Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

TL;DR

This paper introduces two Chebyshev spectral methods, the Tau and Collocation approaches, for efficiently computing atmospheric acoustic normal modes, outperforming previous Legendre-Galerkin methods in speed.

Contribution

The paper presents two novel Chebyshev spectral methods for solving atmospheric acoustic normal modes, transforming the problem into a simple eigenvalue problem and demonstrating improved computational efficiency.

Findings

Both methods successfully compute normal modes in atmospheric acoustics.

The proposed methods are faster than previous Legendre-Galerkin spectral methods.

Numerical experiments confirm the effectiveness of the methods.

Abstract

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a harmonic point source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev--Tau and Chebyshev--Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs were developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After obtaining the eigenvalues and eigenvectors, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre--Galerkin spectral method proposed previously.