Etude numérique des modes et fréquences propres d'une cavité à l'aide de la fonction de Green

1979 ◽  
Vol 57 (2) ◽  
pp. 208-217
Author(s):  
Jacques A. Imbeau ◽  
Byron T. Darling
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Green's Function ◽  
Normal Modes ◽  
Preceding Paper ◽  
Double Precision ◽  
Prolate Spheroidal ◽  
Normal Functions ◽  
Normal Frequency ◽  
High Degree

We apply the methods developed in our preceding paper (J. A. Imbeau and B. T. Darling. Can. J. Phys. 57, 190(1979)) for calculating the Green's function of a cavity to obtain the normal modes and normal frequencies of the cavity. As the frequency of the driving point source approaches that of a normal frequency the response (Green's function) of the cavity becomes infinite, and the form of the Green's function is dominated by the normal mode. There is also a 180° reversal of phase in passing through a resonance. In this way, we are able to calculate the normal frequencies of prolate spheroidal cavities to the full precision employed in the calculations (16 significant digits for double precision of the IBM-370). The Green's functions and the normal functions are also obtainable to a high degree of precision, except in the immediate vicinity of the surface of the cavity where they suffer a well-known discontinuity.

Impact of point source on fissured poroelastic medium: Green’s function approach

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Shishir Gupta ◽  
Soumik Das ◽  
Rachaita Dutta
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Half Space ◽  
Porous Layer ◽  
Green's Function ◽  
Anisotropic Fluid ◽  
Complex Frequency ◽  
Sh Wave ◽  
Content Type ◽  
Porous Half Space

Purpose The purpose of this paper is to investigate the mathematical model comprising a heterogeneous fluid-saturated fissured porous layer overlying a non-homogeneous anisotropic fluid-saturated porous half-space without fissures. The influence of point source on horizontally polarized shear-wave (SH-wave) propagation has been studied intensely. Design/methodology/approach Techniques of Green’s function and Fourier transform are applied to acquire displacement components, and with the help of boundary conditions, complex frequency equation has been constructed. Findings Complex frequency relation leads to two distinct equations featuring dispersion and attenuation properties of SH-wave in a heterogeneous fissured porous medium. Using MATHEMATICA software, dispersion and damping curves are sketched to disclose the effects of heterogeneity parameters associated with both media, parameters related to rigidity and density of single porous half-space, attenuation coefficient, wave velocity, total porosity, volume fraction of fissures and anisotropy. The fact of obtaining classical Love wave equation by introducing several particular conditions establishes the validation of the considered model. Originality/value To the best of the authors’ knowledge, effect of point source on SH-wave propagating in porous layer containing macro as well as micro porosity is not analysed so far, although theory of fissured poroelasticity itself has vast applications in real life and impact of point source not only enhances the importance of fissured porous materials but also opens a new area for future research.

Etude numérique de la fonction de Green scalaire d'une cavité à l'aide d'une nouvelle équation intégrale

1979 ◽  
Vol 57 (2) ◽  
pp. 190-207
Author(s):  
Jacques A. Imbeau ◽  
Byron T. Darling
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Double Precision ◽  
Integration Formula ◽  
Algebraic Equations ◽  
Equation Intégrale ◽  
Exterior Region ◽  
Von Neumann ◽  
Linear Algebraic Equations

We study numerically, with the aid of an IBM-370 computer, the Green's functions of a cavity afforded by the solutions of a new integral equation (B. T. Darling and J. A. Imbeau. Can. J. Phys. 56, 387 (1978)). A number of prolate spheroidal cavities whose eccentricities cover the complete range zero to one are employed, and the solutions are subject to the Dirichlet and von Neumann conditions at the surface. We use the Gauss–Legendre integration formula to replace the integral equation by a set of linear algebraic equations. The Green's function is evaluated by substituting the solution of this set in the formula of Helmholtz, using the same integration formula. Criteria for the optimization of this procedure also are developed and employed. The Green's function can be determined to high precision except in the immediate vicinity of the surface of the cavity where it suffers a well-known discontinuity. We also explore the use of the Helmholtz formula itself in the exterior region as an integral equation to obtain the Green's function of the cavity. We find that although the precision of the solution is much less than that afforded by the precedingly mentioned integral equation the precision is still within the range of practical application. All calculations used double precision arithmetic (16 significant digits on the IBM-370).

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