SCATTERING OF COMPRESSION WAVES BY SPHERICAL OBSTACLES

Geophysics ◽  
1959 ◽  
Vol 24 (1) ◽  
pp. 30-39 ◽  
Author(s):  
Leon Knopoff
Keyword(s):  
Wave Beam ◽  
Wave Length ◽  
Incident Beam ◽  
P Wave ◽  
Rigid Sphere ◽  
S Wave ◽  
P Waves ◽  
Compression Waves ◽  
Plane P Waves ◽  
Special Case

The scattering of plane P waves by a spherical obstacle is formulated. A calculation is given for the special case of scattering by a perfectly rigid sphere in which the medium outside has a Poisson’s ratio of [Formula: see text]. The range of sizes of obstacles used in the calculation includes radii very small compared with wave length and radii comparable to the wave length. For incident P waves, scattered P and S are computed with shifts in time phase occurring in both with respect to the incident beam. For small obstacles, the scattered S wave is generally broadside to the scattered P‐wave beam.

scholarly journals Phased array compaction cell for measurement of the transversely isotropic elastic properties of compacting sediments

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA113-WA123 ◽  
Author(s):  
Kurt T. Nihei ◽  
Seiji Nakagawa ◽  
Frederic Reverdy ◽  
Larry R. Myer ◽  
Luca Duranti ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Phased Array ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
P Waves ◽  
Array Measurements ◽  
Experimental Apparatus ◽  
Marine Sediment Sample ◽  
Plane P Waves

Sediments undergoing compaction typically exhibit transversely isotropic (TI) elastic properties. We present a new experimental apparatus, the phased array compaction cell, for measuring the TI elastic properties of clay-rich sediments during compaction. This apparatus uses matched sets of P- and S-wave ultrasonic transducers located along the sides of the sample and an ultrasonic P-wave phased array source, together with a miniature P-wave receiver on the top and bottom ends of the sample. The phased array measurements are used to form plane P-waves that provide estimates of the phase velocities over a range of angles. From these measurements, the five TI elastic constants can be recovered as the sediment is compacted, without the need for sample unloading, recoring, or reorienting. This paper provides descriptions of the apparatus, the data processing, and an application demonstrating recovery of the evolving TI properties of a compacting marine sediment sample.

Numerical simulation of azimuthal acoustic logging in a borehole penetrating a rock formation boundary

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. D283-D291 ◽  
Author(s):  
Peng Liu ◽  
Wenxiao Qiao ◽  
Xiaohua Che ◽  
Xiaodong Ju ◽  
Junqiang Lu ◽  
...  
Keyword(s):  
Domain Model ◽  
P Wave ◽  
S Wave ◽  
Angle Variation ◽  
P Waves ◽  
Acoustic Logging ◽  
S Waves ◽  
Rock Formation ◽  
Logging Tool ◽  
Difference Time

We have developed a new 3D acoustic logging tool (3DAC). To examine the azimuthal resolution of 3DAC, we have evaluated a 3D finite-difference time-domain model to simulate a case in which the borehole penetrated a rock formation boundary when the tool worked at the azimuthal-transmitting-azimuthal-receiving mode. The results indicated that there were two types of P-waves with different slowness in waveforms: the P-wave of the harder rock (P1) and the P-wave of the softer rock (P2). The P1-wave can be observed in each azimuthal receiver, but the P2-wave appears only in the azimuthal receivers toward the softer rock. When these two types of rock are both fast formations, two types of S-waves also exist, and they have better azimuthal sensitivity compared with P-waves. The S-wave of the harder rock (S1) appears only in receivers toward the harder rock, and the S-wave of the softer rock (S2) appears only in receivers toward the softer rock. A model was simulated in which the boundary between shale and sand penetrated the borehole but not the borehole axis. The P-wave of shale and the S-wave of sand are azimuthally sensitive to the azimuth angle variation of two formations. In addition, waveforms obtained from 3DAC working at the monopole-transmitting-azimuthal-receiving mode indicate that the corresponding P-waves and S-waves are azimuthally sensitive, too. Finally, we have developed a field example of 3DAC to support our simulation results: The azimuthal variation of the P-wave slowness was observed and can thus be used to reflect the azimuthal heterogeneity of formations.

