Developments of parabolic equation method in the period of 2000–2016

Xu, Chuanxiu; Tang, Jun; Piao, Shilong; Liu, Jiaqi; Zhang, Shi-Zhao

doi:10.1088/1674-1056/25/12/124315

Cited by 8 publications

(3 citation statements)

References 93 publications

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“…However, in 1972, Hardin and Tappert proposed a split-step Fourier method that could use computers to solve parabolic equations efficiently and deal with underwater acoustic propagation problems [30]. Since then, the parabolic equation method has been widely studied, and various numerical solutions algorithms have emerged [29,31]. At present, the parabolic equation method has become a very important tool to approximately solve problems in electromagnetic wave propagation.…”

Section: Introductionmentioning

confidence: 99%

General Formulas of Effective Earth Radius and Modified Refractive Index

Feng

Huang

2021

IEEE Access

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The effective earth radius can transform the curved ray path into a straight line, and the modified refractive index can transform the spherical ground into a plane. Using them can simplify the propagation modeling of electromagnetic waves in the atmosphere, so they are widely used in the calculation of long-range propagation of electromagnetic wave, and adopted by the ITU Recommendations and the IEEE Standards. In this paper, upon employing simple geometric methods, we provide general formulas for the effective earth radius and the modified refractive index, pointing out that they both originate from the use of conformal transformations. In turn, we prove that for both quantities the formulas contained in the ITU Recommendations and the IEEE Standards are special cases of the general formulas given in this paper. Our results provide a solid mathematical background to the concepts of effective earth radius and modified refractive index, and pave the way for their use beyond the propagation of electromagnetic waves in the earth atmosphere, e.g. to the propagation of other types of waves in generic media.INDEX TERMS Effective earth radius, modified refractive index, conformal transformation, wave propagation, ray tracing.

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Section: Introductionmentioning

confidence: 99%

General Formulas of Effective Earth Radius and Modified Refractive Index

Feng

Huang

2021

IEEE Access

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“…Over the past decades, substantial efforts have been dedicated to the study of theoretical and numerical aspects of the PE, leading to an increased number of practical applications. Extensive literature reviews of existing PE models and case studies are available in Lee et al (2000) and Xu et al (2016).…”

Section: Introductionmentioning

confidence: 99%

An iterative three-dimensional parabolic equation solver for propagation above irregular boundaries

Khodr

Azarpeyvand

Green

2020

The Journal of the Acoustical Society of America

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This paper describes the development of an iterative three-dimensional parabolic equation solver that takes into account the effects of irregular boundaries and refraction from a layered atmosphere. A terrain-following coordinate transformation, based on the well-known Beilis-Tappert mapping, is applied to the narrow-angle parabolic equation in an inhomogeneous media. The main advantage of this approach, which has been used in two dimensions in the past, is the simplification of the impedance boundary condition at the earth's surface. The transformed initialboundary value problem is discretized using the Crank-Nicholson marching scheme in the propagating direction and second-order finite-differences in the transversal plane. The proposed method relies on an efficient iterative fixedpoint solver, which involves the inversion of tridiagonal matrices only. The accuracy of the method is evaluated through a comparison with boundary element simulations in a homogeneous atmosphere above a Gaussian hill. Results show that transversal scattering occurs in the shadow zone of the obstacle where the two-dimensional parabolic equation underestimates the pressure amplitude. The model is particularly suited for the simulation of infrasound in a three-dimensional environment with realistic topographies. V

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“…Readers are referred to a review article by Xu et al for a summary of the relevant studies. [4] Also, there have been a variety of 3D models, including but not limited to 3D PEs, [4][5][6][7][8][9][10][11][12][13] 3D coupled-mode methods, [14][15][16] hybrid methods, [17][18][19][20] and analytical methods. [21][22][23][24][25][26] Nevertheless, studies that incorporate both bottom elasticity and 3D propagation are very scarce.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation

Tang¹,

Piao²,

Zhang³

2017

Chinese Phys. B

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A three-dimensional (3D) parabolic equation (PE) model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid-elastic interfaces.

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Developments of parabolic equation method in the period of 2000–2016

Cited by 8 publications

References 93 publications

General Formulas of Effective Earth Radius and Modified Refractive Index

General Formulas of Effective Earth Radius and Modified Refractive Index

An iterative three-dimensional parabolic equation solver for propagation above irregular boundaries

Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation

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