Plane Compressional Waves Research Articles

Caliper Effect on Borehole Coupling

The mechanism of low-frequency Borehole Coupling from the literature is here applied to a borehole of non-uniform radius. It is noted that in addition to the pressure signal in a fluid-filled borehole coincident with the passage of an external plane compressional wave, there are waves created at any change in radius which arrive later as coherent noise. This is illustrated first with a single step-change in radius. An example using the caliper log from an oil well shows that these delayed waves create a noise train of substantial amplitude relative to the direct compressional wave. It is pointed out that a down-hole pressure source will radiate a noise train due to caliper changes which will follow the direct compressional wave with exactly the same waveform as that encountered in Borehole Coupling.

Diffraction of elastic waves by an elliptic crack—II

A recently developed integral equation technique is used to obtain a low frequency solution for the diffraction of a plane compressional or shear wave by an elliptic crack embedded in an elastic medium. The mixed boundary value problem is reduced to a coupled system of integro-differential equations. A formal power series solution for the coupled system of integro-differential equations is developed. Attention is focussed on the farfield scattered amplitudes and the dynamic stress intensity factor. The limiting values when the ellipse degenerates into a circle agree with those of a circular crack.

Axisymmetric soil–structure interaction by global–local finite elements

AbstractA version of the global–local finite element method is presented for studying dynamic steady‐state soil–structure interaction wherein the soil medium extends to infinity. Herein, only axisymmetric behaviour is considered. In this approach, conventional finite elements are used to model the structure and some portion of the surrounding soil medium considered to be homogeneous and isotropic. A complete set of outgoing waves in the form of spherical harmonics for the entire space is used to represent the behaviour in the half‐space beyond the finite element mesh and these are termed the global functions. Full traction and displacement continuity is enforced at the finite element mesh interface with the outer region. On the free surface of the half‐space in the outer field, traction‐free surface conditions are enforced by demanding that a sequence of integrals of the weighted‐average tractions must vanish. Numerical examples are presented for the response of different shaped foundations, resting on the free surface or at various submerged levels, due to a normal seismic plane compressional wave. Plots of differential scattering cross‐sections show the angular distribution of the energy (its directional nature) of the scattered field.

When reflection coefficients are zero

At certain angles of incidence, the amplitude of a plane compressional wave reflected from a plane interface separating two solids is zero. Here I consider as a function of incidence angle those combinations of densities, compressional‐wave velocities, and shear‐wave velocities that give rise to a vanishing reflection coefficient.

HIGH-FREQUENCY SCATTERING OF A NORMALLY INCIDENT PLANE COMPRESSIONAL WAVE BY A PENNY-SHAPED CRACK

The axisymmetric problem for the scattering of time-harmonic elastic waves by a penny-shaped crack may, with the aid of Hankel transforms, be reduced to a pair of functional equations involving the transformed stresses and displacement discontinuities in the plane of the crack. The case of a normally incident plane compressional wave is considered, and the solution of one of the equations is obtained using a modified form of the generalized Wiener-Hopf technique for solving scattering problems involving finite strip boundaries. The solution is valid at high frequencies and enables one to derive rigorous high-frequency asymptotic approximations of the far field, uniform for all angles of observation, together with the forward scattering cross-section.

Acoustic intensity by intuition

For a plane sinusoidal acoustic wave, we know that average intensity is directed perpendicular to the wave front and uniform in magnitude. For two identical plane waves at right angles, intensity must be directed at 45° and be increased by √. Wrong! For a standing wave, we realize that average intensity is zero. For two standing waves at right angles, the average intensity must surely be zero. Wrong again! Measurements in a 4‐ft by 8‐ft enclosure confirmed the circulating flow of energy which can exist. In an elastic solid, average intensity is perpendicular to the wave front for both compressional and shear plane waves. When a plane shear wave and a plane compressional wave at the same frequency travel in the same direction, one would expect the direction of average energy flow to be perpendicular to the wave fronts. Actually, the direction of average intensity as a function of distance oscillates about the perpendicular. It is clear that intuition based on linear theory may lead one astray when applied to a nonlinear phenomenon like intensity.

The scattering of ultrasonic waves in polycrystalline materials with texture

The theory of ultrasonic propagation in polycrystals with independent and uniformly distributed orientations of the grains presented in previous papers [J. Acoust. Soc. Am. 72, 1021–1031 (1982); 73, 1160–1163 (1983)] is generalized to calculate the scattering coefficients and the phase and group velocities of plane compressional and shear waves in textured polycrystals. The calculation was done for plane waves in polycrystals of cubic symmetry with rolling texture in second-order perturbation theory using the assumption that the changes in the material constants from grain to grain are small. In the limit texture equal to zero the analytical results are exactly the same as those for untextured polycrystals previously presented. Numerical calculations are carried out for some examples.

On: “Reflection and transmission of plane compressional waves by R. D. Tooley, T. W. Spencer, and H. F. Sagori, (GEOPHYSICS, 30, 552, April, 1965).

One of the most widely reproduced figures for the partition of energy of a seismic wave at an interface between two media is from Tooley et al.’s 1965 paper. Recently we referred to this paper and noticed some anomalies in the curves shown there in Figure 11. We looked at the values of the curves at zero angle of incidence and noted that while the values of a P-wave velocity ratio of 2, 3, and 4 agree with the values given by the simplified formula {(V2−V1)/(V2+V1)}**2, the values shown for the P-wave velocity ratios less than 1 do not agree. For a ratio of 0.25, the coefficient at zero angle should be 0.36; it is shown as about 0.72. A ratio of 0.5 should give a coefficient of 0.11; 0.52 is shown. There are similar discrepancies for all the other ratios less than 1.

