Theory of Propagation of Sound in a Half-Space of Variable Sound Velocity under Conditions of Formation of a Shadow Zone

Abstract

This paper treats the propagation of sound from a point source situated at some depth below the surface of a half-space in which the sound velocity decreases with depth. There is then a “limiting ray” which forms the boundary of a “shadow zone” into which no rays penetrate. Our chief interest is in the intensity of the diffracted radiation that penetrates into the shadow zone and its dependence on distance from source and on frequency. In Part I a detailed study is made of propagation of sound in a medium in which sound velocity decreases at a constant rate a with depth. It is found that inside the shadow zone the intensity I of monochromatic sound of frequency f falls off essentially exponentially with distance from the shadow boundary: I(P)=I(B)R0Re−Ar, A=5.93ca1f1, (A) where B denotes the point on the shadow boundary at the depth of the point of observation P, R0, and R the ranges of B and P, r=R−R0, and c the sound velocity. The intensity on the shadow boundary I(B) decreases with decreasing frequency on account of interference between the source and its image. As a result there exists at each point inside the shadow zone a frequency of optimum penetration. The solution inside the shadow zone is given in terms of normal modes. It is shown that in case of explosive sound the travel time for a point in the shadow zone is such as to indicate that part of the trajectory is covered near the surface with the surficial sound velocity. Details are given for computing the pressure-time curve in case of an exponentially decaying pulse at the source. In order to study the effect of a variable sound velocity gradient with depth, an investigation is made in Part II, using asymptotic methods, of propagation of sound in media in which c=c0/(1+αz+βz2)12. It is found that Eq. (A) is still valid for extremely high frequencies, where a now denotes the surficial velocity gradient. For moderate and low frequencies the decrement may be less or greater than that given by Eq. (A), depending on whether the velocity gradient decreases or increases with depth. In general, for high frequencies the decrement is largely determined by the surficial velocity gradient, while for low frequencies it is increasingly dependent on the velocity distribution in the deep layers. The general theory of propagation of sound in media with arbitrary vertical variation of sound velocity is developed from the stand-point of normal modes and asymptotic solutions. It is emphasized that for certain media the normal mode solution needs to be supplemented by branch-line integrals.