The reflexion of internal/inertial waves from bumpy surfaces. Part 2. Split reflexion and diffraction

1971 ◽  
Vol 49 (1) ◽  
pp. 113-131 ◽  
Author(s):  
P. G. Baines
Keyword(s):  
Wave Field ◽  
Logarithmic Singularity ◽  
Fluid Velocity ◽  
Wave Theory ◽  
Dispersive Wave ◽  
Inertial Waves ◽  
Incident Wavelength ◽  
Wave Fields ◽  
Reflected Wave ◽  
Square Root Singularity

This paper considers the linear inviscid reflexion of internal/inertial waves from smooth bumpy surfaces where a characteristic (or ray) is tangent to the surface at some point. There are two principal cases. When a characteristic associated with the incident wave is tangent to the surface we have diffraction; when the tangential characteristic is associated with a reflected wave we have split reflexion, a phenomenon which has no counterpart in classical non-dispersive wave theory. In both these cases the problem of determining the wave field may be reduced to a set of coupled integral equations with two unknown functions. These equations are solved for the simplest topography for each case, and the properties of the wave fields for more general topographies are discussed. For both split reflexion and diffraction, the fluid velocity has an inverse-square-root singularity on the tangential characteristic, and the energy density has a corresponding logarithmic singularity. The diffracted wave field penetrates into the shadow region a distance which is of the order of the incident wavelength. Possibilities for instability and mixing are discussed.

Angle‐dependent reflectivity by means of prestack migration

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1223-1234 ◽  
Author(s):  
C. G. M. de Bruin ◽  
C. P. A. Wapenaar ◽  
A. J. Berkhout
Keyword(s):  
Reflection Coefficient ◽  
Wave Field ◽  
Grid Point ◽  
Depth Level ◽  
Wave Fields ◽  
Prestack Migration ◽  
Reflected Wave ◽  
Zero Offset ◽  
Source Wave ◽  
Angle Dependent Reflectivity

Most present day seismic migration schemes determine only the zero‐offset reflection coefficient for each grid point (depth point) in the subsurface. In matrix notation, the zero‐offset reflection coefficient is found on the diagonal of a reflectivity matrix operator that transforms the illuminating source‐wave field into a reflected‐wave field. However, angle dependent reflectivity information is contained in the full reflectivity matrix. Our objective is to obtain angle‐dependent reflection coefficients from seismic data by means of prestack migration (multisource, multioffset). After downward extrapolation of source and reflected wave fields to one depth level, the rows of the reflectivity matrix (representing angle‐dependent reflectivity information for each grid point at that depth level) are recovered by deconvolving the reflected wave fields with the related source wave fields. This process is carried out in the space‐frequency domain. In order to preserve the angle‐dependent reflectivity in the imaging we must not only add all frequency contributions but we should extend the imaging principle by adding along lines of constant angle in the wavenumber‐frequency domain. This procedure is carried out for each grid point. The resulting amplitude information provides a rigorous approach to amplitude‐versus‐offset related methods. The new imaging technique has been tested on media with horizontal layers. However, with our shot‐record oriented algorithm it is possible to handle any subsurface geometry. The first tests show excellent results up to high angles, both in the acoustic and in the elastic case. With angle‐dependent reflectivity information it becomes feasible to derive detailed velocity and density information in a subsequent stratigraphic inversion step.

The reflexion of internal/inertial waves from bumpy surfaces

1971 ◽  
Vol 46 (2) ◽  
pp. 273-291 ◽  
Author(s):  
P. G. Baines
Keyword(s):  
Plane Wave ◽  
Energy Flux ◽  
Incident Wave ◽  
Wave Number ◽  
Inertial Waves ◽  
Incident Wavelength ◽  
Reflected Wave ◽  
Incident Plane ◽  
The Difference ◽  
Reflecting Surface

When internal and/or inertial waves reflect from a smooth surface which is not plane, there is in general some energy flux which is ‘back-reflected’ in the opposite direction to that of the incident energy flux (in addition to that ‘transmitted’ in the direction of the reflected rays), provided only that the incident wavelength is sufficiently large in comparison with the length scales of the reflecting surface. The reflected wave motion due to an incident plane wave is governed by a Fredholm integral equation whose kernel depends on the form of the reflecting surface. Some specific examples are discussed, and the special case of the ‘linearized boundary’ is considered in detail. For an incoming plane wave incident on a sinusoidally varying surface of sufficiently small amplitude, in addition to the main reflected wave two new waves are generated whose wave-numbers are the sum and difference respectively of those of the surface perturbations and the incident wave. If the incident wave-number is the smaller, the difference wave is back-reflected.

