Wave Function Expansions and Perturbation Method for the Diffraction of Elastic Waves by a Parabolic Cylinder

1967 ◽  
Vol 34 (4) ◽  
pp. 915-920 ◽  
Author(s):  
S. A. Thau ◽  
Yih-Hsing Pao
Keyword(s):  
Wave Function ◽  
Separation Of Variables ◽  
Low Frequency ◽  
Elastic Solid ◽  
Compressional Wave ◽  
Parabolic Cylinder ◽  
Infinite Matrix ◽  
Elastic Solids ◽  
Incident Plane ◽  
Scattered Fields

The dynamic response, including the stresses at the surface, of a rigid parabolic cylinder in an infinite elastic solid is studied for an incident plane compressional wave. The method of separation of variables in parabolic coordinates is used. With the wave function for one of the scattered waves expanded into a series of those for the other wave, the total scattered fields are then determined numerically by inverting a truncated infinite matrix. The same problem is solved also by a recently developed method of perturbation which describes the two waves in elastic solids in terms of wave functions with a common wave speed. With the latter method, the total scattered waves are determined analytically for the various orders of perturbation, and these results supplement the numerical wave function expansion results in the low-frequency range.

Motion of a rigid disc in an elastic solid

1971 ◽  
Vol 61 (6) ◽  
pp. 1717-1729
Author(s):  
A. K. Mal
Keyword(s):  
Exact Solution ◽  
Elastic Solid ◽  
Approximate Solutions ◽  
Compressional Wave ◽  
Fredholm Integral Equations ◽  
High Frequencies ◽  
Incident Plane ◽  
Smooth Contact ◽  
Rigid Disc ◽  
Quantities Of Interest

abstract A rigid circular disc embedded in an infinite, isotropic elastic solid is assumed to be excited by a normally-incident plane, harmonic compressional wave. The disc is assumed to be either in perfectly welded or perfectly smooth contact with the surrounding solid. In each case, the problem is reduced to the solution of Fredholm integral equations of the second kind. For the case of the welded contact, exact numeral solutions valid at any given frequency are given. Approximate solutions valid at low and high frequencies are obtained for both cases and are compared with the exact solution. The displacement and compliance of the disc as well as other quantities of interest are presented graphically as functions of frequency.

ELASTIC WAVES ALONG A CYLINDRICAL BORE

Geophysics ◽  
1962 ◽  
Vol 27 (3) ◽  
pp. 327-333 ◽  
Author(s):  
J. E. White
Keyword(s):  
Elastic Waves ◽  
Low Frequency ◽  
Elastic Solid ◽  
Approximate Expression ◽  
Compressional Wave ◽  
Axially Symmetric ◽  
Low Frequencies ◽  
Phase Velocities ◽  
Pierre Shale ◽  
Limiting Case

This paper concerns axially symmetric solutions for waves propagating along a cylinder in an infinite elastic solid. Solutions are presented describing unattenuated propagation along the axis at phase velocities higher than shear and compressional speeds in the solid, in contradiction to earlier publications. Special attention is given to the limiting case of phase velocity equal to compressional speed in the solid, which at low frequencies very closely approximates the coupling of a fluid‐filled borehole to a plane compressional wave in the surrounding solid. Comparison with some experiments in a uniform section of Pierre shale show excellent agreement at low frequencies. In the low‐frequency limit, these solutions reduce to an approximate expression for borehole coupling published earlier by the author.

Torsional Waves in an Infinite Elastic Solid Containing a Spheroidal Cavity

1972 ◽  
Vol 39 (4) ◽  
pp. 995-1001 ◽  
Author(s):  
S. K. Datta
Keyword(s):  
Shear Stress ◽  
Separation Of Variables ◽  
Low Frequency ◽  
Elastic Solid ◽  
Major Axis ◽  
Cavity Surface ◽  
Legendre Functions ◽  
Low Frequencies ◽  
Spheroidal Cavity ◽  
Torsional Waves

A low-frequency analysis is presented here for the axisymmetric problem of diffraction of torsional waves by an oblate spheroidal cavity in an isotropic homogeneous elastic medium. The method used gives a complete low-frequency expansion of the scattered field in terms of associated Legendre functions, instead of spheroidal wave functions that one gets by the method of separation of variables. This makes the numerical computation much simpler. Graphs and tables are presented for the displacement distribution on the cavity surface and the nonzero shear stress at the end of the major axis of the spheroid. It is found that for low frequencies and for the values of the ratio (b/a) of the minor and major axes of the spheroid considered here the absolute value of the ratio of the nonzero dynamic and static shear stress evaluated at the end of the major axis is independent of b/a for confocal spheroids. An estimate is also given for the radius of convergence of the low-frequency expansion.

