On the Equivalence of Saddle Point Approximations and Ray Theory in Elastic Wave Problems

Abstract

Summary It has been known for some time that the Bromwich method of expansion in negative powers of exponentials of the formal solutions of elastic wave problems for stratified media by wave theory methods gives rise to the various reflected and refracted waves predicted by the ray theory of geometrical optics. It is shown here by application to an example how the saddle point evaluation by the steepest descents method of the various terms in such an expansion of the solution leads to exactly the same expressions for the displacements in the reflected and refracted waves as can be obtained by the geometrical theory. The example chosen is that of the elastic waves produced by a harmonic point source of dilatational waves situated in the middle of an internal stratum bounded on both sides by halfspaces of identical elastic properties, all perfectly elastic, homogeneous and isotropic. In the course of the demonstration, we give the reflection and refraction coefficients for the amplitudes of displacements when a plane P or SV wave is incident at an interface between two elastic media. These are in the form in which they occur in the wave solutions of problems involving reflection and refraction at plane interfaces. I. Introduction Bromwich (1916, p. 425-428) studied the problem of a bar with one end fixed and struck at the other by a particle moving in the direction of the length of the bar. He showed that the expansion of the formal operational solution in negative powers of exponentials leads to a solution in terms of pulses. The method has since been applied by Muskat (1938a, b), Pekeris (1948, p. 46 et seq.), Officer (1951), Newlands (1952, p. 222 et seq.), Honda & Nakamura (1953, 1954) and others to the formal solutions of elastic wave problems for stratified media. Whereas Muskat and others studied problems involving plane boundaries, van der Pol & Bremmer (1937) have discussed the propagation of radio waves from a spherical point source situated above a spherical earth. They show (1937, pp. 844-6; see Bremmer, 1949, Ch. V for further detail) that the saddle point approximation for the reflected field gives an expression which may be obtained from the primary field by multiplying it by a reflection coefficient corresponding to the extra divergence acquired by a thin cone of rays on being reflected from a spherical instead of a plane surface. At the same time, the phase factor has to be