Reflection Coefficient for Plane Waves in a Fluid Incident on a Layered Elastic Half-Space

1983 ◽  
Vol 50 (2) ◽  
pp. 405-414 ◽  
Author(s):  
D. B. Bogy ◽  
S. M. Gracewski
Keyword(s):  
Reflection Coefficient ◽  
Half Space ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Wave Number ◽  
Solid Boundary ◽  
Inviscid Fluid ◽  
Interface Waves ◽  
Plate Models ◽  
Layered Solid

The reflection coefficient is derived for an isotropic, homogeneous elastic layer of arbitrary thickness that is perfectly bonded to such an elastic half-space of a different material for the case when plane waves are incident from an inviscid fluid onto the layered solid. The derived function is studied analytically by considering several limiting cases of geometry and materials to recover previously known results. Approximate reflection coefficents are then derived using various plate models for the layer to obtain simpler expressions that are useful for small values of σd, where σ is the wave number and d is the layer thickness. Numerical results based on all the models for the propagation of interface waves localized near the fluid-solid boundary are obtained and compared. These results are also compared with some previously published experimental measurements.

scholarly journals Dynamic Stress and Displacement in an Elastic Half-Space with a Cylindrical Cavity

2011 ◽  
Vol 18 (6) ◽  
pp. 827-838 ◽  
Author(s):  
İ. Coşkun ◽  
H. Engin ◽  
A. Özmutlu
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Cylindrical Cavity ◽  
Circular Cross Section ◽  
Elastic Half Space ◽  
Least Square ◽  
Polar Coordinates ◽  
Wave Number ◽  
Forcing Function

The dynamic response of an elastic half-space with a cylindrical cavity in a circular cross-section is analyzed. The cavity is assumed to be infinitely long, lying parallel to the plane-free surface of the medium at a finite depth and subjected to a uniformly distributed harmonic pressure at the inner surface. The problem considered is one of plain strain, in which it is assumed that the geometry and material properties of the medium and the forcing function are constant along the axis of the cavity. The equations of motion are reduced to two wave equations in polar coordinates with the use of Helmholtz potentials. The method of wave function expansion is used to construct the displacement fields in terms of the potentials. The boundary conditions at the surface of the cavity are satisfied exactly, and they are satisfied approximately at the free surface of the half-space. Thus, the unknown coefficients in the expansions are obtained from the treatment of boundary conditions using a collocation least-square scheme. Numerical results, which are presented in the figures, show that the wave number (i.e., the frequency) and depth of the cavity significantly affect the displacement and stress.

The Transition Matrix Formalism for the Scattering of an Alluvium on an Elastic Half-Space

2007 ◽  
Author(s):  
Wen-I Liao ◽  
Tsung-Jen Teng ◽  
Shiang-Jung Wang
Keyword(s):  
Half Space ◽  
Transition Matrix ◽  
Plane Waves ◽  
Three Dimensional ◽  
Scattering Matrix ◽  
Elastic Half Space ◽  
Basis Functions ◽  
Matrix Formalism ◽  
T Matrix ◽  
Singular Wave

This paper develops the transition matrix formalism for scattering from an three-dimensional alluvium on an elastic half-space. Betti’s third identity is employed to establish orthogonality conditions among basis functions that are Lamb’s singular wave functions. The total displacements and associated tractions exterior and interior to the surface are expanded in a Rayleigh series. The boundary conditions are applied and the T-matrix is derived. A linear transformation is utilized to construct a set of orthogonal basis functions. The transformed T-matrix is related to the scattering matrix and it is shown that the scattering matrix is symmetric and unitary and that the T-matrix is symmetric. Typical numerical results obtained by incident plane waves for verification are presented.

scholarly journals Plane waves in thermoelasticity with one relaxation time

2001 ◽  
Vol 26 (4) ◽  
pp. 225-232
Author(s):  
Jun Wang ◽  
Wen Dong Chang
Keyword(s):  
Relaxation Time ◽  
General Solution ◽  
Laplace Transform ◽  
Half Space ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Second Sound ◽  
Boundary Temperature ◽  
Explicit Expressions

We apply the thermoelastic equations with one relaxation time developed by Lord and Shulman (1967) to solve some elastic half-space problems. Laplace transform is used to find the general solution. Problems concerning the ramp-type increase in boundary temperature and stress are studied in detail. Explicit expressions for temperature and stress are obtained for small values of time, where second sound phenomena are of relevance. Numerical values of stress and temperature are calculated and displayed graphically.

Exact Transient Response of an Elastic Half Space Loaded Over a Rectangular Region of Its Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 516-522 ◽  
Author(s):  
F. R. Norwood
Keyword(s):  
Half Space ◽  
Real Axis ◽  
Transient Response ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Impulsive Loading ◽  
Rectangular Region ◽  
The Real ◽  
Algebraic Expressions ◽  
Straight Lines

The response of an elastic half space to a normal impulsive loading over one half and also over one quarter of its bounding surface is considered. By a simple superposition the solution is obtained for a half space loaded on a finite rectangular region. In each case the solution was found to be a superposition of plane waves directly under the load, plus waves emanating from bounding straight lines and the corners of the loaded region. The solution was found by Cagniard’s technique and by extending the real transformation of de Hoop to double Fourier integrals with singularities on the real axis of the transform variables. Velocities in the interior of the half space are given for arbitrary values of Poisson’s ratio in terms of single integrals and algebraic expressions.

Dynamic response of circular cavity and crack in anisotropic elastic half-space by out-plane waves

2018 ◽  
Vol 91 ◽  
pp. 100-106 ◽  
Author(s):  
Huanan XU ◽  
Jianwei Zhang ◽  
Zailin Yang ◽  
Guoguan Lan ◽  
Qingyun Huang
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Circular Cavity ◽  
Anisotropic Elastic

Refraction of P- and S-Wave at the Interface of Micropolar Elasticity and Thermoelasticity with Voids

2018 ◽  
Vol 06 (03n04) ◽  
pp. 1850005
Author(s):  
R. Lianngenga ◽  
J. Lalvohbika ◽  
Lalawmpuia
Keyword(s):  
Thermal Expansion ◽  
Half Space ◽  
Plane Waves ◽  
Linear Thermal Expansion ◽  
Elastic Half Space ◽  
Micropolar Elasticity ◽  
S Wave ◽  
Incident Plane ◽  
Energy Ratios ◽  
Refracted Waves

The problem of incident plane waves at the interface of micropolar thermoelastic half-space with voids and micropolar elastic half-space with voids has been attempted. The amplitude and energy ratios of various reflected and refracted waves for the incident [Formula: see text]- and [Formula: see text]-waves are obtained with the help of appropriate boundary conditions at the interface. The effect of linear thermal expansion and microinertia on the amplitude and energy ratios due to the incident [Formula: see text]- and [Formula: see text]-waves are discussed. Numerically and analytically, these amplitude and energy ratios are computed to show the effect of linear thermal expansion and microinertia. It is observed that the effect of linear thermal expansion is less for incident [Formula: see text]-wave and the effect of microinertia is less for incident [Formula: see text]-wave.

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