Application of a Chebyshev Collocation Method to Solve a Parabolic Equation Model of Underwater Acoustic Propagation

Wang, Yongxian; Tu, Houwang; Liu, Wei; Xiao, Wenbin; Liu, Qiang

doi:10.1007/s40857-021-00218-5

Cited by 20 publications

(9 citation statements)

References 22 publications

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“…We show the discretization of the operator L. In the Chebyshev-Tau spectral method, applying Equations ( 16) and (17) to Equation ( 9), we can obtain the discrete forms of the operator L and Equation ( 9) as follows:…”

Section: Discretized Atmospheric Normal Modes By Chebyshev-tau Spectral Methodsmentioning

confidence: 99%

“…They subsequently solved for the normal modes in underwater acoustics using the Legendre-Collocation method and proved that both of the spectral methods had high accuracy [16]. They also applied the spectral methods to the parabolic approximation of underwater acoustics [13,17,18]. The results of these studies indicated that it is feasible to apply spectral methods for the calculation of underwater acoustics, and in many cases, it has higher accuracy than the finite difference method.…”

Section: Introductionmentioning

confidence: 99%

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Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

Wang

Xiao

et al. 2021

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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.

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Section: Discretized Atmospheric Normal Modes By Chebyshev-tau Spectral Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

Wang

Xiao

et al. 2021

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“…In solving the range-independent normal modes, COUPLE employs the Galerkin method; however, finite difference method such as Kraken are traditionally used 18 . In recent years, many studies have begun to solve the acoustic propagation model by applying more accurate spectral methods [19][20][21][22][23][24][25] . In 1993, Dzieciuch 26 first used the Chebyshev-Tau spectral method to solve for the normal modes of the water column.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical algorithm for solving range-dependent underwater acoustic waveguides based on a direct global matrix of coupled modes and the Chebyshev-Tau spectral method

Tu¹,

Wang²,

Chao³

et al. 2021

Preprint

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Sound propagation in a range-dependent ocean environment has long been a matter of widespread concern in ocean acoustics. Stepwise coupled modes is one of the main methods to solve range-dependent acoustic propagation problems. Underwater sound propagation satisfies a Helmholtz equation, the solution of which represents the core of computational ocean acoustics. Due to its high accuracy in solving differential equations, the spectral method has been introduced into computational ocean acoustics in recent years and has achieved remarkable results. However, the existing underwater acoustic propagation algorithms based on the spectral method can calculate only range-independent ocean acoustic waveguides, which excludes applications in more general range-dependent environments. In this paper, a complete and efficient algorithm is designed using an improved global matrix of coupled modes to solve the range dependence of the ocean environment and using the Chebyshev-Tau spectral method to precisely solve the eigenmodes in a stepped range-independent stair. Based on this algorithm, a complete and efficient numerical program is developed, and the numerical simulation results verify that this algorithm is extremely computationally fast and accurate for various range dependence and seabed environments.

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“…The sound pressure results throughout an entire space can be obtained after adding boundary conditions to constrain the Helmholtz equation [6]. Recently, Wang et al [7] used the onedimensional spectral method to correctly solve the parabolic equation model. Due to the high accuracy and fast convergence speed of the spectral method, it has greatly promoted the development of ocean acoustic calculation.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Method for Two-Dimensional Ocean Acoustic Propagation

Wang

Zhu

et al. 2021

JMSE

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The accurate calculation of the sound field is one of the most concerning issues in hydroacoustics. The one-dimensional spectral method has been used to correctly solve simplified underwater acoustic propagation models, but it is difficult to solve actual ocean acoustic fields using this model due to its application conditions and approximation error. Therefore, it is necessary to develop a direct solution method for the two-dimensional Helmholtz equation of ocean acoustic propagation without using simplified models. Here, two commonly used spectral methods, Chebyshev–Galerkin and Chebyshev–collocation, are used to correctly solve the two-dimensional Helmholtz model equation. Since Chebyshev–collocation does not require harsh boundary conditions for the equation, it is then used to solve ocean acoustic propagation. The numerical calculation results are compared with analytical solutions to verify the correctness of the method. Compared with the mature Kraken program, the Chebyshev–collocation method exhibits higher numerical calculation accuracy. Therefore, the Chebyshev–collocation method can be used to directly solve the representative two-dimensional ocean acoustic propagation equation. Because there are no model constraints, the Chebyshev–collocation method has a wide range of applications and provides results with high accuracy, which is of great significance in the calculation of realistic ocean sound fields.

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Application of a Chebyshev Collocation Method to Solve a Parabolic Equation Model of Underwater Acoustic Propagation

Cited by 20 publications

References 22 publications

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

An efficient numerical algorithm for solving range-dependent underwater acoustic waveguides based on a direct global matrix of coupled modes and the Chebyshev-Tau spectral method

A Spectral Method for Two-Dimensional Ocean Acoustic Propagation

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