Reflection, Refraction, and Absorption of Elastic Waves at a Frictional Interface: SH Motion

1979 ◽  
Vol 46 (3) ◽  
pp. 625-630 ◽  
Author(s):  
R. K. Miller ◽  
H. T. Tran
Keyword(s):  
Coulomb Friction ◽  
General Procedure ◽  
Frictional Stress ◽  
Elastic Solids ◽  
Sh Waves ◽  
Relative Slip ◽  
Transmission And Reflection Coefficients ◽  
Incident Plane ◽  
Frictional Interface ◽  
Transmission And Absorption

The reflection, refraction, and absorption of obliquely incident plane harmonic antiplane strain (SH) waves at a frictional interface between dissimilar semi-infinite elastic solids is investigated by an approximate analytical approach. The frictional stress at the interface is assumed to depend on the normal stress and the relative slip across the interface, but remains otherwise arbitrary throughout the analysis. General expressions are developed for the transmission and reflection coefficients, and the partitition of incident wave energy into reflection, transmission, and absorption. The special case of bonding by Coulomb friction is examined in detail as an example of application of the general procedure. An exact solution is also presented for the case of bonding by Coulomb friction, and a comparison between approximate and exact solutions provides an indication of the accuracy of the approximate method of analysis.

Delineating a cased borehole in unconsolidated formations using dipole acoustic data from a nearby well

Geophysics ◽  
2021 ◽  
pp. 1-39
Author(s):  
Gu Xihao ◽  
Xiao-Ming Tang ◽  
Yuan-Da Su
Keyword(s):  
Shear Waves ◽  
Low Frequency ◽  
Well Being ◽  
Compressional Wave ◽  
Acoustic Imaging ◽  
Wave Radiation ◽  
Dipole Source ◽  
Compressional Waves ◽  
Single Well ◽  
Cased Borehole

A potential application for single-well acoustic imaging is the detection of an existing cased borehole in the vicinity of the well being drilled, which is important for drilling toward (when drilling a relief well), or away from (collision prevention), the existing borehole. To fulfill this application in the unconsolidated formation of shallow sediments, we propose a detection method using the low-frequency compressional waves from dipole acoustic logging. For this application, we perform theoretical analyses on elastic wave scattering from the cased borehole and derive the analytical expressions for the scattered wavefield for the incidence of compressional and shear waves from a borehole dipole source. The analytical solution, in conjunction with the elastic reciprocity theorem, provides a fast algorithm for modeling the whole process of wave radiation, scattering, and reception for the borehole acoustic detection problem. The analytical results agree well with those from 3D finite-difference simulations. The results show that compressional waves, instead of shear waves as commonly used for dipole acoustic imaging, are particularly advantageous for the borehole detection in the unconsolidated formation. Field data examples are used to demonstrate the application in a shallow marine environment, where dipole-compressional wave data in the measurement well successfully delineate a nearby cased borehole, validating our analysis results and application.

Foreshock wave transmission into the magnetosheath and magnetosphere: results from global hybrid-Vlasov simulations

2021 ◽  
Author(s):  
Lucile Turc ◽  
Markus Battarbee ◽  
Urs Ganse ◽  
Andreas Johlander ◽  
Yann Pfau-Kempf ◽  
...  
Keyword(s):  
Solar Wind ◽  
Mach Number ◽  
Cone Angle ◽  
Low Frequency ◽  
Compressional Wave ◽  
Wave Transmission ◽  
Wave Power ◽  
Solar Wind Flow ◽  
Magnetic Pulsations ◽  
Wave Properties

