On the scattering of short surface waves by a finite dock

1968 ◽  
Vol 64 (4) ◽  
pp. 1109-1129 ◽  
Author(s):  
F. G. Leppington
Keyword(s):  
Wave Train ◽  
Normal Incidence ◽  
Great Depth ◽  
Travelling Wave ◽  
Reflection And Transmission ◽  
Large Wave ◽  
Transmission Coefficients ◽  
Weakly Coupled ◽  
Wave Trains ◽  
Induced Velocity

AbstractA sinusoidal travelling wave-train is at normal incidence upon a two-dimensional finite dock fixed on the surface of a body of water of great depth, and the problem investigated herein is that of finding the limiting form of the induced velocity potential for short waves. Of particular interest are the amplitudes of the wave-trains reflected and transmitted towards infinity by such an obstacle. The potential is expressed as a sum of coupled semi-infinite dock potentials, whence results a pair of weakly coupled integral equations for the solution. This formulation of the problem is shown to be amenable to an approximate solution for large wave-numbers, and the first few terms are derived in formal expansions for the reflection and transmission coefficients.

scholarly journals SWRT: A package for semi-analytical solutions of surface wave propagation, including mode conversion, across transversely aligned vertical discontinuities

2018 ◽  
Vol 7 (1) ◽  
pp. 101-112 ◽  
Author(s):  
Arjun Datta
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Mode Conversion ◽  
Normal Incidence ◽  
Seismic Wave Propagation ◽  
Reflection And Transmission ◽  
Computing Power ◽  
Transmission Coefficients ◽  
Surface Wave Propagation ◽  
Core Functionality

Abstract. We present a suite of programs that implement decades-old algorithms for computation of seismic surface wave reflection and transmission coefficients at a welded contact between two laterally homogeneous quarter-spaces. For Love as well as Rayleigh waves, the algorithms are shown to be capable of modelling multiple mode conversions at a lateral discontinuity, which was not shown in the original publications or in the subsequent literature. Only normal incidence at a lateral boundary is considered so there is no Love–Rayleigh coupling, but incidence of any mode and coupling to any (other) mode can be handled. The code is written in Python and makes use of SciPy's Simpson's rule integrator and NumPy's linear algebra solver for its core functionality. Transmission-side results from this code are found to be in good agreement with those from finite-difference simulations. In today's research environment of extensive computing power, the coded algorithms are arguably redundant but SWRT can be used as a valuable testing tool for the ever evolving numerical solvers of seismic wave propagation. SWRT is available via GitHub (https://github.com/arjundatta23/SWRT.git).

Amplitude‐variation‐with‐angle behavior of self‐similar interfaces

Geophysics ◽  
1999 ◽  
Vol 64 (6) ◽  
pp. 1928-1938 ◽  
Author(s):  
Kees Wapenaar
Keyword(s):  
Normal Incidence ◽  
Reflection And Transmission Coefficients ◽  
Well Logs ◽  
Amplitude Variation ◽  
Reflection And Transmission ◽  
Velocity Function ◽  
Transmission Coefficients ◽  
Step Functions ◽  
Self Similar ◽  
Amplitude Variation With Angle

Amplitude‐variation‐with‐angle (AVA) analysis is generally based on the assumption that the medium parameters behave as step functions of the depth coordinate z, at least in a finite region around the interface. However, outliers observed in well logs often behave quite differently from step functions. In this paper, outliers in the acoustic propagation velocity are parameterized by functions of the form [Formula: see text]. The wavelet transform of this function reveals properties similar to those of several outliers in real well logs. Moreover, this function is self‐similar, according to [Formula: see text], for β > 0. Analytical expressions are derived for the acoustic normal incidence reflection and transmission coefficients for this type of velocity function. For oblique incidence, no explicit solutions are available. However, by exploiting the self‐similarity property of the velocity function, it turns out that the acoustic angle‐dependent and frequency‐dependent reflection and transmission coefficients are self‐similar as well. To be more specific, these coefficients appear to be constant along curves described by [Formula: see text], where p is the raypath parameter and ω the angular frequency. The singularity exponent α that is reflected in these curves may prove to be a useful indicator in seismic characterization.

scholarly journals SWRT: A package for semi-analytical solutions of surface wave propagation, including mode conversion, across transversely aligned vertical discontinuities

