THE CAGNIARD METHOD IN COMPLEX TIME REVISITED¹

The Cagniard‐de Hoop method is ideally suited to the analysis of wave propagation problems in stratified media. The method applies to the integral transform representation of the solution in the transform variables (s, p) dual of the time and transverse distance. The objective of the method is to make the p‐integral take the form of a forward Laplace transform, so that the cascade of the two integrals can be identified as a forward and inverse transform, thereby making the actual integration unnecessary. That is, the exponent (–sw(p)) is set equal to –sτ, with τ varying from some (real) finite time to infinity. As usually presented, the p‐integral is deformed onto a contour on which the exponent is real and decreases to –∞ as p tends to infinity. We have found that it is often easier to introduce a complex variable τ for the exponent and carry out the deformation of contour in the complex τ‐domain. In the τ‐domain the deformation amounts to ‘closing down’ the contour of integration around the real axis while taking due account of singularities off this axis.

Typically, the method is applied to an integral that represents one body wave plus other types of waves. In this approach, the saddle point of w(p) that produces the body wave plays a crucial role because it is always a branch point of the integrand in the τ‐domain integral. Furthermore, the paths of steepest ascent from the saddle point are always the tails of the Cagniard path along which w(p) →∞. That is, the image of the pair of steepest ascent paths in the p‐domain is a double covering of a segment of the Re τ‐axis in the τ‐domain. The deformed contour in the p‐domain will be the only pair of steepest ascent paths unless the original integrand had other singularities in the p‐domain between the imaginary axis and this pair of contours. This motivates the definition of a primary p‐domain, i.e. the domain between the imaginary axis and the steepest descent paths, and its image in the τ‐domain, the primary τ‐domain. In terms of these regions, singularities in the primary p‐domain have images in the primary τ‐domain and the deformation of contour on to the real axis in the τ‐domain must include contributions from these singularities.

This approach to the Cagniard‐de Hoop method represents a return from de Hoop's modification to Cagniard's original method, but with simplifications that make the original method more tractable and straightforward. This approach is also reminiscent of van der Waerden's approach to the method of steepest descents, which starts exactly the same way. Indeed, after the deformation of contour in the τ‐domain, the user has the choice of applying asymptotic analysis to the resulting ‘loop’ integrals (Watson's lemma) or continuing to obtain the exact, although usually implicit, time‐domain solution by completing the Cagniard‐de Hoop analysis.

In developing the method we examine the transformation from a frequency‐domain representation of the solution (ω) to a Laplace representation (s). Many users start from the frequency‐domain representation of solutions of wave propagation problems. In this case issues arising from the movement of singularities under the transformation from ω to s must be considered. We discuss this extension in the context of the Sommerfeld half‐plane problem.