On the Green function of the Lord–Shulman thermoelasticity equations

2019 ◽  
Vol 220 (1) ◽  
pp. 393-403 ◽  
Author(s):  
Zhi-Wei Wang ◽  
Li-Yun Fu ◽  
Jia Wei ◽  
Wanting Hou ◽  
Jing Ba ◽  
...  
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Parabolic Type ◽  
Second Order ◽  
P Wave ◽  
Thermal Mode ◽  
S Wave ◽  
Order Tensor ◽  
P Waves ◽  
Thermal Conductivities

SUMMARY Thermoelasticity extends the classical elastic theory by coupling the fields of particle displacement and temperature. The classical theory of thermoelasticity, based on a parabolic-type heat-conduction equation, is characteristic of an unphysical behaviour of thermoelastic waves with discontinuities and infinite velocities as a function of frequency. A better physical system of equations incorporates a relaxation term into the heat equation; the equations predict three propagation modes, namely, a fast P wave (E wave), a slow thermal P wave (T wave), and a shear wave (S wave). We formulate a second-order tensor Green's function based on the Fourier transform of the thermodynamic equations. It is the displacement–temperature solution to a point (elastic or heat) source. The snapshots, obtained with the derived second-order tensor Green's function, show that the elastic and thermal P modes are dispersive and lossy, which is confirmed by a plane-wave analysis. These modes have similar characteristics of the fast and slow P waves of poroelasticity. Particularly, the thermal mode is diffusive at low thermal conductivities and becomes wave-like for high thermal conductivities.

P-wave slowness surface approximation for tilted orthorhombic media

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. C99-C112 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Phase Shift ◽  
Ray Tracing ◽  
Approximate Formula ◽  
P Wave ◽  
Slowness Surface ◽  
P Waves ◽  
Time Approximation ◽  
Numerical Examples ◽  
Validity Range ◽  
Plane P Waves

We have developed an analytic and approximate formula for vertical slowness components of down- and upgoing plane P waves in 3D tilted orthorhombic media. A perturbation method and Shanks transform were used to derive the approximation for slowness surface of P waves in tilted orthorhombic media. We have also quantitatively described the validity range of the radial horizontal slowness components for the proposed formula. The validity range was affected by the strength of the anellipticity of an orthorhombic medium: the stronger the anellipticity, the smaller the validity range. Numerical examples determined that the proposed formula is accurate for tilted orthorhombic media with weak to strong anellipticity. We have also evaluated in detail the application of the proposed formula on calculating the P-wave intercept time in the [Formula: see text] domain for horizontally layered, tilted orthorhombic models. Our formula is useful for ray tracing, phase-shift migration, and [Formula: see text] domain intercept time approximation for tilted orthorhombic media.

scholarly journals Estimation of Geodynamic Properties using Seismic Techniques at a Steel Rolling Factory, Northwestern Gulf of Suez, Egypt

2020 ◽  
Vol 8 (6) ◽  
pp. 1785-1794
Keyword(s):  
Wave Velocity ◽  
Seismic Waves ◽  
Seismic Refraction ◽  
Industrial Area ◽  
P Wave ◽  
Gulf Of Suez ◽  
S Wave ◽  
P Waves ◽  
Steel Rolling ◽  
P Wave Velocity

