 , 0~8<n. These analytic continuations are also explained in C.4 of Appendix C. Continuations involving branches with higher branch numbers follow from the last two examples in C.4 of Appendix . c with the only difference being that more branch points are involved. However, the details of continuations about individual branch points is the same. It is noted that the continuation described by Figs. 16 and 17 is called a dilatational continuation while ~he -113- continuation described by Figs. 18 and 19 is called an equivoluminal continuation. These names for the continua- tions are explained as follows: If a given real branch is continued in the x-plane, as shown in Figs. 16 and 18, so that the path circles all of the branch points from "the set of positive branch points 11 which are common to this continuation of the branch, then i t is called a dilatational continuation if the branch continues onto a dilatational branch on x=e i8 , 0~8~TI. Likewise 1 it is called an equivoluminal continuation if the branch continues onto an equivoluminal branch on x=e ie , 0~8~TI . These names are used both for continuations on closed paths, as in Figs. 16 and 18, and also for unclosed paths. The continuation described by Figs. 14 and 15 is not designated in this manner simply because it involves both a dilatational and an equivoluminal branch on x=e ie , 0~8~TI. However, this is the only such case since only these branches share a branch point on x=e ie , 0<8<TI. A point of interest is that equivoluminal continuations involve branch points from 11 the set of positive branch points" which have j+k constant while dilatational continuations involve those with k-j constant. This assumes that the branch points are designated by the triplets (±,j,k) so that they are also zeros of the functions + l:ijk in (3.1-9). -114- In Fig. 20 the branch points from "the set of positive branch points," which are common to neighboring real branches, are shown on a-1 a+l~x<l. The branch points in Fig. 20 are designated by points on vertical lines connecting the real branches which share the branch points. the points are labeled by the triplets Some of (±,j,k) identifying them as zeros of the functions ~~k in (3.1-9). Fig. 20 serves to illustrate the equivoluminal and dilatational continuations. If continuations about the branch points, as in Figs. 16 and 18, are visualized, an equivoluminal continuation involves starting at x=:~i on a real branch nn(X), such as the symmetrical branches n 3 ,n 5 , n 6 ,n , ... or the asymmetrical branches n ,n ,n ,n ,n , ... , 8 2 4 5 7 8 and stair-stepping down a branch at every branch point encountered by continuing toward x=l on the real x-axis. Similarly, a dilatational continuation involves starting a-1 at x=a+l on a real branch nn(X), such as the symmetrical branches n ,n ,n ,n , ... or the asymmetrical branches 1 2 4 7 n1 ,n 3 ,n 6 , ... , and stair-steppin~ up a branch at every branch point encountered by continuing toward x=l on the real x-axis. A close relationship exists between the type of continuation just described and the value of the real branch a-1 at x=a+l" For all but a few of the lowest branches (this -115- depends on the value of v) , those branches which take the a-1 values at x=a+l given by the first of (B.1-5) in Appendix B or by the first of (B.1-6) lead to equivoluminal continuations. Likewise, those branches which take the values at a-1 x=a+l given by the second of (B.1-5) or by the second of (B.1-6) lead to dilatational continuations. -116- CHAPTER 4 THE APPROXIMATE REPRESENTATION OF THE HIGH-FREQUENCY RESPONSE The modal solution (2.3-15) can now be replaced by an equivalent representation which makes the high-frequency response much more accessible. By utilizing the analytic continuations of Section 3.4, the modal solution is found to be equivalent to integrals over the integration paths CE and c 0 in Fig. 21, and these integrals involve the equivoluminal and dilatational branches from Subsection 3.2.2 respectively. The series representations in Section 3.3 for the equivoluminal and dilatational branches are then used in these integrals to obtain approximations to the high-frequency response of the plate. The geometry of the resulting wave fronts in the plate is shown in Fig. 22 and a discussion is given relating the various wave fronts to the frequency spectrum in the x,n-space or in the w,Kspace. This method requires expansions of the integrands, which are much like those used by Rosenfeld and Miklowitz [l] to obtain wave front expansions. However, relatively convenient representations of the high-frequency response are found here, which appear to be valid without the -117- restriction of nearness to the wave front as is required for wave front expansons . A specific representation is further approximated near a wave front to make a comparison with an amplitude in the wave front expansion given by Rosenfeld [13]. 4.1. THE MODAL SOLUTION ON THE EQUIVOLUMINAL AND DILATATIONAL BRANCHES In this section the modal solution (2 . 3-15), with the exception of the Rayleigh modes n=O, is shown to be equivalent to a sum of integrals over the equivoluminal branches plus a sum of integrals over the dilatational branches of Subsection 3.2.2. This is done so as to make use of the series representations in Section 3.3 to approximate the high-frequency response of the plate. The Rayleigh modes are not incl uded because this method offers no advantage over the usual w 1 K representation for these modes. The integrals (2 . 3-lSa) separate into two parts con- sisting of that part which results from the dilatational potential ¢ through the first term in each of the functions U in (2.3-16) and that part which results from the equivoluminal potential of the functions u. ~ through the second term in each -118- With some foresight about the task of identifying the wave fronts, it is helpful, and perhaps necessary, to write those parts of the integrals (2.3-lSa} which result from the dilatational potential ~ a~ the imaginary part of integrals of the form (4.1-la} Similarly , those parts of (2.3-lSa} which res.ult from the equivoluminal potential \jJ are written as the imaginary part of integrals of the form 1 c F\jJ(X,nn(X},1';} [ sin (Ks}] e . lWT dx (4.1-lb) cos(Ks} In both (4.1-la} and (4.1-lb}, C is the integration path shown in Fig. 2 and K and w are given by (2.3-7) with n=nn(X}, n=l,2,3, . . . . The bracket term in (4.1-lb} just means that either sin(Ks} or cos(Ks} may occur in the integrand as in (2.3-lSa) . These forms are to be used to examine the response of the right half of the plate , s>O. As was mentioned, the representation (2.3-15) is not a form which readily yields information about the wave fronts or about the high-frequency response in general. Thus, it is expected to be advantageous to write the response as a sum of integrals over the equivoluminal and -119- dilatational branches since these branches seem closely related to the equivoluminal and dilatational wave fronts respectively. By using the forms (4.1-la) and (4.1-lb) v either the symmetrical or asymmetrical parts of the representation (2.3-15) can be written as the imaginary part of sin F (Kl;)] . ]ax 1jl [ COS (Kl;) Loo + f[ . e1WT F ¢sin (WT) e 1 Kl; + F \jJ [sin m=O,l C (Kl;)] e . ]ax 1 WT (4.1-2) COS(Kl;) D where the integration paths CE and CD are shown in the first sketch of Fig. 21. The justification of the equiva- lence of (4.1-2) and the modal solution (2.3-15) will be given shortly. F,,,(x,n 'I' n On CE' Fcp=Fcp (x,nn(X) ,r;) and Fl/J= (x) ,r;) where n (x), n=l,2,3, •.. , are the equivon luminal branches shown in Fig. 9 or analytic continuations of these branches. On CD' Fcp=Fcp(x,nm(X) ,r;) and F~= Fl/J(x vnm(X) ,r;) where nm(X) are the dilatational branches shown in Fig. 10 or analytic continuations of these branches . The lower limit on the sum in the second term of (4.1-2) is from m=O for the symmetrical case and from m=l for the asymmetrical case. Again, w and K are given -120- by (2.3-7) with n=nn(X) being the equivoluminal branches in the first integral of (4.1-2) and n=nm(X) being the dilatational branches in the second integral. a-1 The first sketch of Fig. 21 shows CE going from x=a+l to x_ =e 1 ie_ 1 given by (A-la) on the path c as discussed in Appendix A. 1 on which IRl=l From x_ , CE proceeds counter1 clockwise on the unit circle lxl=l to the point x=-1. CE is meant to be such that all of the branch points common to the particular equivoluminal branch n (x) which belong to n "the set of positive branch points" of Subsection 3.1.3 are a-1 to the right of the portion of CE from x=a+l to x=x_ 1 . Likewise, all of the branch points common to the particular branch which belong to "the set of negative branch points" should be to the left of this portion of CE. If this is not possible due to the location of some of the branch points, small indentions can be. made in CE so as to leave all of these branch points on the proper side of the integration path. The first sketch of Fig. 21 also shows the integration . f rom x=a+l a-l to x_ exac tl y as d i' d pa th c D going 1 cE and the same comments hold for those branch points common to the particular dilatational branch nm(X). c 0 then continues clockwise on the unit circle lxl=l toward the point x=l. A small indention in c 0 about x=l must be included because -121the integrals in (4.1-2) fail to exist for certain values of ;,s,T as 8+0 on x=e seen from (2.3-7), i8 , 0<8<n. This difficulty can be (3.2-8) and Fig. 10 showing that on the dilatational branches w=ilwl and K=ilKI with both wand K unbounded as 8+0 on x=e i8 , 0<8<TI, and limw-a. 8+0K ratio of velocities gi~en by (2.1-2). a is the The indention in c 0 passes above all of the branch points from "the set of positive branch points" which are common to the particular dilatational branch nm(X). to the real x-axis. Yet C0 approaches x=l tangent This is possible since the locus of this infinite set of branch points is also tangent to the real x-axis. Justification of the equivalence of the representations (4.1-2) and (2.3-15) still explanation. require~ considerable This is not done rigorously since it involves rearranging infinite series which are not expected to converge everywhere due to the presence of singular wave fronts. However, the representation (4.1-2) can be shown to involve precisely the same integrals over the same branches as does (2.3-15) and that is done in the follow:ing. For each equivoluminal branch, the path CE is deformed onto the real x-axis and the unit circle x=e 0<8<TI . i8 This is done as in Fig. 18 so that the deformed a-1 path goes from x=a+l along the real x-axis in a positive -122- direction with vertical indentions for the branch points from "the set of positive branch points" which are common to this particular equivoluminal branch. Finally, x=l is reached and the deformed path remains on x=e ie , o~e~rr, to the end point x=-1. Likewise, the integration path c0 is deformed onto the segment a-l<x<l of the real x-axis for each dilatational a+l- branch with similar vertical indentions for the branch points common to the particular branch. Examples of these d e f orma t'ions o f CD are s h own in · F'igs. 14 and 16 from x=a+l a-l to x=l. By the Cauchy-Goursat theorem, these deformations of the integration paths CE and c0 do not change the values of the integrals. (4.1-2) is then equivalent to sums of integrals over all equivoluminal and dilatational continuations of the branches n (x) / n=l,2,3, . . . , with the equivoluminal continn uations extending from x=:~i to x=l and then to x=-1 while t he · 1 continuations · · a-l t 0 x= 1 . d i· 1 atationa ex t en d f rom x=a+l The equivoluminal and dilatational continuations are defined in Subsection 3.4.2 and example~ are shown in Figs. 19 and 17 respectively. (4.1-2) then involves integrals over all of the equivoluminal branches on x=eie, 0<8<rr, which is identical on both C in (2.3-15) and the path CE deformed as described -123- for (4.1-2). This portion of the integrals is then identi- cal in (2.3-15) and (4.1-2). The remaining portion of (4.1-2) involves integrals over the equivoluminal and dilatational continuations of all branches n (x), n=l,2,3, ... , on a-l <x<l plus the branch n a+ 1 - line integrals around the various branch points from "the set of positive branch points." Referring to Fig. 20 and to the description of equivoluminal and dilatational continuations from x=:~i to x=l which is given in Subsection 3.4.2, it is seen that (4.1-2), with the paths CE and c 0 deformed, covers every portion of every real branch on a-1 a+l~x~l. This fact along with the fact that the equivo- luminal branches are COVered On x=eie I osesnl means that the same branches have been covered in (4.1-2) as have been in (2.3-15). Only the integrals on the indentions about the branch . t 1 as shown in Figs. 14, 16 and 18 poin s a d.Jacent to a-1 a+lSX~ remain. Every branch point belonging to "the set of posi- tive branch points" is easily seen from Fig. 20 to be involved twice in these integrals--once on an equivoluminal branch and once on a dilatational branch. It is then a simple matter to show that the two branch line integrals about a particular branch point cancel each other. The integrals on the small circular paths can also be shown to cancel. Also, by making a simple expansion about the -124- branch points and by using (3.1-13) it is seen that cosn± 1 2 XRcosxn=o(jx-xcj ) near a branch point x=xc· (cosn±xRcosxn) -1 Since are the only singular terms at the branch points in the integrands of (2.3-lSa), the integrals on the small circular paths give zero contribution as the paths are shrunk to the branch point. Thus, (4.1""'.2) results from (2.3-15) by a rearrangement of the series representation. The representation (4.1-2) is taken as the response of the plate to the loading (2.1-7). To justify this form as a solution of the govern- ing equations of motion (2.1-4) requires the procedure discussed at the end of Section 2.2. 