Wave and Extra-wide-angle Parabolic Equations for Sound Propagation in a Moving Atmosphere PDF Download
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Author: Vladimir E. Ostashev
Publisher:
ISBN:
Category : Differential equations, Partial
Languages : en
Pages : 16
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Book Description
The narrow-angle parabolic equation (NAPE) with the effective sound speed approximation (ESSA) is widely used for sound and infrasound propagation in a moving medium such as the atmosphere. However, it is valid only for angles less than 20° with respect to the nominal propagation direction. In this paper, the wave equation and extra-wide-angle parabolic equation (EWAPE) for high-frequency (short-wavelength) sound waves in a moving medium with arbitrary Mach numbers are derived without the ESSA. For relatively smooth variations in the medium velocity, the EWAPE is valid for propagation angles up to 90°. Using the Padé (n,n) series expansion and narrow-angle approximation, the EWAPE is reduced to the wide-angle parabolic equation (WAPE) and NAPE. Versions of these equations are then formulated for low Mach numbers, which is the case that is usually considered in the literature. The phase errors pertinent to the equations considered are studied. It is shown that the equations for low Mach numbers and the WAPE with the ESSA are applicable only under rather restrictive conditions on the medium velocity. An effective numerical implementation of the WAPE for arbitrary Mach numbers in the Padé (1,1) approximation is developed and applied to sound propagation in the atmosphere.
Author: Vladimir E. Ostashev
Publisher:
ISBN:
Category : Differential equations, Partial
Languages : en
Pages : 16
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Book Description
The narrow-angle parabolic equation (NAPE) with the effective sound speed approximation (ESSA) is widely used for sound and infrasound propagation in a moving medium such as the atmosphere. However, it is valid only for angles less than 20° with respect to the nominal propagation direction. In this paper, the wave equation and extra-wide-angle parabolic equation (EWAPE) for high-frequency (short-wavelength) sound waves in a moving medium with arbitrary Mach numbers are derived without the ESSA. For relatively smooth variations in the medium velocity, the EWAPE is valid for propagation angles up to 90°. Using the Padé (n,n) series expansion and narrow-angle approximation, the EWAPE is reduced to the wide-angle parabolic equation (WAPE) and NAPE. Versions of these equations are then formulated for low Mach numbers, which is the case that is usually considered in the literature. The phase errors pertinent to the equations considered are studied. It is shown that the equations for low Mach numbers and the WAPE with the ESSA are applicable only under rather restrictive conditions on the medium velocity. An effective numerical implementation of the WAPE for arbitrary Mach numbers in the Padé (1,1) approximation is developed and applied to sound propagation in the atmosphere.
Author: David Keith Wilson
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Author: Michael D. Collins
Publisher: Springer Nature
ISBN: 1493999346
Category : Science
Languages : en
Pages : 135
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Book Description
This book introduces parabolic wave equations, their key methods of numerical solution, and applications in seismology and ocean acoustics. The parabolic equation method provides an appealing combination of accuracy and efficiency for many nonseparable wave propagation problems in geophysics. While the parabolic equation method was pioneered in the 1940s by Leontovich and Fock who applied it to radio wave propagation in the atmosphere, it thrived in the 1970s due to its usefulness in seismology and ocean acoustics. The book covers progress made following the parabolic equation’s ascendancy in geophysics. It begins with the necessary preliminaries on the elliptic wave equation and its analysis from which the parabolic wave equation is derived and introduced. Subsequently, the authors demonstrate the use of rational approximation techniques, the Padé solution in particular, to find numerical solutions to the energy-conserving parabolic equation, three-dimensional parabolic equations, and horizontal wave equations. The rest of the book demonstrates applications to seismology, ocean acoustics, and beyond, with coverage of elastic waves, sloping interfaces and boundaries, acousto-gravity waves, and waves in poro-elastic media. Overall, it will be of use to students and researchers in wave propagation, ocean acoustics, geophysical sciences and more.
