The two-component representation of time-harmonic elastic body waves in the high- and intermediate-frequency regimes

1997 ◽  
Vol 101 (1) ◽  
pp. 52-65 ◽  
Author(s):  
Larissa Ju. Fradkin ◽  
Aleksei P. Kiselev
Keyword(s):  
Elastic Body ◽  
Intermediate Frequency ◽  
Body Waves ◽  
Time Harmonic ◽  
Two Component ◽  
Component Representation ◽  
Representation Of Time

Reciprocity and Related Topics in Elastodynamics

2006 ◽  
Vol 59 (1) ◽  
pp. 13-32 ◽  
Author(s):  
Jan D. Achenbach
Keyword(s):  
Elasticity Theory ◽  
Elastic Body ◽  
19Th Century ◽  
Wave Motion ◽  
Review Article ◽  
Integral Representations ◽  
Time Harmonic ◽  
Reciprocity Relations ◽  
Elastic Systems ◽  
The 19Th Century

Reciprocity theorems in elasticity theory were discovered in the second half of the 19th century. For elastodynamics they provide interesting relations between two elastodynamic states, say states A and B. This paper will primarily review applications of reciprocity relations for time-harmonic elastodynamic states. The paper starts with a brief introduction to provide some historical and general background, and then proceeds in Sec. 2 to a brief discussion of static reciprocity for an elastic body. General comments on waves in solids are offered in Sec. 3, while Sec. 4 provides a brief summary of linearized elastodynamics. Reciprocity theorems are stated in Sec. 5. For some simple examples the concept of virtual waves is introduced in Sec. 6. A virtual wave is a wave motion that satisfies appropriate conditions on the boundaries and is a solution of the elastodynamic equations. It is shown that combining the desired solution as state A with a virtual wave as state B provides explicit results for state A. Basic elastodynamic states are discussed in Sec. 7. These states play an important role in the formulation of integral representations and integral equations, as shown in Sec. 8. Reciprocity in 1-D and full-space elastodynamics are discussed in Secs. 910, respectively. Applications to a half-space and a layer are reviewed in Secs. 1112. Section 13 is concerned with reciprocity of coupled acousto-elastic systems. The paper is completed with a brief discussion of reciprocity for piezoelectric systems. There are 61 references cited in this review article.

Finite-element simulations of Stoneley guided-wave reflection and scattering at the tips of fluid-filled fractures

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. T23-T36 ◽  
Author(s):  
Marcel Frehner ◽  
Stefan M. Schmalholz
Keyword(s):  
Finite Element ◽  
Elastic Body ◽  
Deformation Behavior ◽  
Guided Waves ◽  
Amplitude Ratio ◽  
Crack Tip ◽  
Two Dimensions ◽  
Guided Wave ◽  
Body Waves ◽  
Crack Geometry

The reflection and scattering of Stoneley guided waves at the tip of a crack filled with a viscous fluid was studied numerically in two dimensions using the finite-element method. The rock surrounding the crack is fully elastic and the fluid filling the crack is elastic in its bulk deformation behavior and viscous in its shear deformation behavior. The crack geometry, especially the crack tip, is resolved in detail by the unstructured finite-element mesh. At the tip of the crack, the Stoneley guided wave is reflected. The amplitude ratio between reflected and incident Stoneley guided wave is calculated from numerical simulations, which provide values ranging between 43% and close to 100% depending on the type of fluid filling the crack (water, oil or hydrocarbon gas), the crack geometry (elliptical or rectangular), and the presence of asmall gas cap at the cracktip. The interference of incident and reflected Stoneley guided waves leads to a node (zero amplitude) at the tip of the crack. At other positions along the crack, this interference increases the amplitude. However, the exponential decay away from the crack makes the Stoneley guided wave difficult to detect at a relatively short distance away from the crack. The part of the Stoneley guided wave that is not reflected is scattered at the crack tip and emitted into the surrounding elastic rock as body waves. For fully saturated cracks, the radiation pattern of these elastic body waves points in every direction from the crack tip. The emitted elastic body waves can allow the detection of Stoneley guided wave-related resonant signals at distances away from the crack where the amplitude of the Stoneley guided wave itself is too small to be detected.

The Study of Elastic Symmetry and Anisotropy of Elastic Body Waves for Carbonatic Rocks

1995 ◽  
Author(s):  
S. A. Vyzhva ◽  
G. T. Prodajvoda ◽  
V. V. Korol ◽  
A. A. Kulickov ◽  
P. J. Cholach
Keyword(s):  
Elastic Body ◽  
Body Waves ◽  
Elastic Symmetry

An evaluation of electromagnetic methods in the presence of geologic noise

Geophysics ◽  
1987 ◽  
Vol 52 (8) ◽  
pp. 1106-1126 ◽  
Author(s):  
Perry A. Eaton ◽  
Gerald W. Hohmann
Keyword(s):  
Half Space ◽  
Survey Design ◽  
Three Dimensional ◽  
Intermediate Frequency ◽  
Resolving Power ◽  
Step Response ◽  
Maximum Value ◽  
Time Harmonic ◽  
Electromagnetic Methods ◽  
Noise Measurements