4.2. THE HIGH-FREQUENCY APPROXIMATION 4.2.1. The High-Frequency Expansion of the Modal Solution. The representation (4.1-2) is to be approximated by using the series representations for the equivoluminal and dilatational branches from Section 3.3. Before proceeding with this approximation, it is convenient to write portions of the integrands of (4.1-2) or (2. 3-lSa) as sums of exponential functions. A product of the terms sin {un) eiK~ and the first terms of u in .(2.3-16) and of [sin (Kl;)] . e cos (KO 1WT . and the second terms of u are required. -125- In order to write these products, the phase functions ¢i and ~i' i=l,2, .•. ,8, are defined as ~ (f; , q •X) = ~I; [ ( x-:~i) (:~i-x) 1, r 1 (4.2-1) 1 1 2 -(-1) i ~ a-1 a+l [(x-a+l )(--x a-1 )] 2 where the upper sign applies for i=l,2,3,4, the lower sign applies for i=S,6,7,8 and 7,8. ~i ~ is replaced with -~ for i=3,4, is called the dilatational phase function and ~i is called the equivoluminal phase function since they are to be related to the dilatational potential ¢ and the equivoluminal potential ~ respectively through (2.2-1) and (2.3-16) defining the functions U. The spatial and time ~i dependence of the response depends strongly on and ~i and the geometry of the wave fronts is determined entirely by these functions since they will contain all of the dependence of the integrals on the variables The phase functions ~, T and ~. (4.2-1) are now used to write the -126required products in two parts where the first and second terms in the expressions (2.3-16) for the functions U are separated to give sinn ( 1-x) z; cosn ( l+x) 2 2 icosn (l-x)z;cosn (l+x) 2 2 sin(Kt;)sinn (l-x)cosn (l+x)r; 2 2 sin(Kt;)cosn(l-x)sinn (l+x}l; 2 2 ( 4. 2-2) where (Ski) = -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 l l -1 l -1 (4.2-2a) -1 1 1 -1 1 -1 1 -1 The form (4.2-2) is deduced by simply writing the trigonometric functions in their exponential forms. -127- The portion of the integrands in (4.1-2) or (2.3-15) which contain the branches n can be expanded by using the series representations in Section 3.3. From (2.3-lSa) and (4.2-2), the only portion of the integrands which involve n in~R- is composed of the expressions e e in~R- (cosn±xRcosxn) -1 . (cosn±xRcosxn) _1 and Expansions of these . expressions are required for the four sets of branches: the symmetrical and asymmetrical equivoluminal branches and the symmetrical and asymmetrical dilatational branches. The expansion which exhibits the wave fronts is a power series in e '(l) and e-i. e 2in(O) -x n and (0) -2in (O) for the equivoluminal branches and in ei(l+x)n(O) for the dilatational branches . n(O) is the first term in the series representation for given by (3.3-14) , tively. (3.3-16), n (3.3-23) and (3.3-28) respec- These expansions leave the branch numbers n and m as a linear multiple in the argument of the exponential functions making it an easy matter to sum over the branch numbers. The expansions are then simple in form and the only difficulty is in calculating the coefficients in the power series. It is necessary to write the expressions cosn±xRcosxn from the denominators of the integrals in the forms -128- cosn±xRcosxn (4.2-3a) 1 -in(1+ R i(l-x>n) ·(1+l±xRe-i(l-x)n 2in) cosn±xRcosxn = ~ -X e i(l- ) e l±xRe x n (4.2-3b) for these expansions where (4.2-3a) is to be used when n is on an equivoluminal branch and (4.2-3b) is to be used when n is on a dilatational branch. The expansion 4 -i(l-x)n(O) -2in(O) ) + O le en I ,je en I ( ] (4.2-4) is for the symmetrical equivoluminal branches nn' n=l,2,3, (O) is . . b y (3 .J~ ') 3) an d "'~£is . . by h ... ,were nen given given (4.2-1) . The error term in (4.2-4) means that there are possible errors o le ( -in (O) ( (l-x)p+2q) ) en I of p and q such that p+q=4 . for any combination The coeffici ents A pq in (4.2-4) result from the series representation (3.3-14) for the symmetrical equivoluminal branches and from the form (4.2-3a) . The first ten of these coefficients are -129- AlO = i (qi i -1+x)n~ 1 > l+xR2 l+x AOl = 2 1 2 __1+2 A20 = 2 9, - 2c1-xl~t + 1 - 3x + x ](n~ l) + i (qi R- -l+x> n~ > 2 2 + XR2] ne(1) -·[l+xR\ _ ( 3 _ 2 x )l+xR All = i i+x i l+x 22 ) A02 = (l+xR l+x A30 = -~H - - 3(1-xl~; + 3Cl-3x+x2l~t - Cl-xl c1-6x+x2i](n!1l) 3 [qi~ - 2(1-x)qin + l-3x+x 2]n(e 1 )n(e 2 ) + i(qi -1+x>n<e 3 > N N ~ · = l[l+xR2qi2-2((3-2 )l+xR2 - R2)~ + (9-14 +4 2)l+xR2 2 i+x x i l+x - s (1- )R2J(, (1))2 - i[1+xR2 qi x x \ne l+x x i i x x l+x - (3-2 ) l+xR2 + xR2]ne(2) x l+x ( 3 -S)l+xR x l+x 2 ( 4.2-5) where the terms n~j) are given by (3.3-15) . The same -130- expansion is obviously valid with ~i from (4.2-1) replacing <l>i throughout (4.2-4) and (4.2-5). From (4.2-3a), (4.2-5) and (3.3-i6), it is apparent that the expansion (4.2-4) is also valid for the expression e inn<l>i (cosn -xRcosxn ) n n -1 where n , n=l,2,3, ..• , are the n asymmetrical equivoluminal branches if the coefficients A are replaced by (-l)PA pq pq and n(O) is replaced by n(O) . 1 en en-2 n (O)l results from (3.3-3) for n (O). en en- 2 The expansions involving the dilatational branches are derived in the same manner as for those involving the equivoluminal branches, however, they can also be deduced from (4.2-4) due to the similarity between (4.2-3a) (4.2-3b). and As a result of this, the expansion = -2 [p+q=3 , L A e 1-x p,q=O pq 0 + 0 je ( in~> [<l>i+1+(1+x>p+2q] · 0 i ( 1 +x >n ( ) 2 in ( > ) dm I ,je dm I 4 ] ( 4. 2-6) is for the symmetrical dilatational branches nm(X), m=O,l, 2, ... , where n:) is given by (3.3-24) and 9i is given by i (4.2-1). The coefficients A are given by (4.2-5) with pq . and <l>i replaced by -x and -<l>i respectively and with replaced by -n~j). The terms n~j) n~k) are given by (3.3-25). x -131- R=R(x) remains unchanged. Again the expansion is obviously valid with ~i replacing ¢t throughout. As for the equivoluminal branches, the expansion (4.2-6) is also valid for the expression e -xRcosxn ) -1 m where n in ¢i m (cosnm m, m=l,2,3, ... , are the asymmetrical I dilatational branches if the coefficients A are replaced pq 1 by (-1)PA and n(O) is replaced by n(0) . n (O) 1 results 1 pq dm dm-dm2 ( 0) 2 from (3.3-24) for ndm . The expansions (4.2-4) and (4.2-6) and the similar expansions for the asymmetrical branches are expected to converge uniformly with respect to x on the same regions in the x-plane as did the series representations in Section 3.3 for the equivoluminal and dilatational branches. This is true in part because those series representations were used to develop these expansions, but, it is also true because a geometric series was used to expand the expressions (4.2-3a) and (4.2-3b). convergence is then that An additional condition for 1± Rei(l+x)n -2i X . e n < 1 on the l±xRe-i(l+x)n equi voluminal branches, and the analogous condition with x and n replaced by their negatives as in (4.2-3b) on the dilatational branches. These conditions are certainly not satisfied at a branch point since cosn±xRcosxn=O as was found in Section 3.1, however, the presence of the factors e -2in and e 2in in (4.2-3a) and (4.2-3b) insures that th~se -132- geometric series converge uniformly on practically the same regions as do the series representations of Section 3.3. Also, the same comments hold here as did in Section 3.3 about the regions of convergence becoming larger as the branch numbers n and m are increased . Obviously, the ex- pansion (4.2-6) involving the symmetrical dilatational branch n (x) cannot be valid on all of the integration path 0 CD since this branch has a branch point at x =e 0 by (2.3-8) where 0<8 <e_ 0 ie 0 given 1 1 for O<v22· Very little can be claimed about convergence of the dilatational expansions on the small indention of c 0 about x=l in the first sketch of Fig. 21. The series of Section 3.3 were not shown to converge here. Also, c0 approaches the locus of a set of branch points whose limit point is x=l. However, it is a reasonable assumption that the series do converge on this indention since this locus of branch points probably determines the boundary of the domain of convergence. The fact that the expansioris (4.2-4) and (4.2-6) and the similar expansions for the asymmetrical branches fail to converge to the desired functions on all of the integraa-1 tion paths CE and CD as x=a+l is approached is the error which makes this method approximate. The expansions (4.2-4) and (4.2-6) suggest yet another set of phase functions to simplify the notation. These . are -133- (4.2-7) <P d.Q,pq where <Pi and ~i are defined by (4.2-1). In (4.2-7) the subscripts e and d refer to the fact that these phase functions will be used in the equivoluminal and dilatational expansions respectively. The remaining subscripts refer to integers with i=l,2, .•. ,8 and p,q=0,1,2, . • . • The phase functions and (4.2-6), the forms (4.2-7), the expansions (4.2-4) (4.2-2) and the representations (4.1-2) make it possible to write (2.3-15) for the displacement component u~ as 16rrµ~ ,...., Im~ aI u~ = L., ;. n=l p,q=O x { w 1 . <P i LogR inrr<P t 2 2 rx -2(1-2v)x+l]e e pq . e e pq !~ - ( 1 -x 2) e 2 e.Q,pq LogR .inTI~ e.Q,pq} d e X -134- ()() ()() + Im L: L: m=O p,q=O X { -~ LogR i(m+!)n~ 2 Ix 2 -2(1-2v)x+l]e 2 dtpq e ~pq - (1-x 2 )e -!~at pq LogRe i(m+!)n~dt pq } ax i: f + Im t1 (-1) PA n=l p,q=O t=l x s l-x Pq 2t -3 CE 2 x o<x> 1 . ( 1) ~ LogR 1 n-- n ... 2 rx 2 -2 ( l-2v >x +l] e 2 ell pq . e e.tpq ;f,. { ()() + rmL: m=l x . 2 -:k LogR imn qi rx -2(1-2v)x+l]e 2 d.tpq e dtpq { - (1-x 2 )e -!~ai pq LogRe imn~dt pq} ax (4.2-8) The Rayleigh modes, n=O, are not included in (4.2-8). Also, the expression is approximate in the sense that the -135- expansions (4.2-4) and (4.2-6) have been used as exact representations. The sum over p and q has been written as I if all of the coefficients A and A were known. pq pq The summation and integration order has been changed. This can be justified rigorously only on the portions of the integration paths CE and c 0 where {4.2-4) and (4.2-6) can be shown to converge uniformly with respect to appropriate expressions. n(O) and n(O) en en- 1 x to the from (3.3-3) and 2 n~) and n(O) 1 from (3.3-24) have been used in (4.2-8). dm- 2 The four integrals in (4.2-8) represent integrations over the symmetrical equivoluminal branches, the symmetrical dilatational branches, the asymmetrical equivoluminal branches and the asymmetrical dilatational branches respectively. The terms Sk~ are given by (4.2-2a). As can be seen from (2.3-15), a similar representation holds for the displacement component u~. The symmetrical and asymmetrical integrals for each of the equivoluminal and dilatational cases can be added in (4.2-8) to give -136- 167Tµ~ ar u~ ~ rm}: ()() ()() L: n=l p,q=O ()() ()() + Im :E 'E m=l p,q=O (4.2-9) and similarly for u~. This operation of adding the symmetrical and asymmetr.i:cal terms is expected to cancel all of the extra wave fronts which would have resulted from a singular loading applied at the bottom face of the plate (~=O, ~=-1). It is recalled that such a loading is present in a purely symmetrical or asymmetrical loading. The cancellation must occur -137· as a result of the terms like s Q,+(-l)Ps Q,e 2 1 - i . TI"' 2'*' Q, e pq' etc., vanishing for certain combinations of Q,, p and q and for certain values of the variables~' T, ~ and X· The phase functions in (4.2-9), which are defined by (4.2-7), are complex on the integration paths CE and c 0 and the integrals are more convenient to handle if these paths are deformed so that these phase functions are either real or appear in a simpler form. This will make it possible to sum over the branch numbers n and m and then to recognize singular wave fronts. It is very important to realize that (4.2-9} is now a term-by-term integration and the integrands possess none of the branch points which were of concern before. the only branch point of interest here is x 01 In fact, where R=O, so that it is a branch point of the function LogR. The locus of x 01 as Poisson's ratio v varies is shown in Fig. 4 and the point is described following (A-3) in Appendix A. With the preceding in mind, the integration paths CE and c 0 of (4.2-9) and shown in the first sketch of Fig . 