Reflection and Refraction of a Harmonic SH Wave at the Interface of Two Dissimilar Media With MicroHeterogeneity

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
Burhanettin S. Altan
Keyword(s):  
Incident Wave ◽  
Critical Angle ◽  
Harmonic Wave ◽  
Classical Case ◽  
Wave Number ◽  
Sh Wave ◽  
Sh Waves ◽  
Reflection And Refraction ◽  
Strain Relationship ◽  
The Media

Reflection and refraction of harmonic SH-waves from the interface of two dissimilar media with microheterogeneity is studied. The effect of the microheterogeneity on the overall behavior of the media is taken into account by adding higher order displacement gradients in the stress-strain relationship. It is found that a harmonic wave reflects back with the same angle of the incident wave, like in a classical case. However, it is found that the direction of propagation of the refracted wave is dependent on the wave number. It is also shown that the critical angle for which the incident wave cannot be transmitted to the other half plane is dependent on the wave number.

Reflection of SH waves from anisotropic transition layer

1974 ◽  
Vol 64 (6) ◽  
pp. 1979-1991 ◽  
Author(s):  
V. Thapliyal
Keyword(s):  
Reflection Coefficient ◽  
Incident Wave ◽  
Transition Layer ◽  
Seismic Energy ◽  
Considerable Effect ◽  
Anisotropy Factor ◽  
Sh Wave ◽  
Sh Waves ◽  
Obliquely Incident ◽  
Order Discontinuity

abstract The effects of anisotropy on the reflection of SH-waves (horizontally polarized shear waves) from a transition layer are studied. The transition layer is sand-wiched between two isotropic homogeneous half-spaces and is constituted by a medium which is both anisotropic and inhomogeneous. The SH-wave potentials are obtained for an anisotropic inhomogeneous medium in which both the anisotropy factor (ratio of the horizontal rigidity to the vertical rigidity) and vertical velocity vary with depth. An expression for the reflection coefficient of SH waves is obtained when the material mentioned above forms a finite transition zone between two isotropic homogeneous half-spaces. For further generalization, a second-order discontinuity along with the first-order on eis being assumed in the material properties, at the boundaries of the transition layer. The mathematical and numerical analyses show that the anisotropy factor, found at the top of the transition layer (N0/M0) produces considerable effect on the reflection coefficient for an obliquely incident SH wave. It has been noted that the greater the thickness of the transition layer, the greater is the dependence of the reflection coefficient upon the value of the anisotropy (N0/M0). The minima and maxima of the reflection of seismic energy are found dependent on the value of anisotropy. For greater values of the anisotropy, these maxima and minima shift toward the lower values of the wavelength of the propagating wave (or toward the higher values of the thickness of the transition layer). In fact, the values of the reflection coefficient at which these maxima and minima of seismic energy occur are found greater for the higher values of anisotropy. The effects of anisotropy are found more pronounced for the larger angles of incidence. This remains so until the angle of refraction becomes imaginary. However, no effects of the anistropy factor are found on the reflection coefficients for a normally incident wave. The results, mentioned above, are therefore discussed only for the obliquely incident wave. A geophysically interesting situation has been chosen for studying, quantitatively, the effects of the anisotropy factor on the reflection of SH waves.

SH-wave velocity in a fiber-reinforced anisotropic layer overlying a gravitational heterogeneous half-space

2015 ◽  
Vol 11 (3) ◽  
pp. 386-400 ◽  
Author(s):  
Rajneesh Kakar
Keyword(s):  
Dispersion Equation ◽  
Half Space ◽  
Love Wave ◽  
Elastic Half Space ◽  
Wave Number ◽  
Fiber Reinforced ◽  
Sh Wave ◽  
Dimensionless Wave Number ◽  
Sh Waves ◽  
Content Type

Purpose – The purpose of this paper is to investigate the existence of SH-waves in fiber-reinforced layer placed over a heterogeneous elastic half-space. Design/methodology/approach – The heterogeneity of the elastic half-space is caused by the exponential variations of density and rigidity. As a special case when both the layers are homogeneous, the derived equation is in agreement with the general equation of Love wave. Findings – Numerically, it is observed that the velocity of SH-waves decreases with the increase of heterogeneity and reinforced parameters. The dimensionless phase velocity of SH-waves increases with the decreases of dimensionless wave number and shown through figures. Originality/value – In this work, SH-wave in a fiber-reinforced anisotropic medium overlying a heterogeneous gravitational half-space has been investigated analytically and numerically. The dispersion equation for the propagation of SH-waves has been observed in terms of Whittaker function and its derivative of second degree order. It has been observed that on the removal of heterogeneity of half-space, and reinforced parameters of the layer, the derived dispersion equation reduces to Love wave dispersion equation thereby validates the solution of the problem. The equation of propagation of Love wave in fiber-reinforced medium over a heterogeneous half-space given by relevant authors is also reduced from the obtained dispersion relation under the considered geometry.

