Plane-Strain Diffraction of Transient Elastic Waves by a Circular Cavity

Author: Peck, Jerry Clifford

Year: 1965

Degree: Dissertation (Ph.D.)

Advisor: Miklowitz, Julius

Committee Member: Unknown, Unknown

Option: Aeronautics; Mathematics

DOI: 10.7907/GDNE-E586

Abstract

The plane-strain problem of the diffraction of a transient plane dilatation wave by a circular cavity in an elastic medium is treated. The method used determines the (total) solution only in the shadow zone, i.e., those points which cannot be connected to the source of disturbance by straight-line rays. Numerical results are obtained for the velocities and displacements on the "back" surface of the cavity caused by a step-stress incident wave.

The analysis is based on a method devised by Friedlander (see his book Sound Pulses, Cambridge, 1958) for the analogous acoustic diffraction problem. This method converges most rapidly at short time, in contrast to Fourier series methods. The Friedlander method essentially employs integral transforms on both time and [Theta], the circumferential coordinate. In the shadow zone, the [Theta]-inversion can be performed by residue theory, the residues resulting from poles at the roots of a "frequency equation." The roots are infinite in number, and may be regarded as forming a dispersion spectrum relating the frequencies and angular wave numbers of a series of circumferential propagation modes. The time-transform inversion is carried out by contour integration and subsequent numerical evaluation.

The transient response results are found to compare well with the Fourier-series solutions at moderate to long times, but at short time the differences are marked, as would be expected. The fact that the present technique yields good long-time results suggests it is even more powerful than might be expected. The major limitation of the numerical method is its restriction to the shadow zone.

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