2017 ◽  
Author(s):  
Arjun Datta
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Mode Conversion ◽  
Normal Incidence ◽  
Seismic Wave Propagation ◽  
Reflection And Transmission ◽  
Computing Power ◽  
Transmission Coefficients ◽  
Surface Wave Propagation ◽  
Core Functionality

Abstract. We present a suite of programs that implement decades-old algorithms for computation of seismic surface wave reflection and transmission coefficients at a welded contact between two laterally homogeneous quarter-spaces. For Love as well as Rayleigh waves, the algorithms are shown to be capable of modelling multiple mode conversions at a lateral discontinuity, which was not shown in the original publications or in the subsequent literature. Only normal incidence at a lateral boundary is considered so there is no Love-Rayleigh coupling, but incidence of any mode and coupling to any (other) mode can be handled. The code is written in Python and makes use of SciPy's Simpson's rule integrator and NumPy's linear algebra solver for its core functionality. Transmission side results from this code are found to be in good agreement with those from finite difference simulations. In today's research environment of extensive computing power, the coded algorithms are arguably redundant but SWRT can be used as a valuable testing tool for the ever evolving numerical solvers of seismic wave propagation. SWRT is available via GitHub (https://github.com/arjundatta23/sw_reftrans.git).

scholarly journals Oblique Water Wave Diffraction by a Step

2017 ◽  
Vol 22 (1) ◽  
pp. 35-47 ◽  
Author(s):  
P. Dolai
Keyword(s):  
Surface Water ◽  
Water Wave ◽  
Numerical Estimate ◽  
Wave Train ◽  
Reflection And Transmission Coefficients ◽  
Physical Parameters ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Galerkin Approximations ◽  
Finite Step

Abstract This paper is concerned with the problem of diffraction of an obliquely incident surface water wave train on an obstacle in the form of a finite step. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials are utilized to obtain a very accurate numerical estimate for reflection and transmission coefficients which are depicted graphically. From these figures various interesting results are discussed.

OBLIQUE WAVE SCATTERING BY A RECTANGULAR SUBMARINE TRENCH

2015 ◽  
Vol 56 (3) ◽  
pp. 286-298 ◽  
Author(s):  
RUMPA CHAKRABORTY ◽  
B. N. MANDAL
Keyword(s):  
Water Waves ◽  
Wave Scattering ◽  
Wave Train ◽  
Galerkin Approximation ◽  
Central Line ◽  
Wave Number ◽  
Reflection And Transmission ◽  
Oblique Wave ◽  
Transmission Coefficients ◽  
Linearized Theory

The problem of oblique wave scattering by a rectangular submarine trench is investigated assuming a linearized theory of water waves. Due to the geometrical symmetry of the rectangular trench about the central line $x=0$, the boundary value problem is split into two separate problems involving the symmetric and antisymmetric potential functions. A multi-term Galerkin approximation involving ultra-spherical Gegenbauer polynomials is employed to solve the first-kind integral equations arising in the mathematical analysis of the problem. The reflection and transmission coefficients are computed numerically for various values of different parameters and different angles of incidence of the wave train. The coefficients are depicted graphically against the wave number for different situations. Some curves for these coefficients available in the literature and obtained by different methods are recovered.

scholarly journals Scattering of Oblique Water Waves by an Infinite Step

2018 ◽  
Vol 23 (2) ◽  
pp. 327-338
Author(s):  
P. Dolai ◽  
D.P. Dolai
Keyword(s):  
Surface Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Train ◽  
Reflection And Transmission Coefficients ◽  
Physical Parameters ◽  
Reflection And Transmission ◽  
Infinite Depth ◽  
Transmission Coefficients ◽  
Galerkin Approximations

AbstractThe present paper is concerned with the problem of scattering of obliquely incident surface water wave train passing over a step bottom between the regions of finite and infinite depth. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultra spherical Gegenbauer polynomials are utilized to obtain very accurate numerical estimates for reflection and transmission coefficients. The numerical results are illustrated in tables.

