Two-dimensional scattering of a plane compressional wave by surface imperfections

1969 ◽  
Vol 59 (3) ◽  
pp. 1349-1364
Author(s):  
Ivor K. McIvor
Keyword(s):  
Order Approximation ◽  
Strong Dependence ◽  
Plane Waves ◽  
Compressional Wave ◽  
Elastic Half Space ◽  
Kernel Functions ◽  
The Body ◽  
Angle Of Incidence ◽  
Small Surface ◽  
Surface Imperfections

abstract A perturbation method for treating the scattering of plane waves by small surface imperfections on an elastic half space is presented. The solution to the first order approximation is given as convolution integrals of the surface imperfection with kernel functions defined by Fourier inversion integrals. The evaluation of these integrals is discussed and their asymptotic representations determined. The far field scattered displacements are explicitly obtained for arbitrary imperfections. The scattered field consists of a Rayleigh surface wave and four body phases which at the free surface travel with the speed of dilational or distortional waves. Numerical examples are given. In particular the error in the apparent angle of emergence due to the scattered waves is obtained. The body phases exhibit the familiar 3/2 geometric attenuation, but still may make a significant contribution at moderately long distances. A strong dependence of the magnitude of the error on the angle of incidence is demonstrated.

STUDIES ON SEISMIC WAVES: I. REFLECTION AND REFRACTION OF PLANE WAVES

Geophysics ◽  
1946 ◽  
Vol 11 (1) ◽  
pp. 1-9 ◽  
Author(s):  
C. Y. Fu
Keyword(s):  
Rayleigh Waves ◽  
Seismic Waves ◽  
Plane Waves ◽  
The Body ◽  
Body Waves ◽  
Angle Of Incidence ◽  
Apparent Velocity ◽  
Square Root ◽  
Absolute Value ◽  
Reflection And Refraction

By taking the apparent velocity along the boundary as the parameter instead of the angle of incidence, the equations for the different wave amplitudes may be put in more symmetrical forms. In this way, it is more convenient to discuss both the body waves and the Rayleigh waves at the same time. A difficulty in the plotting of the square root of the wave intensity against the angles is also discussed. When the reflection or refraction coefficient is not real, the meaning of the intensity, as obtained by squaring the absolute value of the latter quantity, needs clarification.

Propagation of SH-Waves Through Non Planer Interface between Visco-Elastic and Fibre-Reinforced Solid Half-Spaces

2017 ◽  
Vol 33 (4) ◽  
pp. 545-557
Author(s):  
B. Prasad ◽  
P. C. Pal ◽  
S. Kundu
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Seismic Waves ◽  
Order Approximation ◽  
Elastic Half Space ◽  
Angle Of Incidence ◽  
Sh Waves ◽  
Transmission Coefficients ◽  
Reflection And Refraction ◽  
Special Cases

AbstractIn the propagation of seismic waves through layered media, the boundaries play crucial role. The boundaries separating the different layers of the earth are irregular in nature and not perfectly plane. It is, therefore, necessary to take into account the corrugation of the boundaries while dealing with the problem of reflection and refraction of seismic waves. The present study explores the reflection and refraction phenomena of SH-waves at a corrugated interface between visco-elastic half-space and fibre-reinforced half-space. Method of approximation given by Rayleigh is adopted and the expressions for reflection and transmission coefficients are obtained in closed form for the first and second order approximation of the corrugation. The closed form formulae of these coefficients are presented for a corrugated interface of periodic shape (cosine law interface). It is found that these coefficients depend upon the amplitude of corrugation of the boundary, angle of incidence and frequency of the incident wave. Numerical computations for a particular type of corrugated interface are performed and a number of graphs are plotted. Some special cases are derived.

Reflection of Thermoelastic Waves From the Insulated Surface of a Solid Half-Space With Time-Delay

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
Nihar Sarkar ◽  
Soumen De ◽  
Narayan Das ◽  
Nantu Sarkar
Keyword(s):  
Free Boundary ◽  
Half Space ◽  
Plane Waves ◽  
Generalized Thermoelasticity ◽  
Elastic Half Space ◽  
Reflection Coefficients ◽  
Angle Of Incidence ◽  
Energy Ratios ◽  
Thermally Conducting ◽  
Stress Free Boundary

Abstract This paper is devoted to study the reflection of thermoelastic plane waves from the thermally insulated stress-free boundary of a homogeneous, isotropic and thermally conducting elastic half-space. A new linear theory of generalized thermoelasticity under heat transfer with memory-dependent derivative (MDD) is employed to address this study. It has been found that three basic waves consisting of two sets of coupled longitudinal waves and one independent vertically shear-type wave may travel with distinct phase speeds. The formulae for various reflection coefficients and their respective energy ratios are determined in case of an incident coupled longitudinal elastic wave at the thermally insulated stress-free boundary of the medium. The results for the reflection coefficients and their respective energy ratios for various values of the angle of incidence are computed numerically and presented graphically for copper-like material and discussed.

Reflection of plane waves from a thermo-microstretch elastic solid under the effect of rotation

2014 ◽  
Vol 92 (6) ◽  
pp. 488-496 ◽  
Author(s):  
Mohamed I.A. Othman ◽  
Ya Qin Song
Keyword(s):  
Harmonic Wave ◽  
Plane Waves ◽  
Elastic Solid ◽  
Elastic Half Space ◽  
Angle Of Incidence ◽  
Governing Equations ◽  
Plane Harmonic Wave ◽  
Continuous Boundary ◽  
Coupled Theory ◽  
Micropolar Material

A reflection of a plane harmonic wave at the interface of thermo-microstretch elastic half space is studied. The formulation is applied to generalized thermo-elasticity theories, the Lord–Şhulman and Green–Lindsay theories, as well as the classical dynamical coupled theory. Using potential function, the governing equations reduce to ten differential equations. Coefficient ratios of reflection of different waves with the angle of incidence are obtained using continuous boundary conditions. By numerical calculations, the variation of coefficient ratios of reflection with the angle of incidence is illustrated graphically in magnesium crystal micropolar material under three theories. Also the effect of frequency and rotation on the coefficient ratios of reflection is illustrated graphically in the context of Lord–Shulman theory.