21 are deformed as shown in the second and third sketches of Fig. 21. The path CE is deformed back onto the original integration path C shown in Fig. 2 except for a vertical indention about the branch point at x 01 . The path c 0 is just collapsed onto the real x-axis on a-l<x<l with an a+l- - -138- identical vertical indention for the branch point at x The point x 01 . 01 where R=O causes difficulties in the representation (4.2-9) with CE and c 0 deformedasdescribed. The integrands in (4.2-9) may be singular at this point depending on the values of the phase functions as can be 1 -24 i LogR e pq , etc. seen from the functions e Actually, it would be desirable to demonstrate mutual cancellation of the branch line integrals about this point as was done in Section 4.1 about the original branch points. It is true that the integrals along the branch cut to x cancel by 01 mutual cancellation when the sum over the branch number is taken with the exception of one line integral belonging to the lowest branch. However, cancellation of the integrals on the small circular path about the branch point x well concealed in (4.2-9) if it is present at all. 01 is The intention here is not to investigate these branch line integrals in detail, but to look at the remainder of the 1 inte. grals on a- <x<l and on x=e a+l- - ie , 0<8<rr, and to try to locate physically meaningful results. The approach to be used is to bring the summation over the · mode or branch numbers n and m inside the integrands of (4.2-9) and to sum over the terms e inrr¢ eNpq n , etc. These are the only quantities in (4.2-9} which contain the branch numbers . This interchange of the sununation and integration -139- order is again done without justification. The summations yield generalized functions if the phase functions (4.2-7) are real and they yield ordinary functions by a simple geometric series if the phase functions are complex with positive imaginary parts. The phase functions (4.2-7) can be examined more conveniently in terms of the variable B = _!_ '1a 2-l 2a i+x (4.2-10) 1 2 x .e 1 2 with x =e i2 on x=e ie , 0<8<rr. Bis real on all of the inte- From (2.3-5) and (2.3-7), B=~ in w a- 1 <x<l, l>B>! Va 2-l terms of the original variables. On a+ - -a 1- - gration path C of Fig. 2. and on x=e ie I o~e~rr, 1-~ 1-~ e a va--1 ~B~O where B=ava--1 cos2· The phase functions (4.2-7) then become ~etpq i [l;~l-B 2 = 'I' etpq = i [T- (-1) (-1) tT_ (2p+2q+l-i;) tl;~l-B 2 - (2p+2q+l:;:l) ~B 2 -~]-2q-l±l ~B 2 -a:;1]-2q-l+I; -140- l ~B2_a ;1 2 [~-V1-B 2 - (-1) i-c + ( 2p+2q+ 1± 1) B] +2q+ l+r,; a '¥ = dipq ( 2p+2q+l+r;) B] +2q+l± 1 (4.2-11) where the upper sign still applies for i=l,2,3,4, the lower sign applies for R.=5,6,7,8 and r; is replaced by -r; for £=3, 4,7,8. In (4.2-11) all of the terms are real and the radi- cals are nonnegative on the integration path C except for 1-x - -1 which is pure imaginary with the x2 imaginary part being negative on x=e a ya ie 2 2 ~B 2 - -1 . -l 2 i -~ . e - = -1~-B = -- Va--1 sm-. a2 a a 2 0<8~7T, where Thus, <P e)Vpq n and . i6 '¥eipq have nonnegative imaginary parts on x=e , oses7T, and a-1 are real on a+l~xsl. · The phase functions ~dipq and '¥dipq <l are only involved on a-1< a+l-X- where they are also real. It is seen from (4.2-11) that '¥eipq is real on all of the integration path c including x=e ie I osesTI, only for the special cases where 2p+2q+l+l=O, which means that p=q=O and i=l,2,3,4 using the fact that p,q~O. These cases will be seen to be identified with the circular equivoluminal wave -141- Likewise, front. ~ is real on all of C for the special n eivpq cases where 2p+2q+l-s=O, which means that p=q=O for Q,=1,2, 5,6 and s=l or p=q=O for Q,=3,4,7,8 and s=-1. When the phase functions (4.2-11) are real, the repre- sentation co • L eimr~ = 12 + n=l can be used where co ~ L..J m=-co i TT o (~-2m) + ~ot~ (4.2-12) o (~-2m) is the Dirac delta function. This is just a Fourier series representation of periodic generalized functions and it results from the representations given by Lighthill r9J. Also, (4.2-12) can be de- rived by differentiating the Fourier series representations co co ~cosnTT<P=-log I 2sin~<P I and ~sinnTT<P which represents a n=l n=l L L sawtooth function. A geometric series gives iTT~ = e iTTcl> 1 -e (4.2-13) when the phase functions have positive imaginary parts. (4.2-13) is also valid when the phase functions are real if the limiting process is carried out carefully so as to produce the generalized functions in (4.2-12). The portion of the integrals (4.2-9) which result from 1 ie integration on :~ 1 ~x~l and on x=e f 0~0~TT1 as shown in the -142- second and third sketches of Fig. 21 can now be written with the sums over the branch numbers, n and m, having been performed. These consist of Cauchy principal value inte- grals due to the presence of the generalized functions cot~ and csc~. The integrals are separated into four distinct sets of integrals, each identified with one of the four phase functions (4.2-11). Thus, by using (4.2-12) and (4.2-13), the quantity in (4.2-9) can be represented as 32TIµ~ ~ I de + I ee + I dd + I ed u~ aI (4.2-14) where fl " 81 . :E L.J p. v. p,q=O i=1 00 1 TI A o pq + (-l)Ps 21 pq . a-1 a+l · + 8Im A . ID {e i e) I -J."'°' n eNpq 0 1-e +l)[s 11 ~ e iTI~. eNpq ae ·'.!0i, 2 e1pq ~~ X2D{X) sin~2(cose-1+2v) iL2(~ e ~ 2 (1-x>rx -2c1-~,»x+11e2 etpq 3 - l.TI~ R, e pq ! (4.2-14a) L: ee = p,q=O 1=1 + 8Re l 11 La _P • v . 00 I J 21 00 L: e -i-~ · TI 2 el eNpq p,q=O 1=1 +l)[ s 11 pq~ e iTI~ . 1TI~ l e1pq P.vf de ~ (4.2-14b) I e pq 1-e L:8 ~ 2 X D(X) pq sin~sin8 i~(~ n TI k ( 1-x ) ( 1-x 2 ) e 2 e1 pq LogR A a-1 a+l A . e o pqlD(eie)I + ( -1) P s -143- 2 . -~ Lo R A' (l+x) [x -2(1-2\))x+l]e 2 d1pq g pq ~ a-1 2 a+l x D(x) 1 (4.2-14c) I 1· L: L 1 1 (l+ ) (l- 2) -P • V. A x . x e 2 ~dnN pq LogR =pq ~ p,q=O 1=1 a-1 2 a+l x DCx> 00 8 1 (4.2-14d) In these integrals the form (A-6) in Appendix A defining y=y(8) has been used along with D(e . 18 >=1D(e . 18 ·(e -2Y) 1 . ) le -144- from (A-6a). The terms sk 1 are defined by (4.2-2a) and the coefficients A are given by (4.2-5) for p+q=0,1,2,3. pq The I coefficients Apq result from Apq as described following (4.2-6). The phase functions ~e!pq' ~e!pq' td!pq and 'dtpq are defined by (4.2-11). The subscripts on these integrals are meant to indicate their origin such as Id e for the . integral resulting from the dilatational potential ¢ with the integration over the equivoluminal branches and similarly for the other three integrals. The Dirac delta functions from (4.2-12) are not present in (4.2-14a) , ... , (4.2-14d) simply because only the imaginary parts of the integrals in (4.2-9) are required and the delta functions always multiply the real parts of a-1 the integrands on a+l~x~l. A special case is described in the following where the Dirac delta functions do remain. The second inte. gral in (4.2-14b) for I ee must be rnodi- fied for the cases p=q=O and 1=1,2,3,4 since ~e!OO is real on x=e ie , O~e~n, and the representation (4.2-12) must be used rather than (4.2-13). Thus, using A =1 from (4.2-5), 00 the second term in (4.2-14b) for Iee must be replaced with -145- (4.2-15) for p=q=O and £=1,2,3,4. The second term in (4.2-14a) for Ide does not need to be modified even though its integrand is singular for some points on the boundaries s=±l for p=q=O. It is just necessary to interpret the integral as a Cauchy principal value at these points and to realize that (4.2-12) is a limiting form of (4.2-13). A similar representation for the displacement component us can be written by making very simple changes in (4.2-14a) , ... y (4.2-14d) and in (4.2-15) as suggested by (2.3-lSa) and (2.3-16). 4.2.2. The Geometry of the Wave Fronts. As usual, when dealing with generalized functions, the integrals (4.2-14a) , ... , (4.2-14d) and (4.2-15) are most conveniently investigated when the phase functions ¢ 0 , . e)(,pq ~e£pq' ¢d£pq and ~d£pq replace variable. x or 6 as the independent This makes it possible to identify wave fronts -146- as singularities in the integrals resulting from a singular point of the generalized function coinciding with a singular point of the remainder of the integrand. For the change of variables to the phase functions, the factor dX replaces =~ dBd<l> dB d<I> dx in the integrals where <I> is any one of the phase functions (4.2-11) and B is defined by (4.2-10). A singular point in the integrands occurs if dB d<I> dx dB (4.2-16) =O From ( 4 . 2-10) , dB dx = _!_ '1a2_1 2a i-x 3 2 x and from (4.2-11) , _!__[ d¢ etpq = _ $ _ dB B2 -~ Vl-B- 2p+2q+l-1'.,;l ~B2_a ;1 2 a -147d'!' a 2 -1 e.R-pq + -2- dB a d<I> d.R,pq =dB -d'l'd.R, pq dB (-1).R, B~ a2~1-B2 + a2;1(2p+2q+l+1;)] a (4.2-18) ddB-o at x=l or at B= 1 ~a 2 -l from (4.2-17), however, X a 1-x in all of this singularity is removed by the factor 3 x2 the integrals, except (4.2-14c) for Idd' giving 2a '1a 2 -l For Idd' the factor 3 -2 1-x 3 dx_ dB x2 2-1) , . (4.2-18) for 2 ( B _a 7 d<I>d.R, d dB pq makes ~ nonsingular. dB dB d<I>d.R,pq 1 'n 2 The point B=!~a -l is a special, however, since it will be seen to be identified with the head wave. The head wave is not singular in the displacements for the loading (2.1-7) used here. -148- Thus, since each of the generalized functions in (4.2-14a) , ... , (4.2-14d) and in (4.2-15) is singular at <I>=2m for any integer m, the condition for a singular wave front is the simultaneous satisfaction of the equations <I> ~ = 2m ( 4. 2-19) d<I> = 0 dB for any integer m where phase functions <I> represents any one of the four (4.2-11). (4.2-19) represents four sets of simultaneous equations resulting from (4.2-11) and (4.2-18) each of which can be solved for~ and ~ to give the spatial geometry of the wave fronts as a function of T. From (4.2-18), it is obvious that for ~>O ~:-o is not possible if t=l,3,5,7 since p,q~O and -1~~~1. These phase functions occur in terms which are singular only for ~<O and, therefore, are associated with wave fronts traveling in the negative ~ direction. The phase functions identi- fied by t=l,3,5,7 and the accompanying terms in the inte. grals are set aside for now and only those identified by t=2 , 4,6,8 are considered. It is helpful to define the normalized coordinates and parameters -149= ~ T L ze = 2p+2q+l-l; pq T zd = 2p+2q+l+l; pg T (4.2-20) 2p+2q+l+l e Qpq = T = 2p+2q+l±l T where the upper sign applies for £=2,4, the lower sign applies for £=6,8 and l; is replaced by -s for i=4,8. Then from (4.2-11) and (4.2-18), simultaneous satisfaction of (4.2-19) gives the following parametric descriptions of the singular wave fronts: for the phase functions ¢e£pq' L = a2 --J1-B2 (B+a2-1Qe ) B 2 mq a (4.2-2la) -150- for the phase functions ~e£pq' (4.2-2lb) for the phase functions ~d£pq' (4.2-21c) and for the phase functions ~d£pq' (4.2-2ld) -151- where t=2,4,6,8 throughout. These wave fronts can be sketched in the L,Z-space by taking a fixed value of Q and letting B vary in the range a-1 a+l~X~l. l~B~~ -Va 2 -l corresponding to Also, there are singular wave fronts in (4.2-15) in the range ~ '1a 2 -l>B>O corresponding to x=eie, 0<8<rr. - . - - a There is some restriction on the quantities involved in (4.2-21a) , ... , (4.2-2ld) since all of the quantities de- fined by (4.2-20) are nonnegative as a result of p,q~O, c;>O, T>O and -1::,C";::,l. However, when p is replaced by the integers ±m, as in z:q' Q~mq' etc., the quantity may be negative since the integers m are only restricted by the fact that (4.2-19) must be satisfied. Fig. 22 shows the equivoluminal wave fronts in dashed lines and they are present in lee given by (4.2-14b), and they result from~ n ex,pq as described by (4.2-2lb). The dilatational wave fronts are shown in solid lines and they are present in ~dtpq Idd given by (4.2-14c), resulting from The sketch is at the Cd time T=6.3 for v=0.3 which gives a= -13:'5~1 . 871 from as described by (4.2-2lc). 05 (2 . 1-2). T=6 . 3 means that an equivoluminal wave has had time to cross the plate ~T=3.15 times while a dilatational wave has had time to cross the plate !aT~S.89 times . The top sketch of Fig. 22 is essentially the wave fronts in the L,Z-space with L=~ measured on the same axis as ~ and T -152- Z=Ze zd measured downward from the horizontal line on mq' -mq which a particular wave front terminates. Actually, this sketch just shows the right half of the plate, the thickness -l~s~l s~O, with successively repeated so that all wave fronts appear as continuous lines. The geometry of the wave fronts is exactly like that for a layered half-space made up of identical layers with refraction but no reflection at the junctions. The second sketch in Fig. 22 shows the superposition of these wave fronts as they appear in the plate. The superposition just consists of folding the first sketch as an accordian at each horizontal line representing the edges of the plate to form a single thickness. 