A Chebyshev Spectral Method for Normal Mode and Parabolic Equation Models in Underwater Acoustics

Houwang Tu,Wenbin Xiao,Qiang Lan,Yongxian Wang,Xian Ma,Wei Liu

doi:10.1155/2020/7461314

Abstract

In this paper, the Chebyshev spectral method is used to solve the normal mode and parabolic equation models of underwater acoustic propagation, and the results of the Chebyshev spectral method and the traditional finite difference method are compared for an ideal fluid waveguide with a constant sound velocity and an ideal fluid waveguide with a deep-sea Munk speed profile. The research shows that, compared with the finite difference method, the Chebyshev spectral method has the advantages of a high computational accuracy and short computational time in underwater acoustic propagation.

Highlights

Over 70% of the Earth’s area is covered by the ocean. e ocean is rich in energy, minerals, and biological resources
Among the many types of spectral methods, the Chebyshev spectral method (CSM) with Chebyshev polynomials as the basis functions has been widely used in studies of computational fluid dynamics problems, chemical measurements, and electricity [18,19,20]. is paper introduces the CSM to solve underwater acoustic sound propagation problems, and the characteristics of the CSM and finite difference method (FDM) are compared in terms of the computational accuracy and speed
The computations of the CSM used to solve the underwater acoustic propagation problem require a large number of iterations. e inversion of the dense matrix and requirement of solving equations multiple times result in large memory overhead. erefore, the CSM and FDM have distinct advantages and disadvantages in terms of the calculation costs. e main disadvantage of the CSM is that the absolute smoothness of the basis function means that it can approximate a smooth original function only

Summary

Introduction

Over 70% of the Earth’s area is covered by the ocean. e ocean is rich in energy, minerals, and biological resources. According to the literature [4,5,6], the FDM can be used to develop normal mode model for the calculation of an underwater acoustic field, and this approach has become popular among researchers. Another study [11, 12] developed a method for an approximate calculation of the underwater acoustic field described by a cylindrical coordinate system using the FDM for a three-dimensional (3D) complex scene and integrated the method into FOR3D. In common scientific and engineering numerical modeling problems, the finite volume method, finite element method, and spectral method are all numerical discrete methods that provide alternatives to the FDM Among these methods, the finite volume and finite element methods are more suitable for discretizing equations in integral form, and control equations in differential form are more commonly used in the field of underwater acoustics. Among the many types of spectral methods, the Chebyshev spectral method (CSM) with Chebyshev polynomials as the basis functions has been widely used in studies of computational fluid dynamics problems, chemical measurements, and electricity [18,19,20]. is paper introduces the CSM to solve underwater acoustic sound propagation problems, and the characteristics of the CSM and FDM are compared in terms of the computational accuracy and speed

Underwater Acoustic Propagation Model

Test Case and Analysis of Results

Methods

Findings

Summary and Outlook