An approximate treatment of high-frequency scattering

1957 ◽  
Vol 53 (3) ◽  
pp. 691-701 ◽  
Author(s):  
D. S. Jones ◽  
G. B. Whitham
Keyword(s):  
Approximate Method ◽  
Energy Flux ◽  
Incident Wave ◽  
High Frequency ◽  
Harmonic Wave ◽  
Incident Beam ◽  
Scattered Wave ◽  
The Body ◽  
Scattering Coefficient ◽  
Wave Number

In some recent work Kodis* considers the scattering of a plane harmonic wave by a circular cylinder and uses variational methods to deduce an asymptotic formula for the scattering coefficient in the case of high-frequency waves. The scattering coefficient, σ, is denned as the total energy flux outward from the cylinder in the scattered wave divided by the energy flux in the beam of the incident wave which falls on the cylinder. In the limit, of geometrical optics, the scattered wave is made up of a wave reflected from the forward half of the cylinder with energy flux equal to that in the incident beam, and a wave behind the cylinder cancelling the incident wave there to form the shadow. Thus σ → 2 as ka → ∞, k being the wave number of the incident wave and a the radius of the cylinder. The next term in the asymptotic expansion for σ is proportional to . It has been determined already from the exact series expression for the field by White (6), Wu and Rubinow (7) and Kear (3). But Kodis gives an approximate method which also supplies this result and, since the coefficient is in good agreement with the values obtained from the exact solution, concludes that his approximate procedure could also be used for more general obstacles. In the present paper, we propose an alternative approximate method which is related to the one given by Kodis but is very much simpler. Also we go on to obtain the correction terms for general obstacles in detail. Various writers (see, for example, Keller (4) and van de Hulst(8)) have suggested previously that the correction is still proportional to , where a is some length defined by the body shape. However, the determination of a and of the numerical coefficient has been limited to two-dimensional obstacles.

Multiple reflection of plane SH waves by a dipping layer

1970 ◽  
Vol 60 (1) ◽  
pp. 15-28
Author(s):  
Hiroshi Ishii ◽  
Robert M. Ellis
Keyword(s):  
Multiple Reflection ◽  
Sh Wave ◽  
Constant Depth ◽  
Sh Waves ◽  
Phase Velocities ◽  
The Media ◽  
Plane Sh Wave ◽  
Plane Sh Waves ◽  
Amplitude Versus ◽  
Frequency Curves

abstract The problem of a plane SH wave incident at the base of a dipping layer has been considered. A solution by multiple reflection for waves incident at any angle perpendicular to strike has been obtained for the amplitude, phase and phase velocity. Numerical values are calculated on the surface. For waves propagating in the up-dip direction the amplitude versus frequency curves for a constant depth to the interface change slowly with increasing dip for dip angles less than 20°. However, for waves propagating in the down-dip direction, the character of the amplitude curves changes rapidly. It is found in these cases that the diffracted wave plays an important role. In addition to satisfying the boundary conditions at the surface and between the media, the diffracted wave must also satisfy additional conditions along a dipping surface lying between the free surface and the boundary between the two media due to the geometrical nature of the reflected wave solution. It is found that the phase velocities vary rapidly with both period of the wave and depth to the interface, particularly for propagation directions close to an angle which corresponds to vertical incidence in the horizontally layered case.

Impacts on the Propagation of SH-Waves in a Heterogeneous Viscoelastic Layer Sandwiched between an Anisotropic Porous Layer and an Initially Stressed Isotropic Half Space

2016 ◽  
Vol 33 (1) ◽  
pp. 13-22 ◽  
Author(s):  
S. Kundu ◽  
P. Alam ◽  
S. Gupta ◽  
D. Kr. Pandit
Keyword(s):  
Wave Propagation ◽  
Dispersion Relation ◽  
Half Space ◽  
Porous Layer ◽  
Separation Of Variables ◽  
Finite Thickness ◽  
Wave Number ◽  
Viscoelastic Layer ◽  
Sh Wave ◽  
Sh Waves

AbstractThe present study deals with the affected behaviour of SH-wave propagation through a viscoelastic layer sandwiched between an anisotropic porous layer of finite thickness and an isotropic half space. The sandwiched viscoelastic layer is considered as heterogeneous medium of finite thickness and isotropic half-space is considered as initially stressed medium. The method of separation of variables has been applied to obtain the dispersion equation of SH-wave in their respective media. The obtained complex dispersion relation has been separated into real and imaginary parts. Moreover, the dispersion relation has been satisfied with the classical condition of Love waves. The effects of heterogeneity, attenuation constant, dissipation factor of viscoelasticity, initial stress (compressive), thickness ratio of two layers and porosity on the propagation of SH-waves have been shown by number of graphs. Graphs have been plotted for the dimensionless phase and damping velocity on the propagation of SH-waves with respect to the dimensionless real wave number. The results may be useful to explore the nature and peculiarity of SH-wave propagation in the viscoelastic structure.