1-~ The only case in which the range -va--l>B>O . a - - or x=e i8 05e5n, produces a wave front is for Qepq =0 in (4.2-2lb), or p=q=O and 1=2,4 from (4.2-20), so that L = =B This is obviously the circular equivoluminal wave front 2 since L + (z: 2 2 [ 0 ) =1 or s + 2m+l+(~l) m=0,1,2, ••.• i s ] .2 =T for 1=2,4 and This wave front is labeled 1 1 ,2' ,3' in Fig. 22 with the singularity on the portion l' ,2~ contained in , -1538 2 the representation (4.2-15) where ! "Va -l>B>O or x=ei a - 0<8<n. The singularity on the remaining portion 2' ,3' is contained in the first term of I in (4.2-14b) for p=q=O ee a-1 and i=2,4 where l>B>!Va 2 -l or =---;;:.<l x < 1. a+ - - -a The preceding is significant because the result indi- cates that only the portion l' ,2' of the circular equivoluminal wave front l' ,2' ,3' in Fig. 22 comes from "the equivoluminal sector" of the w,K-plane as defined by (2.3-11) while all other wave fronts, equivoluminal and dilatational, come from "the dilatational sector 11 of the w,K-plane as defined by (2.3-10) The straight equivoluminal wave front 2 1 ,4' in Fig. 22 is the head wave and it is tangent to the circular equivoluminal wave front l' ,2' ,3'. the same terms, as did l' ,2' ,3'. This wave front results from (4.2-14b) with p=q=O and i=2,4 and (4.2-15), However, it comes about for different reasons than the simultaneous satisfaction of tions (4.2-19). the equa- In this case, the change in the form of 2 the integrand, as the point x=l or B=!..Ja -l is crossed on a the integration path, produces the head wave. This is seen by comparing the first term in (4.2-14b) for p=q=O and i=2,4 with (4.2-15) where the Dirac delta functions o(~ei00-2m) are present. Thus, the head wave is associated with the line w=K which separates "the dilatational sector" -154- from "the equivoluminal sector." The geometry of the head 2 wave can be deduced by setting B=~ Va -l and ~eioo=2m for i=2,4 to obtain ze mo = 1 va 2 (a-L) -1 i from (4.2-11) and (4.2-20) giving 2m+l+(-1) 2~ 1 '1a -l (aT-~) 2 for m=0,1,2, . . . . The remaining equivoluminal wave fronts shown in Fig. 22 are contained in the first term of I described by (4.2-2lb) for the figure. ee in (4.2-14b) as Q~q=f,¢ 1 % 1 % 1 ~0 as labeled in Obviously, from (4.2-20), these values of Qe pq specify p+q and, since only A with p+q=0,1,2,3 are given pq in (4.2-5), only the wave fronts Q~q=O,~'*'* are provided for. The dilatational wave fronts shown in Fig. 22 result from singularities contained in Idd given by (4.2-14c) and they are described by (4.2-2lc) corresponding to Q~q=o,;, !,~ as labeled in the figure. 'T 'T From (4.2-20) and (4.2-2lc), Q~q-O implies that p=q=O and i=6,8 so that L = a 2 V1-B 2 ' -155- L 2 +(z~mo) =a 2 2 which is the circular dilatational wave front Q, 2] or t; 2+[-2m+l- (-1) r; 2 =a 22 T for Q,=6, 8 and m=O ,-1,-2 , • . • . This wave front is 1,2 in Fig . 22 and it, along with 3,4 d 2 d 4 from Q =- and 5 ,6 from Q =-, pq T pq T is perpendicular to one of the boundaries r;=±l so that these are conditions of grazing incidence at these points . The dilatational wave front 7,8 is not at graz i ng incidence at point 7 (r;=-1) since it is a reflection of the portion 2' ,3' of the circular equivolurninal wave front which meets the boundary at an angle of incidence which is less than the critical angle tan I -1 va 1 2 -1 Again , the coefficients A for p +q=0 , 1,2 , 3 are sufficient pq to describe the dilatational wave fronts identified by Qd =O 1 ~ 1 ! 1 §. pq T T T 0 For all of the wave fronts shown in Fig . 22, the points a-1 at t;=O or L=O come from B=l , x=a+l or K=O . The point l' on the circular equivoluminal wave front l' ,2',3' comes from B=O, x=-1 or w_l where both w and K are unbounded on the K equivoluminal branches. 2 The point 2' comes from B=;Va -l, x=l or w=a with w and K bounded on the equi voluminal branches. K The points 5' , 7' ,9' , 11 1 and 13' on the reflected equivoluminal wave fronts each come from a point i n the range l>B> ! Va 2-l , :~~<x<l or from " the dilatational sector'' of the w , K-plane . The points 1 , 3 and 5 on the dilatational wave -156- 2 fronts also come from B=l ..Ja -l, x=l or w_a but with w and a K K unbounded on the dilatational branches. The point 7 on the dilatational wave front 7,8 is not at grazing incidence and it comes from a point in the range :~i>x>l or "the dilatational sector 1 ' l>B>~Va 2 -l, of the w ,K-plane. It is interesting, yet certainly expected, that the number of equivoluminal wave fronts present in the plate is related directly to the number of times that the circular dilatational wave front has crossed the plate at ~=0 and vice versa as seen in the top sketch of Fig. 22. The finite number of wave fronts which are present in the plate at any finite time has the desirable effect of only requiring a finite number of the coefficients A in (4.2-5) to pq account for all of the wave fronts. Then only a finite number of the coefficients n(j) and n(k} for the series e d representations in Section 3.3 are required. is seen that coefficients A pq In (4.2-5) i t with p+q fixed involve n(j) e 1 with j=l,2,3, ... ,p+q, and likewise for A while the series representations pq and na(k). (4.2-14a) , ... , Thus, (4.2-14d) involve infinite sums over p,q=0,1,2, ... f only a finite number of singular terms in (4.2-14b) for I ee and (4.2-14c) for Idd are identified as the representations of the wave fronts. The remaining terms do not vanish, however, they can be shown to be small until the singular wave front -157- with which they are identified is present in the plate. There are wave fronts, other than those shown in Fig. 22, which are predicted by (4.2-2la) , ... , (4.2-2ld), how- ever, they cancel when the superposition of integrals (4.2-14) is taken. These cancellations occur in pairs. (4.2-2la) and (4.2-2lc) predict wave fronts associated with ~e£pq and ~d£pq geometry. respectively, which obviously have the same However, a detailed examination of the integral sum Ide+Idd in (4.2-14) shows that the singularities corresponding to these wave fronts cancel leaving only the dilatational wave fronts shown in Fig. 22. Likewise, (4.2-2lb) and (4.2-2ld) predict wave fronts from ~e£pq and ~d£pq respectively, which have the same geometry. These wave fronts also cancel due to the integral sum I ee +I e d in (4.2-14) leaving only the equivoluminal .. wave fronts shown in Fig. 22. If only the wave fronts are of interest in the response, the only function of the terms Ide and Ied in (4.2-14) is to cancel the extraneous wave fronts. 4.2.3. Discussion of Specific Terms Yielding Wave Fronts. An Example of a Solution Approximated Near a Wave Front. In the representation (4.2-15), the terms containing the Dirac delta functions o(~e~00-2m) are very easy to -158- integrate and they yield an explicit expression, which is singular like an inverse square root in the spatial variables. These terms are singular on the portion l' ,2' of the circular equivoluminal wave fron l' ,2' ,3' in Fig. 22 and they are identically zero on the interior of 1 1 ,2' ,3', being nonzero only on certain regions on the exterior. Hence, this is the part of the representation (4.2-14) which proceeds the circular wave front and, along with the remaining terms in (4.2-15), this makes 1 1 ,2' the only wave front in the plate which is singular as the front is approached from both sides. This is usually called the two-sided equivoluminal (or shear) wave. All of the remaining terms in (4.2-14), other than the delta function terms in (4.2-15), are Cauchy principal value integrals due to the presence of the generalized functions TI cot-~ n , 2 eNpq TI n , etc., and they are not so convenient 2 eNpq csc-~ to evaluate. An example is considered here to illustrate the Cauchy principal value integrals, and also to make a check with the amplitudes of the wave front expansions . given by Rosenfeld [13]. The circular dilatational wave front 1 , 2 in Fig. 22 is considered. The singularity corresponding to this wave front has been found to be present in (4.2-14c) for Idd i with p=q=O and i=6,8. Therefore, with A =1 and the 00 -159- particular values of 8l£ and 8 2£ from (4.2-2a), the terms u~ in the representation (4.2-14) for which contain the circular dilatational wave front are 327fµ~ -P v{ • · I = u~ al 1 (l+x) Ix 2 -2(1-2v)x+l] ~ 3 e -k 2 d600 Lo gRt 'IL. an4'*'d600 2 a-1 a+l x D(x) (4.2-22) The prime on u~ is a reminder that this term is only the part of u~ which contains this particular singularity. 7f 7f The 7f term cot4 ~dSOO results from s 18 csc~dSOO+s 28 cot 2 ~dSOO in (4.2-14c) and s 18 =s 28 =1 from (4.2-2a). term is not singular at ~daoo=4m+2 The fact that the merely shows that no wave front is present as a result of a loading at and l,;=-1. ~=O 7f Likewise, for the term tan4~d 600 . (4.2-22) is only a part of the total response given approximately by (4.2-14), however, no further approximations have been made and this single term is expected to be valid well behind each reflection of the circular dilatational wave front. Of course, convergence of the series -160- representation (4.2-14) is assumed here. Also, other terms in (4.2-14) are identified with singular wave fronts and they must be included to account for the response associated with these wave fronts. From (4.2-11) the phase functions involved in (4.2-22) take the form (4.2-23) <I>aioo = 2 for t=6,8, and it is clear that <I>aioo-+-oo as B-+;Va -1 (x-+l) if ~<aT. Also, from (4.2-18), d<I>dtOO=O at B=_!_ Va 4 T 2 -~ 2 dB 2 a T where <I>aioo=l-(-1) £ 2 ~-Va 2 T 2 -~ 2 from (4.2-23), and this is easily ·verified to be a maximum of the phase functions. The portion of (4.2-22) containing <I>dBOO can be written as 1 P.V.~F(xlcoti•dx . a-1 a+l (4.2-24) -co -161- where x 1 (~) (4.2-23). and x For 2 (~) are the appropriate inverses of 4m+2~1-r;-~a 2 T 2 -E;. 2 ~4m-2 for a particular integer m, the second integral in (4.2-24) must be interpreted as + m '~-mp. V -4m'-2 .J <x (ii))(:: )cot~dil F 2 2 = -4m'-6 '1 2 2 2 2 1-r;-a T(~~ ) P.V. 1 F(x 2 (~)) d~ TI cot ~a~ 4 4m-2 ( 4. 2-25) -oo The last term in (4 . 2-25) is obtained by integrating by parts so that the result is an ordinary integral. The interpretation of the principal value integrals suggested by (4.2-25) must be used on all of the -162- representations (4.2-14a) , ... , (4.2-14d) and (4.2-15) when the phase functions become unbounded since an infinite number of singularities are then present. (4.2-25) is a convenient form since the first term along with the first term in (4.2-24) contain the singularity identified as a portion of the circular dilatational wave front. The part of the integral (4.2-22) containing <l>d 600 contains the remaining portion of this wave front. If only the wave front expansion is required, these terms can be combined and approximated with the nonsingular part i of the integrand being evaluated at <l>=l-(-1)-z'l;_'\/ a 2 -r 2 -E 2 where x1(<l>)=x2(<1>). As a specific example, the unreflected portion of the circular dilatational wave front is considered. This is seen in Fig. 22 continuing from the point 1 through the first thickness -l~'f;~l on 1,2, and it is associated with the phase function <l>=<l>dBOO and the singularity of cot~ at <1>=2m=O. The approximation of (4.2-22) suggested in the preceding is 1+ ) r 2_ (l- ) +lJ -2 ( xx 2 2\)x ! x2D(x) 2 2 2 (Va-r 2 2_E 2 -! (1-'l;-Va -r -E:. )LogR . e I(f:,,-r,'l;) a -E 2 Vc42 fil L . (4.2-26) -163- where (4.2-26a) and x = (4.2-26b) 1 2 2 2]- 2 2 in (4.2-26a) results The singular term [ (¢-1+~) +~ -a T dx from the derivative d¢l in the first term of (4.2-24) and the derivative in the first term of (4.2-25). The integral (4.2-26) is an approximation of the sum of these two terms. It is worth mentioning that, while ~~-0 identifies the 2 wave front, ~O at the point where ddx¢-O or at any other dx2 point except for the wave fronts where cancellation occurs between the wave fronts of ¢etpq and ¢dtpq· The Cauchy principal value integral (4.2-26a) can be evaluated approximately by expanding cot%¢ about ¢=0 and -164- point ~,i; is near the wave front, to give I (~ ,-r , i; ) ~ _ _ _ _-_4_ _ _H (a2T 2_~ 2_ (l-i;) 2) va2"[ 2_~ 2_ <i-i; > 2 ~· -4 H (T-T a f2T Q y'T -T Q ) Q where H(T-T ) is the Heaviside step function. Then 0 (4.2-26), under the same approximation, becomes I (l+x) Ix 2 -2 (l-2v)x+l] 1 2 X D (x) ( 4. 2-27) where (4.2-27a) is the arrival time of the wave front and x is given by (4.2-26b) with T replaced by To· H(T-TO) The coefficient of in (4.2-27) is similar to yT-TQ the quantities that Rosenfeld [13] calculated except that the loading cri;i;=-Io(~)H(T) at i;=l replaces (2.1-7) in that work and the coefficient of the singular term of the dilata tional potential ~ is calculated rather than a -165- displacement term. H ('r--r , With u ~A S 2Av'-r--r 0 tT-T 0 ) representing (4.2-27), us~ 0 H(-r--r ) is the approximate displacement for the 0 loading ass=-Io (s)H(-r) by using the convolution theorem of Laplace transforms. Also, us~~ from (2.2-1) is valid near the wave front since ~ is not singular and this can be integrated to give ¢~- 4 a 2 TQ ~s~A(-r--r 3 3 0) 2 H(-r-T ). 