Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

Abstract

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound 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 are 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 the eigenvalues and eigenvectors are obtained, 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.

Application of a Spectral Method to Simulate Quasi-Three-Dimensional Underwater Acoustic Fields

“…The research conducted by Orszag and Gottilieb illustrated that the numerical error of the spectral method decreases exponentially with an increase in the truncation order [4,5]. Consequently, benefiting from its high precision, the spectral method has been widely used in many numerical simulations of scientific and engineering problems [6][7][8].…”

A Spectral Method for Depth-Separated Solution of a Wavenumber Integration Model in Horizontally Stratified Fluid Acoustic Waveguides

“…Among the methods for numerically solving differential equations, in addition to the widely used finite difference and finite element methods, spectral methods are a kind of niche but efficient new method. Spectral methods have high accuracy and fast convergence speed [14][15][16][17][18][19][20] and have been rapidly developed in acoustics [21,22], especially computational ocean acoustics. In recent years, new algorithms of normal modes [23][24][25][26][27][28][29], coupled modes [30][31][32] and parabolic equation models [33][34][35] based on spectral methods have been successively proposed.…”

A High-Efficiency Spectral Method for Two-Dimensional Ocean Acoustic Propagation Calculations

“…Although the propagation of acoustic waves in the ocean is affected by both seawater and the ocean surface, acoustic waves of a fixed frequency can still be obtained after adding boundary conditions to constrain the elliptic partial differential Helmholtz equation [ 4 ]. Recently, our research group used the one-dimensional spectral method to correctly solve the normal modes in underwater sound propagation [ 5 ] and atmospheric acoustics [ 6 ], demonstrating that the spectral method has the advantages of fast convergence and high accuracy when solving the sound field. However, there are still some problems in this calculation process.…”