Scattering of SH Wave on Periodic Cracks in an Infinite Plane of Piezoelectric/Piezomagnic Composite Materials

2012 ◽  
Vol 166-169 ◽  
pp. 3364-3368
Author(s):  
Wei Shi ◽  
Li Xia Ma
Keyword(s):  
Stress Intensity Factor ◽  
Integral Equation ◽  
Composite Materials ◽  
Piezoelectric Materials ◽  
Infinite Plane ◽  
Sh Wave ◽  
Scattering Problems ◽  
Sh Waves ◽  
Periodic Cracks ◽  
The Fourier Transform

In this paper, the scattering problems of SH waves on periodic cracks in an infinite of piezoelectric/piezomagnic composite materials bonded to an infinite of homogeneous piezoelectric materials is investigated, the Fourier transform techniques are used to reduce the problem to the solution of Hilbert singular integral equation, the latter is solved by Lobotto-Chebyshev and Gauss integral equation, at last, numerical results showed the effect of the frequency of wave, sizes and so on upon the normalized stress intensity factor.

Theory of Short Wave Excitation

1986 ◽  
Vol 30 (03) ◽  
pp. 147-152
Author(s):  
Yong Kwun Chung
Keyword(s):  
Incident Wave ◽  
Characteristic Length ◽  
Finite Depth ◽  
Short Wave ◽  
The Body ◽  
Wave Number ◽  
Floating Body ◽  
Wave Excitation ◽  
Exciting Forces ◽  
Wave Exciting Forces

When the wavelength of the incident wave is short, the total surface potential on a floating body is found to be 2∅ i & O (m-l∅ i) on the lit surface and O (m-l∅ j) on the shadow surface where ~b i is the potential of the incident wave and m the wave number in water of finite depth. The present approximation for wave exciting forces and moments is reasonably good up to X/L ∅ 1 where h is the wavelength and L the characteristic length of the body.

scholarly journals Novel internal measurements of ion cyclotron frequency range fast-ion driven modes

2021 ◽  
Author(s):  
Neal A Crocker ◽  
Shawn X Tang ◽  
Kathreen E Thome ◽  
Jeff Lestz ◽  
Elena Belova ◽  
...  
Keyword(s):  
Cyclotron Frequency ◽  
Harmonic Wave ◽  
Wave Number ◽  
Frequency Range ◽  
High Wave Number ◽  
Minor Radius ◽  
Ion Cyclotron ◽  
Cyclotron Emission ◽  
Edge Localized Modes ◽  
Fast Ion

Abstract Novel internal measurements and analysis of ion cyclotron frequency range fast-ion driven modes in DIII-D are presented. Observations, including internal density fluctuation (ñ) measurements obtained via Doppler Backscattering, are presented for modes at low harmonics of the ion cyclotron frequency localized in the edge. The measurements indicate that these waves, identified as coherent Ion Cyclotron Emission (ICE), have high wave number, _⊥ρ_fast ≳ 1, consistent with the cyclotron harmonic wave branch of the magnetoacoustic cyclotron instability (MCI), or electrostatic instability mechanisms. Measurements show extended spatial structure (at least ~ 1/6 the minor radius). These edge ICE modes undergo amplitude modulation correlated with edge localized modes (ELM) that is qualitatively consistent with expectations for ELM-induced fast-ion transport.

scholarly journals Boussinesq Model and CFD Simulations of Non-Linear Wave Diffraction by a Floating Vertical Cylinder

2020 ◽  
Vol 8 (8) ◽  
pp. 575
Author(s):  
Sarat Chandra Mohapatra ◽  
Hafizul Islam ◽  
C. Guedes Soares
Keyword(s):  
Wave Diffraction ◽  
Perturbation Technique ◽  
Harmonic Wave ◽  
Vertical Cylinder ◽  
Numerical Data ◽  
Second Order ◽  
Wave Number ◽  
Linear Wave ◽  
Cfd Simulations ◽  
Boussinesq Model

A mathematical model for the problem of wave diffraction by a floating fixed truncated vertical cylinder is formulated based on Boussinesq equations (BEs). Using Bessel functions in the velocity potentials, the mathematical problem is solved for second-order wave amplitudes by applying a perturbation technique and matching conditions. On the other hand, computational fluid dynamics (CFD) simulation results of normalized free surface elevations and wave heights are compared against experimental fluid data (EFD) and numerical data available in the literature. In order to check the fidelity and accuracy of the Boussinesq model (BM), the results of the second-order super-harmonic wave amplitude around the vertical cylinder are compared with CFD results. The comparison shows a good level of agreement between Boussinesq, CFD, EFD, and numerical data. In addition, wave forces and moments acting on the cylinder and the pressure distribution around the vertical cylinder are analyzed from CFD simulations. Based on analytical solutions, the effects of radius, wave number, water depth, and depth parameters at specific elevations on the second-order sub-harmonic wave amplitudes are analyzed.

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