0 This coeffi- 3 2 cient of (-r--r ) H(-r--r ) checks with the data points given 0 0 by Rosenfeld [13] for v=0.3 and a-r =8 within the accuracy 0 indicated in that work. Detailed work on the dilatational wave fronts is actu· · bl e w-=2 aa 1 ~va--1 r-;:--1 a 11y more convenient i· f t h e varia 1-x 1 f rorn 2 x (2.3-5) and (2.3-7) is used rather than (4 . 2-10). x or B defined by B is best suited for study of the equivoluminal wave fronts. The first reflection of the dilatational wave front 1,2 in Fig . 22 results from the part of (4.2-22) which contains ~d 600 . This wave front results from the singularity 1 at ¢d 600 =2m=-2 where e -21Pd600LogR =R. The fact that one reflection has introduced the factor R in the integrand at the wave front is consistent with the interpretation of R -166- as the reflection coefficient as suggested in Appendix A. Subsequent reflections of this circular wave front result from the singularities at ~daoo=-4, . 2 ~d 3 600 =-6, etc., in (4.2-22) and this introduces R ,R , ..• in the integrands at the wave fronts. 1 e 2~ £ e pq LogR The same is true for all of the factors 1 and e -°21'd£ pq LogR in the representations (4.2-14b) and (4.2-14c) since, at the wave fronts, they result in the factors Rm, m=0,1,2, •.. , in the integrands. Thus, in the preceding discussion about the geometry of the wave fronts as shown in Fig. 22, the integers m from ~=2m in (4.2-19) are associated with the reflections of wave fronts. These interpretations point out the importance of the ( 0) term Log: R (;).(} in (3.3-3) for nen and Log:R(2() in (3.3-24) l+x 1-x for n (O) since they are the terms which give an accurate dm representation of the reflection process. In contrast, the first terms in (3.3-3) and (3.3-24) contain the branch numbers n and m and the summation over the branch numbers produce the generalized functions which lead to the wave fronts. An interesting feature of the representation (4.2-14) is that the second term in (4.2-14a) on x=eie, -TI (2p+2q+l±7.;) involves the terms e v 0~8~Tif a -1 2 - 2 --B 2 a which result -167- from e i7T¢ 9, e pq and the definition (4.2-11) for ¢ n e)(,pq • This is obviously an exponential dependence on the thickness variable s and it is related to the usual form of the dila- tational potential ¢ which is associated with harmonic equivolurninal waves reflecting from a plane boundary at an angle of incidence greater than the critical angle 1- tan-l - a 2 -1 V The terms 9-=l,3,5,7 in (4.2-9) have not entered this discussion, however, they cannot be neglected. It is re- called that these terms are identified with singular wave fronts only for s<O. The terms could be estimated on the deformed paths shown in the second and third sketches of Fig. 21, however, it seems more practical to deform the integration paths CE and CD as follows: CE is deformed a-1 ie a-1 , 0~8~7T. onto -l~X~a+l and CD onto -l~X~a+i plus x=e There are no difficulties at the singularities x=±l due to the form of the phase functions for 9-=l,3,5,7 and the fact that s>O has been specified . The representation (4.2-14) is quite complex, however, it only involves explicit functions of x and it offers direct access to the high-frequency response and the accompanying singular wave fronts. This is contrasted with the modal solution (2.3-15) in which the complete details of -168- the high-frequency response is all but invisible. In. addi- tion, the representation (4.2-14) is apparently considerably more accurate than wave front expansions, which are only valid near the fronts. -169- BIBLIOGRAPHY 1. Rosenfeld, R. L. and Miklowitz, J., "Wave Fronts in Elastic Rods and Plates," Proc. Fourth U.S. Nat Congr. Appl. Mech., New York, 1962, pp. 293-303. 2. Mindlin, R. D., "Waves and Vibrations in Isotropic, Elastic Plates," Structural Mechanics, First Symp. on Naval Struct. Mech., Pergamon Press, New York, 1960, pp. 199-232. 3. Miklowitz, J., "Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlying a Rigid Half-Space," Trans. ASME, Vol. 84E (J. Appl. Mech.), 1962, pp. 53-60. 4. Mencher, A.G., "Epicentral Displacement Caused by Elastic Waves in an Infinite Slab," J. Appl. Phys., Vol. 24, 1953, pp ~ 1240-1246, 1529. 5. Broberg, K. B. , "A Problem on Stress Waves in an Infinite Elastic Plate," Trans. Roy. Inst., Stockholm, Rep. No. 139, 1959. 6. Davids, N., "Transient Analysis of Stress-Wave Penetration in Plates," J. Appl. Mech. , Vol. 26, 19 59, pp. 651-660. 7. Miklowitz, J., "Elastic Wave Propagation," Applied Mechanics Surveys, Spartan Books, Wash., D.C., 1966, pp. 809-839. 8. Scott, R. A. and Miklowi tz, J. , "Transient Compressional Waves in an Infinite Elastic Plate with a Circular Cylindrical Cavity," Trans. ASME, Vol. 31E (J. Appl. Mech.), 1964, pp. ~27-634. 9. Lighthill , M. J., Fourier Analysis and Generalized Functions, Cambridge University Press , London, 1964. 10. Holden, A. N., "Longitudinal Modes of Elastic Waves in Isotropic Cylinders and Slabs," Bell System Tech. J 01 Vol. 30, Pt. 1, 1951, pp. 956-969. 11. Alfors, L. V., Complex Analysis, McGraw-Hill Book Co., New York, 1953. -170- 12. Onoe, M., McNiven, H. D. and Mindlin, R. D., "Dispersion of Axially Synunetric Waves in Elastic Rods," Trans. ASME, Vol. 84E (J. Appl. Mech.), 1962, pp. 729734. 13. Rosenfeld, R. L., Analysis of Long Compressional Waves in Rods of Arbitrary Cross Section and Elastic Wave Fronts in Plates and Circular Rods, Ph.D. Thesis, California Institute of Technology, Pasadena, California, 1962. 14. Robbins, c. I. and Smith, R. C. T., "A Table of Roots of sinz=-z," Phil. Mag. (Ser. 7), Vol. 39, 1948, pp. 1004-1005. 15. Hillman, A. P. and Salzer, H. E., "Roots of sinz=z," Phil. Mag. (Ser. 7), Vol. 34, 1943, p. 575. -171- z x FIGURE 1 The Line Load Problem. -172- wn (K) (n=l, 2, 3, ••• ) ~l K K Imx lxl=l 3 1 x=-1 a-1 x=-a+l x=l FIGURE 2 Mapping of the Real K-Axis, 0++ 00 , onto the Integration Path c in the x-Plane. Rex -173- Symmetrical Rayleigh Branch Asymmetrical Rayleigh Branch I K rmx 2 x=-1 x=O x=l Rex FIGURE 3 Mapping of the Real K-Axis, 0++ 00 , onto the Integration Path c 0 s for the Symmetrical Rayleigh Branch, and the Integration Path c A for the Asymmetrical Rayleigh Branch in the 0 x-Plane. -174- rmx of Locus Rex \) -0.2 -0.3 -0.4 -0.5 FIGURE 4 The Loci of x01 and x02 , where R(x)=O, and of xR ' where R(x)=oo, in the x-Plane as Poissonws Ratio v Varies. The Variation of xR with v. -175- II I• 1.' R (X) 1 .I . , ~4--....----="'-~ I ,___ _ ______._ _ _ 0 • 5 .I . I I -1 0.5 -0.5 1 x 1 x . I -0.5 • \ . \ v=O \ -1 1 ITTXT ......... -- v=O I ,.,,. 0.5 FIGURE 5 R(x) and (R(x)l -1 i versus Xv -l<x<lv for v=0,0.3,2. 2 7T 7T Y ( e) y(e), the Argument of R(e i8 l ) , versus e for v=0,0.3, 2 . FIGURE 6 2 7T n V=O V=0.3 e I O"\ .._,] I I-' -177- n ;'IT 1 . -1(1 ""----sin -r) X X R l!+lsin-l(lr) X X R FIGURE 7 a-1 Branch Points of the Branches n on O~X~a+l (v=0.3). The Triplets (±,j,k) Identify the Branch Points as Zeros of the + Functions ~;k<x>. . ---- Symmetrical Branches - - .... -1 -178- Asymmetrical Branches 0 a-1 a+l 1 FIGURE 8 The Real Branches on the Real x-Axis for -l~X~l (v=0.3). x -179- I .1' Symmetrical Branches I Asymmetrical Branches I I STI 4TI 3TI 2TI x 7T 8=0 x=l 2 8=7T x=-1 e FIGURE 9 The Equivoluminal Branches on the Unit Circle x=e for 0<8<TI (v=O. 3) • ie -180- \Symmetrical Branches. \ \ ' Asymmetrical . Branches I I 57T 47T 37T 2TI 7T e 8=0 7T 8=TT x=1 2 x=-1 x FIGURE 10 The Dilatational Branches on the Unit Circle for 0<8<'1T (v=0.3). x=e ie -181- Imn Upper Bound cosh(Imn>=IRicosh(xrmn) Lower Bound sinh(Imn)= IR I sinh (Ix I Imn) -1 XR 0 a-1 a+l x FIGURE 11 The Bounds on Imn (v=0.3). The Lower Bound is a Portion of the Symmetrical Rayleigh Branch on -l~x<xR and it is the Asymmetrical Rayleigh Branch on XR<x~O. -182- ' '' \ I Srr \ \ l l - \ 3rr -27T 7T Rex . Synunetrical ~ Branches -- .,_. Asynunetrical Branches FIGURE 12 The Complex Branches on the Real x-Axis for a-1 -l~X~a+l (\i=0.3) • -183- \ Synunetrical Branches \ \ Asynunetrical \ Branches 1!\ ~ '' Synunetrical Branch Points Asynunetrical Branch Points n n v=l --- , ... (+00) 0 0.6 1 0 FIGURE 13 1 The Real Branches on O~x~l for the Special Cases v=2 and v=l. x -184- rmx •Branch Points lxl=l 5 x=-1 6 1 x=O a-1 a+l FIGURE 14 A Path in the x-Plane for a Continuation Over Symmetrical Branches. x=l 8=0 3 ln<ei8) I ln<ei8>1 FIGURE 15 a-1 a+l A 8=0 x=l ----r--- • ~ in Fig. 14. A Continuation Over Symmetrical Branches Corresponding to the Path Ren 31 x co I U1 I I-' -186- rmx • Branch Points 3 x=-1 4 1 •i'-~~~--r~"'-i--~~-+-~--' x=O a-1 a+l FIGURE 16 A Path in the x-Plane for a Continuation Over Asymmetrical Branches. x=l 8=0 2 In (eie) I e_1 / e 3 I / 4 FIGURE 17 a-1 a+l 8=0 x=l x A Dilatational Continuation Over Asymmetrical Branches Corresponding to the Path in Fig. 16. Ren 2 I 00 -.J I I ..... -188- • Branch Points rmx x=-1 a-1 a+l x=l Rex FIGURE 18 A Path in the x-Plane for a Continuation Over Symmetrical Branches. 4 8=7T x=-1 -Iron XR Ren FIGURE 19 8=0 x=l x 3 e_l x=-1 8=TI An Equivoluminal Continuation Over Symmetrical Branches Corresponding to the Path in Fig. 18. a-1 a+l y ln<eie) I e I l.O 00 I ...... -190- n n Asynunetrical Branches Synunetrical Branches I ns1-----~'---'~---4~L--~--~~1 - ( -J;l?) • (-J;3,2) 27T+----+--l~--~~~~~"""1° n21----~-'--~._---k--------L ( -,1,1) TI i----~-=:::::-::==~=~~-../' n11-------1---1---------- 0 a-1 a+l 1 0 a-1 a+l l FIGURE 20 A Schematic Representation of the Branch Points Conunon to a-1 Pairs of Real Branches on a+l~x~l (v=0.3}. -191- rmx 0 x=-1 a-1 a+l x=l Rex x=l Rex x=l Rex rmx Xo1 0 , I x=-1 a-1 a+l 0 x=-1 rmx • a-1 a+l 0 FIGURE 21 The Integration Paths CE and CD. A Deformation of CE. A Deformation of c0 • -192T a l;;=l s=aT s=T s=- l' I e VQ pq =O 3 / / I / / / / 6 / / 6 8I 6 x;;;...-..- 10' 4'-- --- 8 =T 12' 4 ~-----====-~~~~--:~--=.;~~14 I._ --- 10 "(. ~ilatational 2 / Wave Fronts / . 1 umina . 1 / / Equivo Wave Fronts FIGURE 22 The Wave Fronts in the Plate at T=6.3 (v=0.3). -193- APPENDIX A THE FUNCTION R(X) The response of a plate to the loading (2.1-7) or to other loadings is completely characterized by the branches n=n n (x) through the modal solution (2.3-15). In turn, the . branches n are completely characterized by the function R(x) in (2.3-12a) through the frequency equations (2.3-13) and (2.3-14). Hence , i t is necessary to know the behavior of R at all points where the behavior of n is required. R is the same function which occurred in the wave front expansions given by Rosenfeld and Miklowitz [l] . It is interesting, in the context of that work and of the present work, that R is identical to the . reflection coefficient giving the ratio of reflected to incident amplitudes of harmonic waves reflecting from a free boundary. R is a function of the angle of incidence by writing (2.3-12a) as a function of the phase velocity (2.3-1), which is given by c - c P sine. - where c is the velocity of the body wave in i question and e. is the angle of incidence measured from l the . normal of the boundary. R is a rational function of both x and v and, fore , it is analytic with respect to there~ x and v except at the zeros of [R]-l (inspection shows that R is bounded and, -194- hence, analytic as x+oo or as v+ 00 ) . These zeros of [R] -1 and the points where R=0,1,-1, etc., are given in this appendix along with the behavior of R on the real x-axis and on the unit circle lxl=l. R(x) can be written in several different forms each of which exhibits points in the x-plane where it takes on special values. The notation which will be used for these points requires some explanation. Two points, x 0 in (2 . 3-8) and XR in (2 . 3-9), have already been defined and the zero subscript means that this is the n=O point of the synunetrical Rayleigh branch (the complex conjugate of x is also such a point) . 0 The R subscript refers to the fact that xR is the singular point of both Rayleigh branches where both w and K are unbounded (also, n is unbounded here) . With respect to the function R (X) , it will be shown that xR (x) =-1 at x=xo and that [R(x}J -1 =0 at x=xR. Other than for these two points , when a notation is required for a point, subscripts will refer to values taken by R(x). For example, it will be seen that R(x)=O at x=x 01 ,x 02 and that R(x)=-1 at x=x_ 1 ,x_* 1 is used for the complex conjugate of x). The forms taken by R(x) are (the notation x* -195- R (X) = xo ~x) <x-xo1) <x-xo2) ( x-x~) 1 · 2 ( a-l)(a+l ) R-1 = xD(x)(X -l) x-a+l a-1)( R+l = xD 1<x) <x-x_ ) 2 <x-x_* ) 2 1 1 (A-1) 2 ) (1 ) xR-1 = - 8D( 1-v (x) -x X 2 xR+l = DTXT"c1+x) <x-x 0 ) <x-x 0* ) where X_l = e ie -1 = 1 - 2v + 2ilv(l-v) (A-la) TI f l ( x=x-1 maps into . . so th at 0 e -1S2 or 0svs2 t h e Lame~ point s where w=l2K), a is given by (2.1-2) with 3-212s:~is1 for O~vs~ and D(X) is given by (2.3-12b). Comparing the form of R(x) in (A-1) with that in (2.3-12a), the points x=x ,x , _!