An important element of electromagnetic (EM) prospecting is survey design; numerical modeling algorithms may be used to calculate signal‐to‐geologic‐noise (S/N) ratios to compare different survey configurations and measured responses quantitatively. Our models consist of a prismatic three‐dimensional (3-D) target in a conductive half‐space which may contain an overburden conductor; the models are energized by a time‐varying current transmitted in a loop of wire. The signal is the scattered or anomalous response of the target, while the geologic noise is either the response of the half‐space or the anomalous response of the overburden conductor. For typical loop sizes in exploration, the coincident‐loop configuration has a relatively high S/N ratio and thus a relatively high capability to resolve the target in the case of half‐space noise. Measurements made with the horizontal‐loop, moving‐coil configuration can be just as effective if the coil separation is one and one‐half to two times the depth of burial of the target and the transmitting and receiving coils are on opposite sides of the target. For coil positions on one side of the target, the S/N ratio decreases with increasing separation. The advantage in resolving power provided by the coincident loop’s superior S/N ratio diminishes as the size of the loop increases. For the case of noise due to the overburden conductor, the horizontal‐loop configuration with a large coil separation is optimal. If the depth of the target is unknown, the fixed‐loop, roving‐receiver configuration is useful for detecting the target but poor in resolving its depth because its S/N ratio is the least sensitive to the depth. With the fixed‐loop configuration, galvanic effects enhance the detectability of the target in a conductive half‐space, but inhibit detection if an overburden conductor is present. Regarding the S/N ratio, there does not appear to be any advantage in measuring the step response of a 3-D target in a conductive environment versus measuring the impulse response. The shapes of their respective S/N anomalies are essentially the same and the maximum impulse S/N ratio is 10 to 30 percent larger than the maximum step S/N ratio, though it occurs later in time by a factor of about 1.7. Although transient S/N ratios for a 3-D target in a conductive host reach a maximum value and then decrease with increasing time, harmonic S/N ratios do not necessarily reach a maximum value at an intermediate frequency. For all three survey configurations and both types of noise, target depths, and half‐space conductivities studied here, maximum transient S/N ratios are larger than harmonic S/N ratios. Peak step S/N ratios are 30 to 50 percent larger than corresponding in‐phase ratios in the case of half‐space noise, and several times larger in the case of the overburden conductor. A phase rotation of the target’s response due to the conductive host appears to amplify the quadrature S/N ratio relative to the in‐phase S/N ratio. However, in‐phase S/N ratios are always much larger than quadrature S/N ratios over the range of host resistivities used in this study.

Elastodynamic Analysis of Underground Structural Failures Induced by Seismic Body Waves

2012 ◽  
Vol 79 (3) ◽  
Author(s):  
Koji Uenishi
Keyword(s):  
Plane Wave ◽  
Incident Angle ◽  
Body Wave ◽  
Body Waves ◽  
Structural Failure ◽  
Circular Tunnel ◽  
Failure Patterns ◽  
Linear Elastic ◽  
Time Harmonic ◽  
Structural Failures

Scattering of elastic waves by structural inhomogeneities such as cylindrical cavities has been a subject of intensive study for decades. The time-harmonic elastodynamic analysis making use of the wave function expansions is one of the typical approaches for such problems, and since it gives semianalytical solutions that may show the effect of parameters of the problem rather explicitly, it is still repeatedly used in the study of dynamic response of elastic structures including inhomogeneities. Here, motivated by the observation of the unique underground structural failure patterns caused by the 1995 Hyogo-ken Nanbu (Kobe), Japan, earthquake, we analyze scattering of a plane harmonic body wave by a uniformly lined circular tunnel (cylinder), and from the structural failure patterns we evaluate possible mechanical characteristics of the associated incident seismic waves. In the two-dimensional, in-plane time-harmonic elastodynamic model employed, the lined circular tunnel may be located at a finite depth from an approximate flat free surface of a homogeneous isotropic linear elastic medium (half-space), and the plane wave impinges upon the tunnel at an arbitrary incident angle. We compare the effect of P and SV wave incidences by calculating the dynamic amplification of stresses and displacements around this simplified tunnel, and also show the influence of the wavelength and the incident angle of the plane wave, the overburden thickness, and the relative compliance of the linear elastic lining with respect to the surrounding medium. The results suggest that the observed underground structural failures, the exfoliation of the lining concrete and buckling of the reinforcing steel bars on the sidewall as well as the detachment of the subgrade from the invert, might have been induced by the incidence of P waves in a relatively high frequency range.

The attenuation of a Rayleigh wave in a half-space by a surface impedance

1966 ◽  
Vol 62 (4) ◽  
pp. 811-827 ◽  
Author(s):  
R. D. Gregory
Keyword(s):  
Surface Wave ◽  
Rayleigh Wave ◽  
Half Space ◽  
Logarithmic Singularity ◽  
Elastic Half Space ◽  
Body Waves ◽  
Impedance Boundary Condition ◽  
Impedance Boundary ◽  
Elastic Potentials ◽  
Time Harmonic

AbstractA time harmonic Rayleigh wave, propagating in an elastic half-space y ≥ 0, is incident on a certain impedance boundary condition on y = 0, x > 0. The resulting field consists of a reflected surface wave, scattered body waves, and a transmitted surface wave appropriate to the new boundary conditions. The elastic potentials are found exactly by Fourier transform and the Wiener-Hopf technique in the case of a slightly dissipative medium. The ψ potential is found to have a logarithmic singularity at (0,0), but the φ potential though singular is bounded there. Analytic forms are given for the amplitudes of the reflected and transmitted surface waves, and for the scattered field. The reflexion coefficient is found to have a simple form for small impedances. A uniqueness theorem, based on energy considerations, is proved.

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