<p>The foreshock, extending upstream of the quasi-parallel shock and populated with shock-reflected particles, is home to intense wave activity in the ultra-low frequency range.<em> </em>The most commonly observed of these waves are the “30 s” waves, fast magnetosonic waves propagating sunward in the plasma rest frame, but carried earthward by the faster solar wind flow. These waves are thought to be the main source of Pc3 magnetic pulsations (10 – 45 s) in the dayside magnetosphere. A handful of case studies with suitable spacecraft conjunctions have allowed simultaneous investigations of the wave properties in different geophysical regions, but the global picture of the wave transmission from the foreshock through the magnetosheath into the magnetosphere is still not known. In this work, we use global simulations performed with the hybrid-Vlasov model Vlasiator to study the Pc3 wave properties in the foreshock, magnetosheath and magnetosphere for different solar wind conditions. We find that in all three regions the wave power peaks at higher frequencies when the interplanetary magnetic field strength is larger, consistent with previous studies. While the transverse wave power decreases with decreasing Alfvén Mach number in the foreshock, the compressional wave power shows little variation. In contrast, in the magnetosheath and the magnetosphere, the compressional wave power decreases with decreasing Mach number. Inside the magnetosphere, the distribution of wave power varies with the IMF cone angle. We discuss the implications of these results for the propagation of foreshock waves across the different geophysical regions, and in particular their transmission through the bow shock.</p>

Finite strains in an anisotropic elastic continuum

1950 ◽  
Vol 202 (1070) ◽  
pp. 345-358 ◽  
Keyword(s):  
Equations Of State ◽  
Elastic Solid ◽  
Finite Strains ◽  
Free Energy Function ◽  
Elastic Solids ◽  
Elastic Continuum ◽  
Invariance Properties ◽  
Anisotropic Elastic ◽  
Necessary And Sufficient ◽  
Thermodynamic Restrictions

The discussion in a previous paper (Oldroyd 1950), on the invariance properties required of the equations of state of a homogeneous continuum, is extended by taking into account thermodynamic restrictions on the form of the equations, in the case of an elastic solid deformed from an unstressed equilibrium configuration. The general form of the finite strainstress-temperature relations, expressed in terms of a free-energy function, is deduced without assuming that the material is isotropic. The results of other authors based on the assumption of isotropy are shown to follow as particular cases. The equations of state are derived by considering quasi-static changes in an elastic solid continuum; the results then apply to non-ideally elastic solids in equilibrium, or subjected to quasi-static changes only, and to ideally elastic solids in general motion. A necessary and sufficient compatibility condition for the finite strains at different points of a continuum is also derived. As a simple illustration of the derivation and use of equations of state involving anisotropic physical constants, the torsion of an anisotropic cylinder is discussed briefly.

scholarly journals Design and Experimental Validation of a Metacomposite Made of an Array of Piezopatches Shunted on Negative Capacitance Circuits for Vibroacoustic Control

2013 ◽  
Author(s):  
F. Tateo ◽  
M. Collet ◽  
M. Ouisse ◽  
M. N. Ichchou ◽  
K. A. Cunefare
Keyword(s):  
New Technologies ◽  
Control Method ◽  
Low Frequency ◽  
Elastic Solid ◽  
Numerical Approach ◽  
Anomalous Behavior ◽  
Phononic Crystals ◽  
Underlying Structure ◽  
Negative Capacitance ◽  
New Class

In the last few decades, researchers have given a lot of attention to new engineered materials with the purpose of developing new technologies and devices such as mechanical filters, low frequency sound and vibration isolators, and acoustic waveguides. For instance, elastic phononic crystals may come to mind. They are materials with elastic or fluid inclusions inside a matrix made of an elastic solid. The anomalous behavior in phononic crystals arises from interference of waves propagating within an inhomogeneous material. The inclusions inside the matrix cause strong modifications of scattering properties. However, the application of phononic crystals is still limited to sonic frequencies. In fact, band gaps can be generated only when the acoustic wavelength is comparable to the distance between the inclusion. In order to overcome this limitation, a new class of metamaterial has been proposed: meta composite. This new class of material can modify the dynamics of the underlying structure using a bidimensional array of electromechanical transducers, which are composed by piezo patches connected to a synthetic negative capacitance. In this study, an application of the Floquet-Bloch theorem for vibroacoustic power flow optimization will be presented. In the context of periodically distributed, damped 2D mechanical systems, this numerical approach allows one to compute the multimodal waves dispersion curves into the entire first Brillouin zone. This approach also permits optimization of the piezoelectric shunting electrical impedance, which controls energy diffusion into the proposed semiactive distributed set of cells. Experiments performed on the examined structure illustrates the effectiveness of the proposed control method. The experiment requires a rectangular metallic plate equipped with seventyfive piezopatches, controlled independently by electronic circuits. More specifically, the out-of-plane displacements and the averaged kinetic energy of the controlled plate are compared in two different cases (control system on/off). The resulting data clearly show how this proposed technique is able to dampen and selectively reflect the incident waves.

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