Effect of the branch-cut on the transformation between modes and rays

Abstract

In this paper the effect of the branch-cut on the transformation between the modes and rays is discussed. Characteristics of the bottom reflection coefficient on the ζ plane were analyzed in detail and a certain branch-cut was taken. Once the integral path of the generalized ray is deformed to the real axis of the ζ plane, the ray representation consists of ‘‘canonical’’ generalized ray and a ‘‘lateral term.’’ The mode representation and the ray representation are given as follows: Mode representation=∑∞nφn(r,z,z0) +∮b. Ray representation =∑∞lBl(r,z,z0) +𝒥∞l∮bl. It was found that the ‘‘canonical’’ generalized ray Bl and the normal mode φn were still connected by the Poisson summation relation strictly, as proved by our previous paper [Gao and Shang, J. Sound Vib. 80, 105–115 (1982)]. The sum of the ‘‘lateral term’’ of the ‘‘canonical’’ generalized ray was exactly equal to the part of the ‘‘lateral’’ wave in the mode representation:∮b=∑∞l∮bl.