_ are obviously the zeros of (l+x) 3-8(1-v)(l+vx)x 01 02 XR and they will be discussed in more detail presently. A useful property of R, which can be used in conjunc- tion with the forms (A-1)~ is given by ·(1) = [R(X)]- 1 RX (A-2) -196- and this results directly from (2.3-12a). From (A-1) and (A-2), points in the x-plane where R takes on special values are listed as follows: R = 1 at R = -1 at R = 0 at a+l a-1 X = a-1' l, a+l' -l 1 X-' oo R x = Xo1' Xo2' IRJ-l = O at 1 1 (A-3) x = XR' Xo1' Xo2' XR = l at X = 0, +l XR = -1 at X = -1, The three points x=xR' Xo 1 1 Xo1 Xo2 ~-, xR defined by (2.3-9). 0 are roots of D(x)=O with 01 and x 02 in the x-plane and the variation of xR with v for O~v~~ are shown in Fig . 4. ing . that The points x l~Rex The loci of xR' x 01 and x 02 are identified by stat- 01 ~Rex 02 >o and that l>Imx 01 =rmx 02 ~o with the 1 points being complex conjugates if v <v< , real and equal c -2 if v=vc' and real and unequal if O<v<v where v ~0.2631. c c It is also true that l~lx 01 1~1x 02 j>O, the modulus being -197- equal to one only for x 01 with v=O. One exception to (A-3) occurs for v=O in which case x 01 =x_ 1 =+1 and R(l)=-1. Derivatives of R with respect to x at certain points are useful and they result directly from (A-1). RI ( 1) l-2v = --2- for Letting 1 0 < \) < 2 \) R' ( 1) = O for v = 0 R'(:~i) = a8 2(a+l) (a-1) RI (-1) = 2 1-v (A-4) , R' (x_l) = 0 ~x (i) x=o = 1 L(!) dx2 R 2 = -8(1-v) 2 x=O d ax <xR> = 16(1-v) x=O -198- The forms in (A-1) and knowledge of the zeros of D(x) determine the inequalities -1 < R < 1 if -1 < [R] - l < 1 a-1 a+l < X < 1 a-1 if -1 < X < a+l (A-5) -1 < xR < 1 if 0 < X < 1 -1 < [XR)-l < 1 if -1 < X < 0 The inequalities are evident in Fig. 5 where R(x) and [R(x)J -1 are sketched on the real x-axis, values v=0,0.3, -l~x~l, for the 1 2 of Poisson's ratio. · circ · i e x=e ie , R (x > ta k es t h e simp · 1 e f orm 0 n th e unit (A-6) where y(8) = 2[8-ArgD(e i8 )] (A-6a) with D(x) given by (2.3-12b). satisfies y(-8)=-y(8) and O~y(8)~'IT with the equalities being taken only at 8=0,8_ ,TI where 1 y(O) = y(TI) = 0, y(8_ ) ='IT 1 (A-7) -199- For the special case v=O and the same range Os_8s_TI, y(6) satisfies y(-8)=2TI-y(8) and 0S,y(8)S,7T with the equalities being taken only at 8=0,7T where y(O)=TI, y(TI)=O. The func- tion y(e) is sketched in Fig. 6 for v=0,0.3, 21 and the angle e_ 1 =cos -1 (l-2v) is identified as the point where y(e_ 1 )=7T in each case . There is an additional path, other than the unit circle x=e ie , on which IRl=l. A portion of this path will a-1 be called c 1 and it connects the two points x=a+l and ie_ 1 x=x_ 1 =e . given by (A-la). This can be shown by making a-1 local expansions of R(x) about x=a+l and x=x_ 1 using R(:~i)=l and R(x_ 1 )=-l from (A-3) and R' (x_ 1 )=0 from (A-4). The path Cl lies entirely interior to the quarter disk Rex> 0 , Imx > 0 , I x I < 1, with the exc.e ption of its endpoints , for Poisson's ratio in Os.vs.;. R (X) Thus, since jR(x) l=l, = will be taken for the form of R(x) when xCc 1 • (A-8) The exact forms of c 1 and of the real function y 1 (x) are very difficult to calculate, but, they are not important for this work. It can be stated with certainty that c 1 is a smooth path and that -200- (A-9) Also, y 1 (x) is a monatonic increasing function of x for dR (x_ 1 ) dy (x_ 1 ) 1 x E: c 1 (except at x_ 1 where dx =dx =O from (A-4)) a-1 as x moves from x_ 1 to a+l on c 1 . y 1 (x) is monatonic on c 1 since, if it were not, there would be some point x E.::: c 1 dR ( ) dy 1 (X) ) where~ X =~=0. However, from the form -1 dx ax (2.3-12a) of R(x), there can be no more than six points (x1x where dR(x)=O dx , and it is obvious from Figs. 5 and 6 and from (A-2) and (A-6) that four such points are on the real x-axis and that two more, x_ circle lxl=l. ~~(x)=O. explain c 1 and x_* , are on the unit 1 Hence, there are no other points where One other aspect which clarifies (A-9) andhelps to 1 and y (x) is that, if x traverses the closed 1 a-1 path a+l+l on the real x-axis, l+x_ 1 on the unit circle a-1 Ix I =l and x_ 1+a+l on c 1 , then the argument of R(x) increases by an amount 2TI due to the presence of one simple zero of R(x) at x closed path. Imx 01 01 , which then must be interior to the 1 This, of course, is with v <v< so that c -2 >0 as shown in Fig. 4. It is evident from the form (2.3-12a), showing that * and from - (A-2) that if xis any point on c 1 R(x)=R * (x), then also jRj=l at the points x * c 1 into a closed curve on which 1 x 1 and __... x jR(x) l=l. This completes -201- APPENDIX B ADDITIONAL INFORMATION ABOUT THE BRANCHES n=n B.l. n {x) THE BRANCHES AT SPECIAL POINTS It is possible to find exact solution pairs x,n of the frequency equations at some points in the x-plane or for certain values of n. Such solution pairs will be helpful in understanding the branches n as continuous analytic functions of X· The results for several points are listed here without indicating the simple calculations. Use has been made of the properties of the function R(X) given in Appendix A and the property of analyticity of the branches as shown in Section 3.1. (1) The case n=O. The trivial solution n=O for all x is not inter- esting since it implies that w=K=O from (2.3-7). However, expansion of the symmetrical frequency equation shows that lnl<<l is possible only near the points x=x ,x * where 0 XR(x)=-1 from (A-3). The expansion about x=xo is ~2i80 n2 = - 3ie v(l-v) -ie 0 _[f+V lr::v<x-xo> + 0 0 3 = v(l-v) e lI=V (1 x-xo 12) _fI+Vce-e O> + o(le-e 0 12 ) (B.1-1) -202- the second expression being specialized to x=ei For 8>8 0 8 with 8~8 0• this is obviously an expansion of the symmetrical Rayleigh branch n 0 (x) about the n=O,x=xo (or w=K=O) point and, indeed, the branch has a square root branch point at x=x . 0 For 8<8 0 , (B.1-1) gives a continuation of the sym- metrical Rayleigh branch about the branch point. Likewise, the asymmetrical frequency equation only admits branches with In I<< 1 near the point x=O. The expan- sion about x=O takes the form n 2 (B.1-2) For x<O this is an expansion of the asymmetrical Rayleigh branch n (x) about the n=x=O (or w=K=O) point and this 0 branch has a square root branch point · at x=O. (2) The point x=l. Directly from the symmetrical frequency equation (2.3-13), the branches take the values nn (1) = mr n=l, 2 t 3 t • • • I (B.1-3) and the branches are analytic at x=l since the derivative dnn(x) (3.1-3) exists with dx The asymmetrical frequency equation (2.3-14) can be rearranged and written as -203- tann = f (x , n ) where f (x , n > = Rsin(x-l)n 1-Rcos (x-1) n Assuming that bounded solutions n exist in a neighborhood of x=l and using (A-4), results in f(l,n) = I R v2 n ( 1) = -l-2\!T) Hence, asymmetrical solutions nn(l), n=lv2,3, ••• , exist and they satisfy 2 tann ( 1) = '--v-n ( 1) l-2v from which n n (1) can be shown to be real. (B.1-4) Only the posi- tive set of solutions nn(l) satisfying ( 2 n;l)rr~nn(l)<nrr for 0<v~ 1 2 are takenv and ( ) = (2n-l)rr + 2(1-2v) + O(n-3) nn 1 2 (2n-l)rrv2 1 for n>>l and O<v<2· dnn(X) The expression (3.1-4) for the derivative dx x=l -204- of the asymmetrical branches takes the ~ indeterminant form due to (B.1-4) and (A-4) and the derivative cannot be shown to exist from this form. However, the form tann=f(x,n) of the asymmetrical frequency equation gives and this expression for the derivative does exist at x=l with the assumption that bounded solutions n (x) exist in n a neighborhood of x=l with n (1) satisfying (B.1-4). n dnn (x) Carrying out this calculation gives dx hence, the branches are analytic at x=l and they are tangent to the hyperbolas n nn ( 1) l+x at this point as were the nTI symmetrical branches tangent to the hyperbolas n=l+x at x=L These preceding values of the symmetrical and asymmet- w rical branches at x=l map into the -=a points on the K branches in the w,K-plane. (3 ) a-1 . . h point x=a+l" Te The symmetrical solutions nn(:~i)• n=l,2,3, •.• , with O<n <n < ••• , comprise the two sets 1 2 -205- ~ i+x q=l,2,3, .•• , a-1 x=a+l (B.1-5) n(:~i) = (2p-1)7T 1-x p=l, 2, 3, . • • • a-1 x=a+l The asymmetrical solutions nn(:~i), n=l,2,3, • • . , with O<n 1 <n 2 < ••• , comprise the two sets 7T n(:~i) = ( 2q-l) i+x q=l, 2 f 3 f • • • I a-1 x=a+l (B.1-6) n(:~i) = ~ i-x p=l,2,3, ••.• a-1 x=a+l a-1 The point x=a+l corresponds to cut-off K=O and, just as in the w,K variables, the branches have separated into two sets at this point; those in .(B.1-5) and (B.l-6) with 1-x in the denominator correspond to motions of the plate which were called the simple thickness-stretch modes and those with i+x in the denominator correspond to motions which were called the simple thickness-shear modes by Mindlin [ 2] • ( 4) The point x=O . The symmetrical frequency equation admits -206- solutions n (O), m=l,2,3, ..• , satisfying m sinn(O) = -n(O) (B.1-7) n (O) can be taken with Ren (O)>O, Imn (O)>O, and m m m O<ln 0 1<1n 1 1< . .. , however, the complex conjugates are equally good solutions. The first ten roots of (B.1-7) are tabulated by Robbins and Smith [14), and for m>>l 4 nm(O) = ( m;l)TI + iLog(4m-l)TI + o(~Logm) The asymmetrical frequency equation admits solutions nm(O), m=0,1,2, . •• , satisfying sinn (O) = n (O) which can also be taken with Ren m (B . l-8) (O)>O, Iron m=l,2,3, ... , and O=ln 1<1n 1<1n 2 1< . . . 0 1 . m (0)>0 for n (0)=0 is the 0 x=O point of the asymmetrical Rayleigh branch which was just discussed. The first ten roots of (B.1-8) are tabu- lated by Hillman and Salzer £15], and for m>>l 4 1 nm ( 0) = ( m~ ) TI + iLog ( 4m+ 1) TI + o ( ~Logm) These solutions at x=O map into w=O and K complex by (2.3-7) and they are related to the degenerate case of the static solution in the plate . -207- (5) The point x=XR· Both frequency equations admit solutions m=l, 2, 3, • . . • (6) (B.1-9) The point x=-1. As in the case of the asymmetrical branches at x=l, the symmetrical frequency equation (2.3-13) can be written as tann = _ Rsin(l+x>n 1-Rcos(l+x>n which allows solution values at x=-1 for the symmetrical branches nm(-1), m=0,1,2, .•. , satisfying tann (-1) = 1-vn (-1) 2 {B.1-10) The root n (-l)=-iJn (-l) J is pure imaginary and it is 0 0 identified as the point on the symmetrical Rayleigh branch where it crosses the equivoluminal tangent (x=-1 or w=K). Other than for the conjugate of n (-l), all other roots of 0 (B.1-10) are real, and the positive set which satisfies 2 , h m'""<n m (-l)<( m+l)'IT f or m= 1 , 2 , 3 , ..• , i·s taKen were 2 = (2m+l) 'IT 2 4 + O(m-3) (2m+l) 'IT (1-v) -208- for rn>> 1. Analogous to the point x=l on the asynunetrical 0 branches, the derivative (3.1-3) takes the 0 form at x=-1 for the synunetrical branches and cannot be shown to exist. However, the form given above for the synunetrical frequency equation allows calculation of the derivative giving x=-1 =~ 2 m (-1) where nm (-1) satisfies (B.1-10). Hence, these symmetrical brances are analytic at x=-1. The asymmetrical branches ·take the values = m'IT nm (-1) ffi=l, 2 I 3 f • • • t (B.l-11) and they are analytic at x=-1 with the derivative (3.1-4) dnm 1 taking the value dx -"T117T· (7) The point x=x_ • 1 At this point, the synunetrical frequency equation is satisfied by each of the sets (2n+l)7T = l+x_1 = (2n+l) 7T n=O, 1, 2, •.. , (B.1-12) 2 v'l-v . e_1 ffi'JT. 2m7T = -ie 1-x -1 IV -1-- 2 m=l, 2, 3, • • • • (B.1-13) -209- The asynunetrical frequency equation is satisfied.by each of the sets . e _1 2mr l+x_1 mr = --e -i-2- n=l, 2, 3 v • •• , (B.1-14) v'l-v . e_l = ( 2m-l) TI = l-x-1 ( 2m-l) TI. -J.-2ie 1 m= 1 , 2 , 3 , . • . • ( B . 1-15 ) 2lv This separation of the branches into two sets at x=x-1 is closely related to the separation which occurs at a-1 x= a+l. (8) The points x=x 01 ,x 02 . These points are not on the real x-axis unless O<v<v - - c (v c is described in Appendix A), however, they are interesting points since both frequency equations admit solutions n=l,2,3, • • . . (B. l-16) These points can be made to move off the real axis, as in Fig. 4, by increasing v, and the same values are retained by n(x 01 > and n(x 02 >. (9) Points x which are real and rational. Exact solution pairs, x,n, satisfying the fre- quency equations can be found at these particular points -210- by finding the zeros of polynomials. Such results are im- portant since they give exact solution pairs at points o t her than those already considered. m This case is illustrated by taking x=- where n=2,3, n 4I • • • f m . m=±l,±2,±3, . . . , and - 1 <~1 n The symmetrical fre- quency equation (or, likewise, the asymmetrical equation} reduces to m at x=n where P n- 1 Cz) is a polynomial of degree n-1 in z. Further, if both m and n are odd, the frequency equation takes the simpler form n-1 where Qn-l is a polynomial of degree ~ 2 -. An obvious solu- -2- tion of these equations , which is of considerable importance for sketching the branches, results from sin(~~=O . Therefore 1 the pair m x =n n = nTI (B . 1-17) for n=2,3,4, • .. and m=±l,±2,±3, .•• satisfies both the symmetrical and asymmetrical frequency equations. The points -211- (B . 1-17) can also be recognized as the common points of the hyperbolas n=~l TI and n=~ for p=n-m and q=m+n. These are -x l+x the hyperbolas which form the grid as discussed in Subsection 3 . 2 . 