Experimental observations of edge waves

1995 ◽  
Vol 288 ◽  
pp. 1-35 ◽  
Author(s):  
S. J. Buchan ◽  
W. G. Pritchard
Keyword(s):  
Wave Field ◽  
Edge Wave ◽  
Marginal Stability ◽  
Initial Growth ◽  
Primary Wave ◽  
Instability Mechanism ◽  
Edge Waves ◽  
Wave Instability ◽  
Wave Fields ◽  
Reflected Wave

The main purpose of this paper is to provide some carefully documented experimental observations of the subharmonic generation of edge waves over a plane beach by waves normally incident on the beach from the distant ocean. In order to establish experimentally the details of the subharmonic instability mechanism, it is important first to determine the properties of the primary wave field whose stability is to be investigated. Thus, a detailed appraisal has been made of the wave field established in the tank and over the beach in the absence of edge waves. These data have been particularly useful in defining the amplitudes of the incident- and reflected-wave fields at the toe of the beach, the magnitudes of which are central to the triggering of the edge-wave instability. Details are presented of the edge-wave field over the beach, as well as marginal stability curves and initial growth rates of the instability, the latter two of which are compared with theoretical estimates obtained from extant theories of the instability mechanism. Some experiments are also described in which edge waves were established as modes forced by topographical imperfections at the beach.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

Interaction of High Frequency Lamb Waves with Surface-Mount Sensor Adhesives

2013 ◽  
Vol 558 ◽  
pp. 489-500 ◽  
Author(s):  
Patrick Norman ◽  
Claire Davis ◽  
Cédric Rosalie ◽  
Nik Rajic
Keyword(s):  
Wave Field ◽  
Lamb Waves ◽  
Base Material ◽  
Adhesive Layer ◽  
Scattered Wave ◽  
Phase Lag ◽  
Wave Fields ◽  
Surface Mount ◽  
Frequency Regime ◽  
Fibre Bragg Gratings

The application of Lamb waves to damage and/or defect detection in structures is typicallyconfined to lower frequencies in regimes where only the lower order modes propagate in order to simplifyinterpretation of the scattered wave-fields. Operation at higher frequencies offers the potentialto extend the sensitivity and diagnostic capability of this technique, however there are technical challengesassociated with the measurement and interpretation of this data. Recent work by the authorshas demonstrated the ability of fibre Bragg gratings (FBGs) to measure wave-fields at frequencies inexcess of 2 MHz [1]. However, when this work was extended to other thinner plate specimens it wasfound that at these higher frequencies, the cyanoacrylate adhesive (M-Bond 200) used to attach theFBG sensors to the plate was significantly affecting the propagation of the waves. Laser vibrometrywas used to characterise the wave-field in the region surrounding the adhesive and it was found that theself-adhesive retro-reflective tape applied to aid with this measurement was also affecting the wavefieldin the higher frequency regime. This paper reports on an experimental study into the influence ofboth of these materials on the propagating wave-field. Three different lengths of retro-reflective tapewere placed in the path of Lamb waves propagating in an aluminium plate and laser vibrometry wasused to measure the wave-field upstream and downstream of the tape for a range of different excitationfrequencies. The same experiment was conducted using small footprint cyanoacrylate film samplesof different thickness. The results show that both of these surface-mount materials attenuate, diffractand scatter the incoming waves as well as introducing a phase lag. The degree of influence of thesurface layer appears to be a function of its material properties, the frequency of the incoming waveand the thickness and footprint of the surface layer relative to the base material thickness. Althoughfurther work is required to characterise the relative influence of each of these variables, investigationsto date show that for the measurement of Lamb Waves on thin structures, careful considerationshould be given to the thickness and footprint of the adhesive layer and sensor, particularly in the highfrequency regime, so as to minimise their effect on the measurement.