The objective of the current investigations is to estimate the dynamic geotechnical properties necessary for evaluating the conditions of the subsurface in order to make better decisions for economic and safe designs of the proposed structures at a Steel Rolling Factory, Ataqa Industrial Area, Northwestern Gulf of Suez, Egypt. To achieve this purpose, four seismic refraction profiles were conducted to measure the velocity of primary seismic waves (P-waves) and four profiles were conducted using Multichannel Analysis of Surface Waves (MASW) technique in the same locations of refraction profiles to measure the velocity of shear waves (S-waves). SeisImager/2D Software Package was used in the analysis of the measured data. Data processing and interpretation reflect that the subsurface section in the study area consists of two layers, the first layer is a thin surface layer ranges in thickness from 1 to 4 meters with P-wave velocity ranges from 924 m/s to 1247 m/s and S-wave velocity ranges from 530 m/s to 745 m/s. The second layer has a P-wave velocity ranges from 1277 m/s to 1573 m/s and the S-wave velocity ranges from 684 m/s to 853 m/s. Geotechnical parameters were calculated for both layers. Since elastic moduli such as Poisson’s ratio, shear modulus, Young’s modulus, and bulk’s modulus were calculated. Competence scales such as material index, stress ratio, concentration index, and density gradient were calculated also. In addition, the ultimate and allowable bearing capacities

Circularly polarized shear waves used for VSP measurements

Geophysics ◽  
1992 ◽  
Vol 57 (4) ◽  
pp. 643-646
Author(s):  
Hans A. K. Edelmann
Keyword(s):  
Shear Wave ◽  
Velocity Ratio ◽  
Shear Waves ◽  
Signal Amplitude ◽  
P Wave ◽  
Polarization Analysis ◽  
S Wave ◽  
P Waves ◽  
Additional Processing ◽  
Wave Recording

If shear waves are to be recorded, all other types of waves (including P waves) have to be regarded as noise. All data processing applied later is limited in its success, not so much by the character of the signal, but by the character of the noise superimposed on the signal. Therefore an optimum method for simultaneous P‐ and S‐wave recording does not exist per se. All efforts made in the field that help to enhance the relatively weak S‐wave signal enhance the possibility of a more detailed interpretation such as polarization analysis. In the course of shear‐wave investigations over a period of more than ten years, simultaneous P‐ and SV‐wave recording has yielded fairly good results for velocity ratio determination, but has never produced satisfying results for polarization analysis because of the interfering P‐wave events. When generating pure SH‐waves, however, P‐wave arrival amplitudes in a shot record can, under favorable conditions, be kept well below the SH‐wave amplitude (−40 dB). Through additional processing, a ratio of P‐ to SH‐signal amplitude of −60 dB can be reached. The improvement achieved by making separate shear‐wave recordings, obviously, must be weighed against the additional costs caused by these recordings.

scholarly journals Numeric Simulation of Acoustic-Logging of Cave Formations

Energies ◽  
2020 ◽  
Vol 13 (15) ◽  
pp. 3908
Author(s):  
Fanghui Xu ◽  
Zhuwen Wang
Keyword(s):  
Excitation Frequency ◽  
Scattered Wave ◽  
P Wave ◽  
Borehole Wall ◽  
Stoneley Wave ◽  
S Wave ◽  
P Waves ◽  
Acoustic Logging ◽  
Source Distance ◽  
First Arrival

The finite difference (FD) method of monopole source is used to simulate the response of full-wave acoustic-logging in cave formations. The effect of the cave in the formation of borehole full-waves was studied. The results show that the radius of cave is not only linearly related to the first arrival of the compressional wave (P-wave), but also to the energy of the shear wave (S-wave). The converted S (S–S wave) and P-waves (S–P wave) are formed when the S-wave encounters the cave. If the source distance is small, the S–S and S–P waves are not separated, and the attenuation of the S-wave is not large, due to superposition of the converted waves. The S–P wave has been separated from the S-wave when the source distance is large, so the attenuation of the S-wave increases. The amplitude of the P and S–waves changes most when the distance of the cave to the borehole wall reaches a certain value; this value is related to the excitation frequency. The amplitude of the Stoneley wave (ST wave) varies directly with the radius of cave. If the radius of the cave is large, the energy of ST wave is weak. The scattered wave is determined by the radius and position of the cave. The investigation depth of a monopole source is limited. When the distance of the cave to the borehole wall exceeds the maximum investigation depth, the borehole acoustic wave is little affected by the cave. In actual logging, the development of the cave can be evaluated by using the first arrival of the P-wave and the energy of the S and ST waves.