1 . The remaining solution points, for x real and rationaL are given by the roots of the polynomials 2 Qn _ 1 (cos ( Pn-l(cos(~)) or ~n)) . Thus , by solving nothing more difficult 2 than for the roots of quadratic equations , the frequency 1 1 1 equations can be so l ved exactly at the points x=±5,±3,±2, ±~,±~,and, by solving for the roots of cubic equations, at 1 +1 +3 +-5 +-3 etc the Pol. nts x=+-71-41-71-7 1-41 . 3 For example, at the point x=- , the symmetrical fre5 quency equation reduces to sinn + Rsin(-;n) = where R=R(-~)>1 for O~v~~- n (-~) = SmTI , This equation has the solutions m=l, 2, 3 , ... n (-~) = SmTI ± 5cos -1{ 1 [ 2~]!} , 3+R-(5+2R+R) 212 -212- m=0,1,2, ... , (only the plus sign for m=O), m=O , 1, 2, ... , where all of the radicals are positive and the inverse functions take their smallest real positive values. These solutions indicate a great deal about the branches n in the interval -l<x<XR· For instance, in the example chosen exact solution points of the synunetrical frequency equation are found both for w and K pure imaginary and for w and K complex through (2.3-7). B.2. THE BEHAVIOR OF THE BRANCHES UNDER THE MAPPING x+! x The properties of the equivoluminal and dilatational branches, which are derived in Subsection 3.2.2, make possible some general statements about the branches n in the complex x-plane. Use can also be made of the fact that in B.l the branches are shown to be analytic functions of x at x=l and x=-1 for the cases where the branches are bounded at these points. From (3.2-11) and (3.2-17) and from tany(-e) 2 -tan~ 2 -213- (y(-8)=-y(8) from Appendix A), the moduli of the equivoluminal branches are even functions of e. That is, if In n I= fn(8) satisfies either (3.2-11) or (3.2-17) for n=l,2,3, ..• , then fn(-8)=fn(e) . the unit circle x=e (3.2-8). Thus, the form of the branches nn(x) on ie is nn(e ie .e )=fn(8)e Using the fact that fn(8) -12 from (1) of is an even function of e results in the interesting property .e 1- = fn(8)e 2 or e for 0~8<TI -ie nn (e -ie ) = f .e -1- n (8)e 2 = n n (eie) (B.2-1) (fn(8) is singular as 8+n) . The analytic continuations of the equivoluminal branches onto the lower half unit circle are given by (B.2-1). More significantly, the property (B.2-1) also must be valid for continuations off the unit ci~cle x=eie since the equality of these analytic functions on the unit circle implies equality everywhere in their domain of analyticity. This being the case , the equivoluminal branches have the property (B . 2-2) -214- which just defines the behavior of the branches nn(x) . under the mapping x~!. . x The property (B.2-2) makes it possible to show that w and K defined by (2.3-7) are invariant under the mapping 1 x~X. That is, if n in (2.3-7) is an equivoluminal branch then w(i)=w(x) and K(i)=K(X). This depends on the defini- tion of the radicals in (2.3-7). A suitable definition which leaves the point x=l and the x-plane near x=l free l 2 of branch cuts is to take x as real and positive on x>O with the branch cut on the negative real x-axis. Simi- larly, the radical in K is taken to be real and positive · 1 a-l a+l wit · h t h e b ranc h cut on th ereon th e interva a+l<x<a-l mainder of the real x-axis. These also are definitions which make w and K real and positive on the integration path c shown in Fig. 2 for n=n n (x), n=l,2,3, . . . . 1 This invariance of w and K under the mapping x+-- lends x support to the statement in Section 2.3 to the effect that if w and K are completely known as a function of x on the unit disk lxl~l then they are known everywhere. Likewise, the moduli of the dilatational branches given by (3.2-12) and (3.2-18) are even functions of 8 with respect to the point 8=n. Hence, from (2) of (3.2-8), the dilatational branches have the property -215- -e -ien (e -ie) = n (eie) for 0<8~rr (B. 2-3) which gives the analytic continuations onto the lower half unit circle . (B . 2-3) becomes (B.2-4) when the dilatational branches are continued off the unit circle. w and K in (2.3-7) are again invariant under the 1 i'f the positive real x-axis is taken for the mapping x-+x 1 a-1 a+l 2 branch cut of x and the segment ~x<-- is taken for the a+l- -a-1 branch cut of the radical in K. The property (B.2-2) can be used to verify the value of the derivative of the equivoluminal branches at x=l as found in B .1. x=l =-ln (-1) results directly from 2 m (B.2-2) and the fact that both the symmetrical and asymmetrical equivoluminal branches are analytic at x=l. dnm(x) wise, from (B.2-4), dx Like- =~nm(-1) for the symmetrical x=-1 and asymmetrical dilatational branches. B.3. THE COMPLEX BRANCHES ON THE REAL x-AXIS For the purpose of examining the complex branches on the real x-axis, n is written as -216- (B.3-1) where nR=Ren and n 1 =Imn. The symmetrical frequency equa- tion (2.3-13) becomes ~ (B. 3-2) where the real parts have been equated using the fact that x and R(x) are real. As usual, the asymmetrical frequency equation is obtained from (B.3-2) by replacing -R(X) with R (X) • A simple case results from setting nR=O in (B.3-2) and in the corresponding asymmetrical frequency equation. These solutions, n=in , are identified as the Rayleigh 1 branches and they will be discussed in the context of the entire set of complex branches. By using the fact that :~i~x<l -l<R(x)~l on the interval (from (A-5) in Appendix A), it is possible to show that no complex solution of the form (B.3-1) exists with n 1 IO which satisfies (B . 3-2) on this interval. This is done by assuming that a solution with n IO exists and then 1 using the inequalities 1Rsinhxn 1 l<lsinhn 1 and jRcoshxn 1 1< 1 coshn 1 to obtain a contradiction . The same proof is also valid for the asymmetrical branches. -217- The existence and form of the complex branches are a-l more apparent on - 1 ~X<a+l 1' f · ( B. 3 - 2) t h e equations are rearranged so that nR and n 1 can be discussed as separate functions. The first equation of (B.3-2) is multiplied by sinhxn 1 and squared, the second equation is multiplied by coshxn 1 and squared, and the results are added to give (B.3-3) This equation is a result of forcing (B.3-2) into a form which is very similar to the original frequency equations (2.3-13) and (2 . 3-14) with n =Imn replacing n and the radi1 cal replacing ±R(x). Actually, the sign of R(x) has been lost in the squaring operations so that (B . 3-3) is valid for both the symmetrical and asymmetrical complex branches, and (B . 3-2) or the analogous asymmetrical frequency equations must be used to identify the particular branches . Similarly , the pai r (B.3-2) can be manipulated to give 2 which can be combined with (B.3-3) to eliminate sin 2nR= 2 2 4sin nR(l-sin nR) 1 resulting in -218- sin 2 (l+X)nR = (B.3-4) The pair (B.3-3) and (B.3-4) are especially convenient forms from which to determine both the symmetrical and asymmetrical sets of complex branches. The following approach can be used to find exact solution points on the complex branches or to verify the existence and form of the branches: (O~sin 2 nR~l) in (B.3-3), which can a constant parameter then be solved for n 2 (1) sin nR is considered as 1 as a function of x and the parameter; (2) this result is substituted into (B . 3-4) to give nR as a function of x and the parameter; (3) the common points of 2 this function nR and the constant values of nR from sin nR= constant give exact solution points x,nR' and n . given by evaluating the function n 1 is then 1 i n (1) at this point X· The preceding procedure depends on the fact that there exists a solution n 1 2 of (B.3-3) for a fixed value of sin nR and such a solution may not exist. as If (B.3-3) is written -219- the left side of this equation is a strictly increasing function of n 1 and the right side is a nonincreasing func- a-1 tion of n 1 for -l<x<a+l· x=O is an exceptional point with complex solutions already known to exist satisfying (B.1-7) and (B.1-8) in Appendix B. Also, the point x=-1 gives a limiting case which cannot be included here. Thus, a single solution n >0 exists if, and only if, 1 1 fil < nI =O which is equivalent to requiring that 2 2 . 2 l-x R sin nR > 2 1 -x (B. 3-5) From the first of (B.3-2) r this also gives a bound on . 2 XnR Wh l.CH ' ......lS Sl.n -220- for n >0, and this can be combined with (B.3-5) to give 1 2 2 . 2 1 1-x R sin xnR > ~ 2 2 R 1-x (B.3-6) On -l<x<O, both (B.3-5) and (B.3-6) are always satis2 fied since x 2 R >1 by (A-5); hence, complex branches exist on this interval by this criterion. a-1 However, on O<x<a+l the inequalities (B.3-5) and (B.3-6) are not satisfied for all x and nR~O. The limiting case n +0 on the complex 1 branches can only occur at points x,nR determined by (B . 3-5) and (B.3-6) with equalities replacing the inequalities. This is not meant to include all of the points on the real branches considered in Subsection 3.2.1 where n 1 =o, but only the limiting points of the complex branches . But, (B.3-5) and (B . 3-6) with equalities are identical to the two equations (3.1-7) with no distinction between the symmetrical and asymmetrical cases. Thus, just as is ex- pected from a local approximation like (3.1-13), the branch points on a-1 O<x~a+l are common points of the real and complex branches. Bounds on the functions n 1 result directly from (B . 3-3) by observing that the radical on the right side of 2 that equation is a nondecreasing function of sin nR for O_<sin 2 nR_<l if R2 _>1 , which is the case on -l<x<a-l. - -a+l Hence , -221- evaluating the right side of (B.3-3) for sin 2nR=O and 2 sin nR=l results in the lower and upper bounds respectively for sinhn 1 , and also for n 1 since sinhn is an increasing 1 function of n1 • The bounds are then defined by the in- equalities which is equivalent to the pair of inequalities sinhn 1 > jRjsinh!xln 1 with n ~ , (B.3-7) 1 ~o throughout. The bounds (B.3-7) are taken (the equalities hold) by the imaginary part of the complex branches at various points in the Rex,Ren-plane; hence, the bounds are also envelopes with n 1 =Imn being tangent to the curves at these points. . ta k en b y n Th e 1 ower b oun d is 1 1' f . 2nR= 0 an d sin 2 sin (l+x)nR=O, which results from (B.3-4) under the condition that the first of (B.3-7) with the equality, or sinhn =jRjsinhlxln , is satisfied. Actually, a solution 1 1 n 1 >0 of the first of (B.3-7) with the equality exists only on -l~x<O where lxRl~l by (A-5) in Appendix A. Hence, the -222- · t e 1 ower b oun d f or n on O'.S_X'.S.a+l a-l is · appropria 1 The upper bound is taken by n 1 · t JUS -o • n 1= 2 if sin nR=l and 2 sin (l+x)nR=O, which results from (B.3-4) and the second of (B.3-7) with the equality, or coshn 1 =1Rlcoshxn . 1 This equation has a solution n >0 on all of the interval 1 a-1 a+l where -l~x< I RI :::_l. The envelopes or bounds defined by (B.3-7) are shown in Fig. 11. Both the upper and lower bounds are obviously unbounded as x+xR where R(x) is also unbounded by (A-3). The asymptotic approximations of the bounds n 1 are easily derived and they both take the form (B.3-8) as x+xR on the real x-axis. Hence, the imaginary part of every branch must have this approximation. The equations (B.3-7) for the bounds can be expanded near x=-1 by using R(-1)=1 from (A-3) and R' (-1)= 1 ~~ from (A-4) in Appendix A to show that n for the lower bound satisfies 1 at x=-1, and that n 1 for the upper bound satisfies -223- 1-v ~I at x=-1. = cothn 1 The lower bound equation at x=-1 is identical to (B.1-10) with n=in . 1 The solutions, n (-l), of these equa1 tions can easily be shown to fall above and below the value l~v with a finite separation that depends on Poisson's ratio v. 2 The Rayleigh branches are a special case where sin nR= 2 sin (l+x)nR=O due to nR=o and the lower bound is an exact solution of the frequency equations. This can also be seen by setting n=in 1 in the frequency equations (2.3-13) and (2.3-14). To be precise, the symmetrical Rayleigh branch n=in 1 with n 1 IO only exists on -1sx<xR where the asymmetrical Rayleigh branch n=in exists on xR<x<O where xR(x)>l. 1 xR(x)~-1, and with n IO only 1 These comments result from the forms of the frequency equations and they are a reminder that the lower bound in (B.3-7) has resulted from squaring operations which obscure the differe.nce between the symmetrical and asymmetrical frequency equations. The lower bound in Fig. 11 is then a sketch of a portion of the syrnmetrical Rayleigh branch on -l~x<xR and of the asymmetrical Rayleigh branch on xR<x~O. If the negative solution n 1 is taken in each case and if the radicals in (2.3-7) are taken as positive imaginary, then both w and positive. K are real and The positive imaginary values of the radicals -224- result from a natural continuation above their respective . a-1 branch points, x=O and x=a+l· The asymptotic approximation of n 1 =Imn as x+xR is . given by (B.3-8) for all of the complex branches and the complete approximation for n=nR+in , as derived from the 1 frequency equations (2.3-13) and (2.3-14), is ~ l n - l+x[qn+iLog(R(xl>J + o\x-xRI -~:~R) (B.3-9) as x+xR with q=0,2,4, •.