## Abstract

Attention is directed to the examination of the reflexion and refraction of a point-source field at a plane surface which separates two media of distinct acoustical properties. While some aspects of this problem are not new, this paper may be regarded as a demonstration of the relation between certain fundamental solutions of the wave equation and singular integrals of parametric plane waves. There are two halves in the discussion. In the first it is shown that the potential of a steadily moving point source may be written in the form of an integral superposition of parametric plane waves regardless of the source velocity. By taking each (real or complex) plane wave to be reflected and refracted independently at the interface between two media, we may construct the total field due to a buried point source which moves at a constant distance from the interface with a constant velocity. By a limiting process the field for a source which moves along the interface may also be determined. The single integrals which arise here are singular in form, the integrands are uniform functions of the velocity of the source regardless of whether it is subsonic or supersonic, but the formal integration path is different in the two cases. All source velocities between zero and infinity are considered. The second half of the discussion is concerned with the field of a fixed point source of step function strength as a function of time. The plane wave representation for this singularity is a double integral which is singular. Expressions for reflected and refracted source fields are again easily found. The extension to the case of a moving point source of step function strength as a function of time is also described and so is the effect of the plane interface. Several aspects of interest are to be mentioned here. The singular surfaces which arise both as characteristics and as envelopes of characteristics of the wave equation, and which are surfaces of discontinuity in either potential or its normal derivative, appear with ease as part of the singular integrals. The field structures known as head waves are also picked out as particular residue contributions to the reflected field, for a source in the medium of smaller sound velocity. The behaviour of this head wave field close to the wave surface is calculated, and the separate stages in the formation of the head wave are described. One expression is found for the moving source of step function strength regardless of the source velocity. This expression contains a residue term which represents the potential of a steadily moving source of constant strength. In the case of subsonic motion this residue need only be considered a long time after the source has been set up, except in the immediate vicinity of the source. In the supersonic case the residue term is responsible at all times for the conical field of the moving source which caps the spherical field due to the initial appearance of the source. A source is a particular type of singularity which is defined only away from a boundary. It is confirmed that a source of unit strength at the common plane surface between two media is a singularity which causes the same mass flux into each medium.

## Footnotes

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- Received March 26, 1962.

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