scholarly journals Comparison of recent S-wave indicating methods

2018 ◽  
Vol 29 ◽  
pp. 00019
Author(s):  
Katarzyna Hubicka ◽  
Jakub Sokolowski
Keyword(s):  
Real Data ◽  
The Body ◽  
Body Waves ◽  
Identification Accuracy ◽  
P Wave ◽  
S Wave ◽  
P Waves ◽  
S Waves ◽  
Mode Decomposition ◽  
Wave Arrival

Seismic event consists of surface waves and body waves. Due to the fact that the body waves are faster (P-waves) and more energetic (S-waves) in literature the problem of their analysis is taken more often. The most universal information that is received from the recorded wave is its moment of arrival. When this information is obtained from at least four seismometers in different locations, the epicentre of the particular event can be estimated [1]. Since the recorded body waves may overlap in signal, the problem of wave onset moment is considered more often for faster P-wave than S-wave. This however does not mean that the issue of S-wave arrival time is not taken at all. As the process of manual picking is time-consuming, methods of automatic detection are recommended (these however may be less accurate). In this paper four recently developed methods estimating S-wave arrival are compared: the method operating on empirical mode decomposition and Teager-Kaiser operator [2], the modification of STA/LTA algorithm [3], the method using a nearest neighbour-based approach [4] and the algorithm operating on characteristic of signals’ second moments. The methods will be also compared to wellknown algorithm based on the autoregressive model [5]. The algorithms will be tested in terms of their S-wave arrival identification accuracy on real data originating from International Research Institutions for Seismology (IRIS) database.

Relation between P- and S-wave corner frequencies in the seismic spectrum

1974 ◽  
Vol 64 (6) ◽  
pp. 1621-1627 ◽  
Author(s):  
J. C. Savage
Keyword(s):  
High Frequency ◽  
Body Wave ◽  
Fault Model ◽  
P Wave ◽  
Corner Frequency ◽  
Rupture Propagation ◽  
S Wave ◽  
P Waves ◽  
Fault Surface ◽  
S Waves

abstract A comprehensive set of body-wave spectra has been calculated for the Haskell fault model generalized to a circular fault surface. These spectra are used to show that in practice the P-wave corner frequency (ƒp) may exceed the S-wave corner frequency (ƒs) when near-sonic or transonic rupture propagation obtains. The explanation appears to be that in such cases ƒs is so large that it is not identified within the recorded band, but rather a secondary corner is mistaken for ƒs. As a consequence of failing to detect the true asymptotic trend, the high-frequency falloff of the spectrum with frequency is substantially less for S waves than for P waves. This explanation appears to be consistent with the demonstration by Molnar, Tucker, and Brune (1973) that ƒp may exceed ƒs.

Stationary phase approximation in focal mechanism determination

1970 ◽  
Vol 60 (4) ◽  
pp. 1221-1229
Author(s):  
Umesh Chandra
Keyword(s):  
Stationary Phase ◽  
Wave Spectrum ◽  
P Wave ◽  
Phase Approximation ◽  
Wave Polarization ◽  
S Wave ◽  
Polarization Data ◽  
P Waves ◽  
Depth Ranges

abstract Tests of the stationary phase approximation method applied to P waves for the determination of focal mechanisms have been carried through for eight earthquakes selected from different geographic locations and depth ranges. The results are found to be in close agreement with the solutions obtained from S-wave polarization data for four earthquakes and in reasonable agreement for three earthquakes. In general, however, the P polarities are more consistent with S-wave polarization solutions than with the solutions obtained by the present method. The stationary phase solutions agree with the P-wave spectrum solutions determined in a previous study. The method is applicable to shallow-focus earthquakes, and to earthquakes of large magnitude in which methods using S-wave polarization data and P-wave spectra are difficult to apply.

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