Rayleigh-Bloch waves trapped by a periodic perturbation: exact solutions

TL;DR

This paper derives exact solutions for Rayleigh-Bloch waves in a 2D Helmholtz setting with a periodic perturbation, providing explicit dispersion relations and characterizing wave trapping.

Contribution

It introduces a method to construct exact quasiperiodic solutions for Rayleigh-Bloch waves with a specific refractive index profile, including a simple dispersion relation formula.

Findings

Exact solutions for Rayleigh-Bloch waves are obtained.

The dispersion relation for these waves is explicitly derived.

Wave decay orthogonal to the periodic structure is characterized.

Abstract

Exact solutions describing the Rayleigh-Bloch waves for the two-dimensional Helmholtz equation are constructed in the case when the refractive index is a sum of a constant and a small amplitude function which is periodic in one direction and of finite support in the other. These solutions are quasiperiodic along the structure and exponentially decay in the orthogonal direction. A simple formula for the dispersion relation of these waves is obtained.