Helmholtz equation in a semi-infinite strip with impedance boundary conditions of the third and fifth orders

TL;DR

This paper addresses boundary value problems for the Helmholtz equation in a semi-infinite strip with complex impedance boundary conditions involving higher-order derivatives, proposing an exact solution technique using Laplace transforms and Riemann-Hilbert problems.

Contribution

It introduces a novel method employing two Laplace transforms to solve fluid-structure interaction problems with higher-order impedance boundary conditions.

Findings

Exact solutions via Riemann-Hilbert problems are obtained.

Existence and uniqueness depend on the Riemann-Hilbert problem index.

Method models wave propagation in semi-infinite waveguides.

Abstract

Two boundary value problems for the Helmholtz equation in a semi-infinite strip are considered. The main feature of these problems is that, in addition to the function and its normal derivative on the boundary, the functionals of the boundary conditions possess tangential derivatives of the second and fourth orders. Also, the setting of the problems is complimented by certain edge conditions at the two vertices of the semi-strip. The problems model wave propagation in a semi-infinite waveguide with membrane and plate walls. A technique for the exact solution of these fluid-structure interaction problems is proposed. It requires application of two Laplace transforms with respect to both variables with the parameter of the second transform being a certain function of the first Laplace transform parameter. Ultimately, this method yields two scalar Riemann-Hilbert problems with the same coefficient and different right-hand sides. The dependence of the existence and uniqueness results of the physical model problems on the index of the Riemann-Hilbert problem is discussed.