Surface and interfacial anti-plane waves in micropolar solids with surface energy

2020 ◽  
pp. 108128652096564
Author(s):  
Mriganka Shekhar Chaki ◽  
Victor A Eremeyev ◽  
Abhishek K Singh
Keyword(s):  
Surface Wave ◽  
Plane Wave ◽  
Numerical Study ◽  
Plane Waves ◽  
Angular Frequency ◽  
Elastic Half Space ◽  
Surface Strain ◽  
Theoretical Frameworks ◽  
Algebraic Form ◽  
New Type

In this work, the propagation behaviour of a surface wave in a micropolar elastic half-space with surface strain and kinetic energies localized at the surface and the propagation behaviour of an interfacial anti-plane wave between two micropolar elastic half-spaces with interfacial strain and kinetic energies localized at the interface have been studied. The Gurtin–Murdoch model has been adopted for surface and interfacial elasticity. Dispersion equations for both models have been obtained in algebraic form for two types of anti-plane wave, i.e. a Love-type wave and a new type of surface wave (due to micropolarity). The angular frequency and phase velocity of anti-plane waves have been analysed through a numerical study within cut-off frequencies. The obtained results may find suitable applications in thin film technology, non-destructive analysis or biomechanics, where the models discussed here may serve as theoretical frameworks for similar types of phenomena.

Interaction of a shear wall with the soil for incident plane SH waves

1972 ◽  
Vol 62 (1) ◽  
pp. 63-83
Author(s):  
M. D. Trifunac
Keyword(s):  
Incidence Angle ◽  
Closed Form Solution ◽  
Shear Wall ◽  
Elastic Half Space ◽  
Form Solution ◽  
Angle Of Incidence ◽  
Sh Waves ◽  
Incident Plane ◽  
Interaction Equation ◽  
Plane Sh Waves

Abstract The closed-form solution of the dynamic interaction of a shear wall and the isotropic homogeneous and elastic half-space, previously studied only for vertically-incident SH waves, is generalized to any angle of incidence. It is shown that the interaction equation is independent of the incidence angle, while the surface-ground displacements heavily depend on it. For the two-dimensional model studied, it is demonstrated that disturbances generated by waves scattering and diffracting around the rigid foundation mass are not a local phenomenon but extend to large distances relative to the characteristic foundation length.

Contribution of Inertia on Venous Flow in the Lower Limb during Stationary Gait

2004 ◽  
Vol 20 (2) ◽  
pp. 115-128 ◽  
Author(s):  
Jean-Thomas Aubert ◽  
Christian Ribreau
Keyword(s):  
Blood Flow ◽  
Gait Cycle ◽  
Strong Dependence ◽  
Venous Blood ◽  
Kinematic Model ◽  
The Body ◽  
Axial Component ◽  
Venous Blood Flow ◽  
Collapsible Tubes ◽  
Body Forces

Blood flows toward the heart through collapsible vessels, the veins. The equations of flow in collapsible tubes in motion show a strong dependence on body forces resulting from gravity and acceleration. This paper analyzes the contribution of body forces to venous blood flow during walking on level ground. It combines the biomechanics of gait and theory of collapsible tubes to point out that body forces due to gravity and limb acceleration cannot be overlooked when considering the determinants of venous blood flow during locomotion. The study involved the development of a kinematic model of the limb as a multi-pendulum arrangement in which the limb segments undergo angular displacements. Angular velocities and accelerations were determined and the body forces were calculated during various phases of the gait cycle. A vascular model of the leg's major venous system was also constructed, and the accelerations due to body and gravity forces were calculated in specific venous segments, using the data from the kinematic model. The results showed there were large, fast variations in the axial component (Gx–Mx) of the body forces in veins between the hip and the ankle. Acceleration peaks down to –2G were obtained at normal locomotion. At fast locomotion, a distal vein in the shank displayed values of (Gx–Mx)/G equal to –3.2. Given the down-to-up orientation of the x-axis, the axial component Mx was usually positive in the axial veins, and Mx could shift from positive to negative during the gait cycle in the popliteal vein and the dorsal venous arch.

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 Reflection at Free Surface in Micropolar Thermoelastic Solid with two Temperatures

2013 ◽  
Vol 18 (1) ◽  
pp. 217-234 ◽  
Author(s):  
K. Sharma
Keyword(s):  
Half Space ◽  
Plane Surface ◽  
Plane Waves ◽  
Relaxation Times ◽  
Specific Model ◽  
Angle Of Incidence ◽  
Amplitude Ratios ◽  
Two Temperatures ◽  
Generalized Thermoelastic ◽  
Thermoelastic Solid

The present investigation is concerned with the effect of two temperatures on reflection coefficients in a micropolar thermoelastic solid half space. With two relaxation times, reflection of plane waves impinging obliquely at a plane interface of the micropolar generalized thermoelastic solid half space with two temperatures is investigated. The incident wave is assumed to be striking at the plane surface after propagating through the micropolar generalized thermoelastic solid with two temperatures. Amplitude ratios of the various reflected waves are obtained in closed form and it is found that these are functions of angle of incidence, frequency and are affected by the elastic properties of the media. The effect of two temperatures is shown on these amplitude ratios for a specific model.

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.

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