• for the symmetrical branches and q=l,3,5, ... for the asymmetrical branches. In this approxi- mation, Log(R(x))=Log 1 R(x) I +iArg(R(x)) with - TI 3TI 2 Arg(R(x))<-r-· (B.3-9) includes the approximation to the symmetrical Rayleigh branch q=O as x+xR on -l~x<xR where Arg(R(x))=O, and to the asymmetrical Rayleigh branch q=l as x+xR on xR<xsO where Arg(R(x))=rr. Every complex branch has a loga- rithmic branch point at x=xR and this feature is not unique to the Rayleigh branches. (B.3-9) indicates the nature of an analytic continuation about the logarithmic branch point xR for every complex branch. Any symmetrical complex branch, including the Rayleigh branch 1 can be continued onto any other symmetrical complex branch by encircling the branch point x=xR· This is equivalent to passing through the branch cut and -225- onto other sheets of the Riemann surface. The same holds true for the asymmetrical complex branches. A few of the complex branches are sketched in Fig. 12 in the Rex,Ren,Imn-space. This figure and other details about the complex branches are discussed in Subsection 3.2.3. -226- APPENDIX C ADDITIONAL MATERIAL FOR ANALYTIC-CONTINUATIONS C.l. AN EXPANSION OF THE FREQUENCY EQUATIONS FOR 1-v SMALL The function R(x)=R(x 1 v) in (2.3-12a) is expanded in powers of 1-v, where the assumption must be made that l+x is not small since both R and the branches n undergo drastic changes near x=-1 as v+l. Then the frequency equations (2.3-13) and (2.3-14) can be approximated, under the assumpt~o·ns that 1-v, x-t and n-k'IT are small where k=2, 3, 4, .•. , j=±l,±2,±3, ••• and XR<t<l, to give (C.1-1) (C . 1-1) is an approximation to the symmetrical branches for j+k odd and to the asymmetrical branches for j+k even. The requirement xR<t, with xR defined by (2.3-9), only serves to avoid the vicinity of x=-1, but this is not very restrictive since xR+-1 as v+l. Thus, {C.1-1) is valid near x=-1 if 1-v is sufficiently small. In addition, (C.1-1) is valid even if 1-v is not small provided that k>>j which means this approximate -227- frequency equation is useful to examine the upper branches near x=O. The simple cubic (C.1-1) vividly illustrates the nature of the branch points, which merge as v+l, both on -l<x<O for j~l and on O<x<l for j~l. If (C.1-1) is taken to be exact, the branch points are identified by requiring that dx_O· thus, the branch points are located at dn ' n = kTI ± 4(1-v) k (k+j)2 IJk so that if j is negative both branch points. x and n are complex at the The very important feature of this is that when Imx>O at the branch points, identifying these as the approximate location of branch points from "the set of negative branch points," it is also true that Imn>O. Therefore, branch points belonging to "the set of negative branch points" are common only to pairs of branches consisting of a real branch on -l<x<O and a complex branch with Irnn>O. The preceding statement requires a very simple examination of the continuations about the branch points as predicted by (C.1-1). Likewise, i t is seen that complex conjugates of these branch points are common to pairs of -228- branches consisting of a real branch on -l<x<O (the same real branches as those just mentioned) and a complex branch with Irnn<O. The simple cubic {C.1-1) has solutions which illustrate the nature of the complex branches as V+l. On 0SXS1 1 for ~v<l, the complex branches have complex loops between pairs of branch points just as shown in Fig. 12 on a-1 - -a+ 1 O<x<~- . v=l causes the branch points and, hence, the base points of neighboring loops to merge. On -lsxso for v<l, each corn- plex branch has an imaginary part which oscillates but is never zero as is also shown in Fig. 12. v=l causes these minimums in Iron to reach Imn=O at the branch points and the branches become identical with those on osxsl. C.2. CONTINUATIONS OF THE EQUIVOLUMINAL AND DILATATIONAL BRANCHES ABOUT THEIR SINGULAR POINTS, x=-1 AND x=l In Subsection 3.3.3 , the equivolurninal and dilata- tional branches were found to ~e singular at x=-1 and x=l respectively . Continuations of these branches about their singular points is required and this presents a problem since none of the representations in Section 3.3 were shown to be valid except as the singular points were approached on x=e ie , 0<8<TI. To resolve this problem, it is easy to show that the approximation -229- (C. 2-1) n=l,2,3, ... , taken directly from the series representation (3.3-14), is valid for the symmetrical equivoluminal branches, with the error indicated, for O<Arg(l+x)<n. The verification of this approximation is contained in the discussion following the form (3.3-6) of the symmetrical frequency equation. Thus, since the complex branches with Imn<O were found to have no branch points adjacent to the real x-axis with Imx>O, they can be continued slightly off the real x-axis. This continuation for a complex branch 2nn with the dominant term l+x must lead to the equivoluminal branch with the approximation (C.2-1). the fact that This argument uses ~~ must remain finite since no branch points are present, and the continuation must take place away from the singular point x=-1. Also, · it is observed from (3.3-4) that the imaginary part of the approximation (C.2-1) is negative on the real x-axis, which is in good agreement with the imaginary part of the complex branches since they oscillate about -~ on the real x-axis near x=-1. -\) 1 2 The complex branches with Ren>O and Imn>O continue about x=-1 Onto the dilatational branches On x=e ie 1 0<8~TI. The reason for the difference between the continuations of complex branches with imaginary parts of different signs -230- is that only those branches with Imn>O have branch points with Imx>O adjacent to the real x-axis on -l<x<O. Similarly, the asymmetrical equivoluminal branches continue about x=-1 onto the asymmetrical complex branches with Imn<O. Also, from the approximations consisting of the leading terms of (3.3-23) and (3.3-28), the dilatational branches continue about x=l onto the only branches available on the real x-axis, the real branches. C.3. THE NATURE OF "THE SET OF POSITIVE BRANCH POINTS" FOR l v=2· "The set of positive branch points" for v= 1 2 , defined in Subsection 3.4.1, has not been explained yet. The na- ture of these branch points is easily resolved by making analytic continuations about the closed paths described in Subsection 3.4.2, except that these branch points are ignored and remain interior to the closed paths. In doing so, it is found that none of the branches have a singlevalued continuation on the closed path. Hence, one branch point from this subset must be common to each pair of neighboring real branches on :~isxSl since this is required for a single-valued continuation. Also, the details of the continuation about each of these branch points must be the -231- same as for the other members of "the set of positive a-1 branch points" which are adjacent to a+l2X2_1. To be pre- cise, this argument only insures that an odd number of these branch points be common to neighboring branches, however, it was found in Subsection 3.1.3 that the branch points are also zeros of ~~k given by (3.1-9) with ±(-l)k= +l where krr<Ren<(k+l)rr at the branch point. Consequently, it is not a reckless assumption to assign the branch point + of this subset which is a zero of ~jk to the neighboring pair of real branches nk and nk+l on the real x-axis. This gives the required number of branch points to make every branch continue as a single-valued, analytic function on a closed path with none of its branch points interior to or on the path. C.4. EXAMPLES OF ANALYTIC CONTINUATIONS ON CLOSED PATHS A closed path in the x-plane is shown in Fig. 14 and it corresponds to the analytic continuation of the symmetrical branch shown in Fig. 15. Both the path and the corres- ponding values of Tl are indicated by a heavy 1 directed line. The dashed, directed lines in Fig. 15 represent transitions caused by the path going around branch points. the modulus ·e Two axes for Jn (e 1 ) J versus e , 0 .Se ,S1T, are used on the left of -232- Fig. 15 to emphasize the transition from the dilatational branch n (e 0 ·e 1 branch no(e ) =i In ie 0 (e )=lno(e ·e 1 ie .e ) Ie -i2 to the symmetrical Rayleigh .e )le -i2 caused by their common branch ie point at eo, where the point xo=e 0 is given by (2.3-8). The numbers 1,2, .•• ,6 label certain corresponding points in both figures. Branch points are indicated in Fig. 14 by dots. A description of this continuation is now given by be- ginning at point 1 or x=O. complex branch with Imn<O gate of this branch). Point 1 is on a symmetrical (Fig. 12 shows the complex conju- n(O)=Ren(O)+iimn(O) satisfies (B.1-7) and it is the root with the smallest positive real part. On 1, 2, n remains on this complex branch with Iron vanishing at 2, which is the branch point identified as (-,0,1) in Fig. 7. The continuation about the branch point at 2 takes n onto a real branch identified as the symmetrical branch n (x) in Fig. 8. 1 Beyond 2, branch points from "the set of positive branch points" are encountered adjacent to the real x-axis. The continuations about these branch points are indicated by dashed lines in Fig. 15 on a-l <x<l. a+ 1 These continuations have the effect of stair- stepping up the real branches. These branch points form an infinite set with a . limit point at x=l and they are -233- zeros of 6~+l,k given by (3.1-9) 3, • . . . for ±(-l)k=+l and k=l,2, These become too numerous to show near x=l in Fig. 14 and the path just passes above all of them and onto the Unl· t · 1 e x=e ie / ClrC o<8~TI. In Fig. 15 this continuation connects the right and left extremes which are both labeled 3 for x=l or 8=0. On the unit circle, n goes onto the lowest dilatational branch described by (3.2-12) with rn=O, which is shown in Fig. 10. point at 4 or x=x =e 0 ie 0 This branch shares the branch . given by (2.3-8) with the synunetri- cal Rayleigh branch which is described by (3.2-11) with n=O and is shown in Fig. 9. The branch is analytic at 5 (x=-1, 8=n) and i t continues onto the real x-axis still on the synunetrical Rayleigh branch where n is pure imaginary with Imn<O on -l~x<xR · The branch point at 6 or x=xR given by (2.3-9) is avoided by a small indention which causes n to change as predicted by the complex conjugate of (B.3-9) with q=O. The real part of n jumps from zero to its imaginary part is unbounded at XR· and : 1 XR This puts n back on the same complex branch as the beginning point 1 and the continuation to that point is uneventful. A similar closed path in the x-plane is shown in Fig . 16 corresponding to the dilatational continuation (as defined in Subsection 3.4.2) of the asymmetrical branch shown in Fig. 17. In this case, there is a branch point at 1 -234- (x=O) which is conunon to the asymmetrical Rayleigh branch and to the asymmetrical real branch n (x) shown in Fig. 8. 1 The continuation from 1 to 2 (x=l) only encounters branch points adjacent to :~i<x<l and these form another infinite set with a limit point at x=l. . + These are zeros of 6kk k given by (3.1-9) for ±(-1) =+land k=l,2,3, . . . . The de- tails of the continuation about these branch points are the same as just described for the symmetrical continuation shown in Figs. 14 and 15. At 2, n goes onto the asymmetri- cal dilatational branch described by (3.2-18) with m=l, which is shown in Fig. 10. This branch has no branch · t s on th e uni· t circ · 1 e x=e i 8 , poin o<e ~TI , · an d i· t con t inues onto the lowest asymmetrical real branch at 3 (x=-1) . From 3 to 4 on the real x-axis, n remains on this real branch and then at 4 a branch point is encountered and n goes onto a complex branch with Imn>O which has the dominant term i:x as x~-1 and is shown in Fig. 12 . a zero of 6_ 11 branch points." This branch point is and it belongs to "the set of negative At 5 (x=xR), n leaves the complex branch and continues onto the asymmetrical Rayleigh branch as predicted by (B.3-9) with q=l. The real part of n de- creases from ~TI~ to zero on the Rayleigh branch with the · l+xR imaginary part of n unbounded at XR · The continuation then proceeds on the asymmetrical Rayleigh branch from 5 -235- to 1 (x=O), which was the starting point. In Figs. 18 and 19, an equivoluminal continuation (as defined in Subsection 3.4.2) is shown and it involves the symmetrical equivoluminal branch n (e 2 ie )=!n 2 (e ie .e ) je scribed by (3.2-11) with n=2 and shown in Fig. 9. - 12 , de- The con- tinuation is much like the two iust described except that only three branch points from "the set of positive branch points" are involved on :~i~x<l. zeros of ~ + 14 , ~ - The branch points are + 23 and 6 32 respectively as x=l is approached. The continuation then goes onto the equivolurninal branch . ana 1 yt1c . at x= 1 an d n 2 ( e ie) wh'1ch 1s branch at x=-1 which has Imn<O. -f ' ~ina 11y onto a comp 1 ex The complex branch has no branch points from "the set of negative branch points" as was found in C.l. The continuation about x=xR is given by the complex conjugate of (B.3-9) with q=4.