Migration of seismic data by phase shift plus interpolation

Geophysics ◽  
1984 ◽  
Vol 49 (2) ◽  
pp. 124-131 ◽  
Author(s):  
Jeno Gazdag ◽  
Piero Sguazzero
Keyword(s):  
Phase Shift ◽  
Wave Field ◽  
Seismic Data ◽  
Three Dimensional ◽  
Reference Wave ◽  
Second Step ◽  
Lateral Velocity ◽  
Wave Fields ◽  
Shift Method ◽  
Phase Shift Method

Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using ℓ laterally uniform velocity fields. The intermediate result is ℓ reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.

Some Boundary Value Problems of Elastic Media Containing Rigid Inclusions Under Compressive Mechanical Loads

2009 ◽  
Author(s):  
Vahagn Makaryan ◽  
Michael Sutton ◽  
Gurgen Chlingaryan ◽  
Davresh Hasanyan ◽  
Xiaomin Deng
Keyword(s):  
Boundary Value Problems ◽  
Elastic Medium ◽  
Integral Transformation ◽  
Logarithmic Singularity ◽  
Boundary Value ◽  
Compressive Loading ◽  
Geometrical Shape ◽  
Square Root ◽  
End Points ◽  
Square Root Singularity

In this work we discuss some 2D boundary-value problems related to an elastic medium containing a thin rigid inclusion with general geometrical shape located in the interface between two separate elastic half-planes and subjected to compressive loading. Assuming perfect bonding between the inclusion and elastic medium, Fourier and Henkel integral transformation techniques are used to obtain the exact solution for the problem. Explicit forms are presented for arbitrary forms of thin inclusions, demonstrating that the tangent shear stress at the end-points of the inclusion has a square-root singularity. It is also shown that the normal stress has a logarithmic singularity when the end-points of the inclusion are approached from the inside of the inclusion and a square-root singularity when the end-points of the inclusion are approached from the outside of the inclusion. For special, extreme cases the solutions for anti-cracks are also presented.

Fundamental Solutions of Poroelastodynamics in Frequency Domain Based on Wave Decomposition

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Boyang Ding ◽  
Alexander H.-D. Cheng ◽  
Zhanglong Chen
Keyword(s):  
Integral Equation ◽  
Wave Field ◽  
Frequency Domain ◽  
Compressional Wave ◽  
Fundamental Solutions ◽  
Point Force ◽  
Wave Fields ◽  
Wave Effects ◽  
Wave Decomposition ◽  
2D And 3D

Fundamental solutions of poroelastodynamics in the frequency domain have been derived by Cheng et al. (1991, “Integral Equation for Dynamic Poroelasticity in Frequency Domain With BEM Solution,” J. Eng. Mech., 117(5), pp. 1136–1157) for the point force and fluid source singularities in 2D and 3D, using an analogy between poroelasticity and thermoelasticity. In this paper, a formal derivation is presented based on the decomposition of a Dirac δ function into a rotational and a dilatational part. The decomposition allows the derived fundamental solutions to be separated into a shear and two compressional wave components, before they are combined. For the point force solution, each of the isolated wave components contains a term that is not present in the combined wave field; hence can be observable only if the present approach is taken. These isolated wave fields may be useful in applications where it is desirable to separate the shear and compressional wave effects. These wave fields are evaluated and plotted.

scholarly journals An Experimental Investigation of Internal Tide Generation by Two-Dimensional Topography

2008 ◽  
Vol 38 (1) ◽  
pp. 235-242 ◽  
Author(s):  
Thomas Peacock ◽  
Paula Echeverri ◽  
Neil J. Balmforth
Keyword(s):  
Experimental Investigation ◽  
Wave Field ◽  
Internal Tide ◽  
Two Dimensional ◽  
Knife Edge ◽  
Wave Fields ◽  
Schlieren Technique ◽  
Conversion Rates ◽  
Theoretical Predictions ◽  
Secondary Instabilities

Abstract Experimental results of internal tide generation by two-dimensional topography are presented. The synthetic Schlieren technique is used to study the wave fields generated by a Gaussian bump and a knife edge. The data compare well to theoretical predictions, supporting the use of these models to predict tidal conversion rates. In the experiments, viscosity plays an important role in smoothing the wave fields, which heals the singularities that can appear in inviscid theory and suppresses secondary instabilities of the experimental wave field.

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