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Journal Article

# The Reflexion of a Spherical Acoustic Pulse by an Absorbent Infinite Plane and Related Problems

P. E. Doak
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 215, No. 1121 (Nov. 25, 1952), pp. 233-254
Stable URL: http://www.jstor.org/stable/99045
Page Count: 22
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## Abstract

A formal integral solution is given for the problem of the reflexion of a spherical acoustic pulse by an infinite plane interface having an impedance of arbitrary dependence on frequency and angle of incidence. In many cases of practical interest the impedance may be assumed to be independent of angle of incidence, and under this assumption the integral solution is relatively easy to evaluate. A simple exact expression for the reflected pulse, in closed form, is obtained when the wall impedance is purely resistive (i.e. independent of frequency). This solution is a special case of a general type of solution of the wave equation when it is reduced to a rotationally symmetric Laplace's equation in the 'spherical polar' co-ordinates $\bigg[\surd \{(ct/r)^{2}-\sin ^{2}\,\vartheta \}, \ \bigg(\frac{ct\,\cos \,\vartheta}{r}\bigg)\Bigg/\surd \{(ct/r)^{2}-\sin ^{2}\,\vartheta \}\bigg]$. To illustrate the relatively wide range of validity of the assumption of an impedance independent of angle of incidence, when applied to real materials, this exact result is compared with an approximate solution for the case where the interface separates two homogeneous isotropic lossless materials. The formal integral solution is evaluated approximately for wall impedances of the following types: (i) resistance and mass, (ii) resistance and stiffness, (iii) resistance, mass and stiffness. The solutions are compared with corresponding solutions for plane incident waves, and the behaviour of the scattered wave, distinguishing between the spherical and the plane wave, is discussed. Possible applications of the results for acoustic waves to problems in the reflexion of blast waves and of transient radiation by an electric dipole are indicated briefly.

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