Separation of Variables Applied to the Wave Equation in Laterally Inhomogeneous Media

2003 ◽  
Vol 160 (7) ◽  
pp. 1225-1244
Author(s):  
A. Druzhinin
Keyword(s):  
Wave Equation ◽  
Separation Of Variables ◽  
Inhomogeneous Media

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

A new solution method for the wave equation in inhomogeneous media

Geophysics ◽  
1985 ◽  
Vol 50 (10) ◽  
pp. 1541-1547 ◽  
Author(s):  
D. M. Pai
Keyword(s):  
Wave Equation ◽  
Matrix Method ◽  
Solution Process ◽  
Inhomogeneous Media ◽  
Mathematical Basis ◽  
Vertical Coordinate ◽  
Mathematical Algorithm ◽  
Matching Layer ◽  
Eigen Value ◽  
Homogeneous Media

A fundamental mathematical algorithm is presented for solving the wave equation in inhomogeneous media. This method completely generalizes the Haskell matrix method, which is the standard method for solving the wave equation in laterally homogeneous media. The Haskell matrix method has been the mathematical basis for many seismic techniques in exploration geophysics. In the method presented the medium is divided into layers and vertically averaged within each layer. The wave equation, within a layer, is then decoupled into an eigenvalue equation of the horizontal coordinate and a wave equation of the vertical coordinate. The eigen‐value equation is solved numerically. The vertical equation is solved analytically, once the eigenvalues are found. The solution throughout the medium is constructed by matching layer solutions at layer interfaces. The solution process of this method is “modular,” in the sense that each layer corresponds to an independent module and all the modules together form the final, total solution. Such a modular solution process has the following advantages. First, in a 2-D problem, for example, each module is a 1-D problem, which is a much simpler problem numerically than the original full 2-D problem. Second, the module solutions can be used repeatedly to form the solution corresponding to different problems. For example, in modeling. only those layers which differ between two models require recalculation. The solution to plane‐wave diffraction by a cylinder is obtained using this method, and it agrees well with the analytical solution.

A Perturbation Method for Low-Frequency Fluid-Structure Interaction Problems

1973 ◽  
Vol 40 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Y. K. Lou
Keyword(s):  
Wave Equation ◽  
Perturbation Method ◽  
Electromagnetic Scattering ◽  
Perturbation Methods ◽  
Separation Of Variables ◽  
Fluid Structure Interaction ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Fluid Structure ◽  
Structure Interaction

Perturbation methods have been used for electromagnetic scattering and diffraction problems in recent years. A similar method suitable for low-frequency fluid-structure interaction problems is presented. The essence of the method lies in the fact that approximate solutions for fluid-structure interaction problems can be obtained from a set of Poisson’s equations, rather than from the reduced wave equation. The method is particularly useful for those problems where the Poisson’s equation may be solved by the method of separation of variables while the reduced wave equation cannot. As an illustrative example, the vibrations of a submerged spherical shell is studied using the perturbation method and the accuracy of the method is demonstrated.

Separation of variables in the wave equation. Sets of the type (1.1) and Schr�dinger equation

1991 ◽  
Vol 34 (2) ◽  
pp. 122-126
Author(s):  
V. G. Bagrov ◽  
B. F. Samsonov ◽  
A. V. Shapovalov
Keyword(s):  
Wave Equation ◽  
Separation Of Variables ◽  
Dinger Equation
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