Acoustic waves II

2004 ◽  
Author(s):  
Robert E. Newnham
Keyword(s):  
Acoustic Impedance ◽  
Acoustic Waves ◽  
Incident Beam ◽  
Normal Incidence ◽  
Reflection And Transmission ◽  
Reflected Waves ◽  
Transmission Coefficients ◽  
Plane Wave Incident ◽  
Incident Waves ◽  
Acoustic Losses

Acoustic impedance, acoustic losses, acoustic waves in piezoelectric solids, and surface waves are discussed in this chapter, along with a number of nonlinear acoustic phenomena. The reflection and transmission of acoustic waves across a boundary is governed by acoustic impedance. One of the most important boundary value problems in acoustics concerns a plane wave incident upon a planar surface, dividing one medium from another. In the general case of an anisotropic medium, the incident beam consists of three waves (one quasilongitudinal, two quasitransverse), each traveling at a different velocity. Each of the three incident waves will be refracted and reflected at the boundary. If the second medium is also anisotropic, each incident wave will generate three reflected waves and three refracted waves, a total of 27 waves in all. Wave propagation in a polycrystalline solid where there are many grain boundaries becomes very complicated. The simpler case of a pure longitudinally-polarized wave at normal incidence to the boundary provides insight into the more general problem. In this case the reflection and transmission coefficients are governed by the relatively simple acoustic impedance parameter (ρc)1/2 = ρv, where ρ is the density, c the stiffness coefficient, and v the phase velocity. The reflection coefficient R at the interface between medium I and medium II is The MKS unit for acoustic impedance is the Rayl (=kg/m2 s). Atypical value for a solid is about 107 rayls. In many acoustic applications it is desirable to reduce reflection by matching the acoustic impedance of the two media. Lithium tantalate transducers are well-matched to iron, for example. Sound transmission from the transducer to the medium can be enhanced with composite materials or with graded coupling layers. Backing materials are often selected to promote reflection. In this case acoustic impedances are mismatched. Tungsten and air are two commonly used backing materials. In an isotropic material the acoustic impedance is (ρc11)1/2 for longitudinal waves and (ρc44)1/2 for shear waves. For anisotropic materials the wave velocities and acoustic impedance change with direction as indicated earlier.

scholarly journals Approximate reflection coefficients for a thin VTI layer

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. C1-C11 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Normal Incidence ◽  
Reflection And Transmission Coefficients ◽  
Reflection Coefficients ◽  
Angle Of Incidence ◽  
Isotropic Layer ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
P And Sv Waves

We have developed an approximate method to derive simple expressions for the reflection coefficients of P- and SV-waves for a thin transversely isotropic layer with a vertical symmetry axis (VTI) embedded in a homogeneous VTI background. The layer thickness is assumed to be much smaller than the wavelengths of P- and SV-waves inside. The exact reflection and transmission coefficients are derived by the propagator matrix method. In the case of normal incidence, the exact reflection and transmission coefficients are expressed in terms of the impedances of vertically propagating P- and S-waves. For subcritical incidence, the approximate reflection coefficients are expressed in terms of the contrast in the VTI parameters between the layer and the background. Numerical examples are designed to analyze the reflection coefficients at normal and oblique incidence and investigate the influence of transverse isotropy on the reflection coefficients. Despite giving numerical errors, the approximate formulas are sufficiently simple to qualitatively analyze the variation of the reflection coefficients with the angle of incidence.

Free Space Microwave Characterization of Silicon Wafers for Microelectronic Applications

2005 ◽  
Vol 2 (2) ◽  
pp. 35
Author(s):  
Zaiki Awang ◽  
Deepak Kumar Ghodgaonkar ◽  
Noor Hasimah Baba
Keyword(s):  
Complex Permittivity ◽  
Free Space ◽  
Normal Incidence ◽  
Silicon Wafers ◽  
Reflection And Transmission Coefficients ◽  
Frequency Range ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Quarter Wave ◽  
Mode Transitions

A contactless and non-destructive microwave method has been developed to characterize silicon semiconductor wafers from reflection and transmission measurements made at normal incidence using MNDT. The measurement system consists of a pair of spot-focusing horn lens antenna, mode transitions, coaxial cables and a vector network analyzer (VNA). In this method, the free-space reflection and transmission coefficients, S11 and S21 are measured for silicon wafers sandwiched between two Teflon plates of 5mm thickness which act as a quarter-wave transformer at mid-band. The actual reflection and transmission coefficients, S11 and S21 of the silicon wafers are then calculated from the measured S11 and S21 using ABCD matrix transformation in which the complex permittivity and thickness of the Teflon plates are known. From the complex permittivity, the resistivity and conductivity can be obtained. Results for p-type and n-type doped silicon wafers are reported in the frequency range of 11 – 12.5 GHz. The dielectric constant of silicon wafer obtained by this method agrees well with that measured in the same frequency range by other conventional methods.

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