Author: P. Malbéqui
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Author: Ding Lee
Publisher: World Scientific
ISBN: 9789810223038
Category : Science
Languages : en
Pages : 224
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Book Description
This book introduces a comprehensive mathematical formulation of the three-dimensional ocean acoustic propagation problem by means of functional and operator splitting techniques in conjunction with rational function approximations. It presents various numerical solutions of the model equation such as finite difference, alternating direction and preconditioning. The detailed analysis of the concept of 3D, N x 2D and 2D problems is very useful not only mathematically and physically, but also computationally. The inclusion of a complete detailed listing of proven computer codes which have been in use by a number of universities and research organizations worldwide makes this book a valuable reference source. Advanced knowledge of numerical methods, applied mathematics and ocean acoustics is not required to understand this book. It is oriented toward graduate students and research scientists to use for research and application purposes.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 68
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Book Description
Parabolic equations can be used to find approximate solutions to the reduced wave equation. By reducing the equation to first order in the range derivative, the solution can be found by marching forward in range. Several numerical techniques can be applied to the solution of the parabolic equation (PE), including finite elements, finite differences, and a Fourier Transform method known as the split-step PE. The split-step solves for each range increment in two steps. First, it propagates forward through a homogenous atmosphere, using the Fourier Transform. It then applies a multiplicative phase correction for index-of-refraction variations. The split-step method leads to a computationally fast model for two reasons: the range steps are several wavelengths, and the Fourier Transform can be evaluated by a Fast Fourier Transform. One of the difficulties encountered in applying the split-step PE to outdoor sound propagation is accommodation of the complex ground impedance. The Green's function PE is a split-step PE which solves for the one-dimensional height-dependant Green's function in a homogenous atmosphere. This Green's function incorporates the complex ground impedance as a complex, angle-dependant plane-wave reflection coefficient. (AN).
Author: Joseph Bishop Keller
Publisher: Springer
ISBN:
Category : Science
Languages : en
Pages : 314
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Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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The foundations of the modern theory of sound propagation and scattering in a homogeneous and isotropic atmospheric turbulence are developed: The sound scattering cross-section for von Karman spectra of temperature and wind velocity fluctuations is calculated; the rigouros theory of line of sight sound propagation in an atmosphere with Kolmogorov, Gaussian and von Karman spectra of temperature and wind velocity fluctuations is developed; a new theoretical formulation of the interference of the direct wave from source to receiver and that reflected from the ground in a turbulent atmosphere is presented; the sound scattering cross section in an atmosphere with arbitrary profiles of temperature and wind velocity is calculated; some predictions of the modern theory are verified experimentally; correct wideangle parabolic equations for sound waves in a turbulent atmosphere are derived and used for numerical simulations of sound propagation. The modern theory has already been adopted by scientists for calculations of sound propagation in turbulent media and as a basis for development of new acoustic remote sensing techniques of the atmosphere and ocean in several countries and organizations including the U.S. Army Research Laboratory.
Author: Kirk A. Weatherly
Publisher:
ISBN: 9781423536963
Category :
Languages : en
Pages : 96
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Book Description
This thesis examines two implementations of the parabolic equation approximation to the acoustic wave equation aimed at removing three errors inherent to the wide-angle parabolic equation (WAPE) model. First, the selection of the range step size used by the split-step Fourier algorithm affects the convergence of the solution. Second, in certain ocean environments WAPE incorrectly computes the down-range transmission loss. Finally, WAPE does not reproduce the standard normal mode basis set as defined by normal mode theory. A double-precision implementation of the WAPE (DP-WAPE) is developed to evaluate the dependence of solution convergence on the numerical precision of the model. Finally, an implementation that is insensitive to the choice of the reference sound speed (COIPE) is evaluated for its ability to reduce or remove the latter two of these three errors. The stability other WAPE solution was found to be unaffected by the DP-WAPE implementation. The range-step dependence is inherent to the split-step algorithm. The COIPE corrects the transmission loss anomaly and satisfactorily reproduces the standard normal mode basis set.
Author: Michael James White
Publisher:
ISBN:
Category : Sound
Languages : en
Pages : 198
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Book Description
