Transient phenomena in a three-layer waveguide and the analytical structure of the dispersion diagram

TL;DR

This paper analyzes transient wave phenomena in a three-layer acoustic waveguide by transforming the dispersion integral into the complex frequency domain, revealing the structure of the waveguide's Riemann surface and simplifying transient field calculations.

Contribution

It introduces an analytical continuation of the dispersion diagram and studies the Riemann surface structure, including weak link cases, for the first time in this context.

Findings

Simplified expression for transient wave components.

Analytical structure of the dispersion diagram's Riemann surface.

Solution for weak link configurations.

Abstract

Excitation of waves in a three-layer acoustic wavegide is studied. The wave field is presented as a sum of integrals. The summation is held over all waveguide modes. The integration is performed over the temporal frequency axis. The dispersion diagram of the waveguide is analytically continued, and the integral is transformed by deformation of the integration contour into the domain of complex frequencies. As the result, the expression for the fast components of the signal (i.e. for the transient fields) is simplified. The structure of the Riemann surface of the dispersion diagram of the waveguide is studied. For this, a family of auxiliary problems indexed by the parameters describing the links between layers is introduced. The family depends on the linking parameters analytically, and the limiting case of weak links can be solved analytically.