Dynamic stress concentration studies by boundary integrals and Laplace transform

AbstractThe dynamic stress field and its concentrations around holes of arbitrary shape in infinitely extended bodies under plane stress or plane strain conditions are numerically determined. The material may be linear elastic or viscoelastic, while the dynamic load consists of plane compressional waves of harmonic or general transient nature. The method consists of applying the Laplace transform with respect to time to the governing equations of motion and formulating and solving the problem numerically in the transfomed domain by the boundary integral equation method. The stress field can then be obtaind by a numerical inversion of the trasformed solution. The correspondence principle is invoked for the case of viscoelastic material behavious. The method is simplified for the case of harmonic waves where no numerical inversion is involved.

Sound reflection by a coated spherical shell

We have developed an analytical model for the transmission (interior fields), and principally for the reflection (exterior fields) of acoustic waves from a coated spherical shell covered with a layer of absorbent material. We consider an elastic spherical shell covered with a viscoelastic layer, immersed in an acoustic medium and containing a different fluid in its interior. The exact equations of elasticity and viscoelasticity govern the behavior of the spherical shell and the coating layer, respectively, and plane compressional incident waves were assumed. This model permits the quantitative study of the way in which absorbing layers of increasingly higher effectiveness alter the sonar cross section of a coated structure, and it also allows the assessment of the relative importance of the individual modal contributions (i.e., partial waves) contained in it. It was possible to isolate resonance features in these modal contributions which originate in the shell and in the coating. The reduction in level of the waves reflected by spherical bilaminar shells is more clearly understood by means of this model, which will also permit resonance‐interpretation in the light of shell theories.

Scattering of elastic waves by an elastic sphere

The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stess intensity factors, far-field amplitudes and the scattering cross section. It is assumed that the wave length is large compared to the radius of the scatterer. Various elastostatic limits are also presented.

Scattering of elastic waves by an elastic sphere

The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stress intensity factors, far-field amplitudes and the scattering cross-section. It is assumed that the wave length is large compared to the radius of the scatterer. Various elastostatic limits are also presented.

Attenuation of a plane compressional wave by a random distribution of thin curcular cracks : Piau, M Int J Engng Sci, V17, N2, 1979, P151–167

Attenuation of a plane compressional wave by a random distribution of thin curcular cracks : Piau, M Int J Engng Sci, V17, N2, 1979, P151–167

Scattering of elastic waves by circular cylindrical flaws and inclusions

The problems of scattering of plane compressional and shear waves by a class of infinite circular cylindrical flaws and inclusions embedded in an infinite isotropic homogeneous elastic medium are solved. The objects considered are an elastic cylinder, an inviscid-fluid-filled cavity, an empty cavity, and a rigid cylinder. Approximate formulas are derived for the displacement fields, stress tensors, far-field amplitudes, dynamic stress intensity factors, and the scattering cross sections. It is assumed that the wavelength is large compared to the radius of the scatterer. Various elastostatic limits are also presented. Some of these results even in elastostatics appear to be new.

Attenuation of a plane compressional wave by a random distribution of thin circular cracks

The problem of the diffraction of obliquely incident plane compressional waves by a thin circular crack located in an infinite isotropic elastic medium is considered. An analytic estimation of the scattering cross-section is obtained for the Rayleigh limit. The attenuation coefficient associated with a random distribution of such cracks is derived for not too closely-spaced cracks, even when weak scattering density conditions are not satisfied. It is found that, for long wave-lengths, the attenuation varies directly with the average of the sixth power of the radius and inversely with the fourth power of the wave-length, with a coefficient which depends on Poisson's ratio. Comparisons are made with known results for spherical cavities and with the attenuation we could calculate for scalar waves. Conclusions are drawn about attenuation measurements used as a non-destructive test to characterize crack distribution in a sample.

Diffraction of plane elastic waves by ellipsoidal inclusions

The method of matched asymptotic expansions has been used to obtain a low-frequency solution for the diffraction of a plane compressional or shear wave by an elastic ellipsoidal inclusion. Attention has been focused on the scattered farfield, angular dependences of which are illustrated graphically for ellipsoidal cavities of different axes ratios.

Scattering of elastic waves by a cylindrical cavity in a solid

Numerical computations of scattering cross sections are made for plane compressional and shear waves incident normally upon an infinitely long circular cylindrical cavity in a homogeneous isotropic elastic solid. Dependencies of the cross sections upon the variables ka and κa (k and κ are the compressional and shear wave vector amplitudes, respectively, and a is the cylinder radius) and upon the material parameter κ/k are discussed, along with the relative contributions of the various components of the total cross sections. Computations are made over the range 0⩽ka⩽6 for incident compressional waves, and 0⩽κa⩽10 for incident shear waves. Both sets of computations are done for a number of values of κ/k corresponding to various host materials.

Scattering of long-wavelength elastic waves by a cylindrical obstacle in a solid

The scattering of plane shear and compressional waves incident normally upon an infinitely long circular cylindrical obstacle in an isotropic homogeneous elastic solid medium is examined in the case in which the incident wavelength is large compared to the obstacle diameter. The scatterer can be either an empty cavity, a nonviscous-fluid-filled cavity, or an elastic solid cylinder. Explicit approximate expressions for the scattering cross sections are derived from the general theory formulated by White. The cross sections are proportional to (ka)3, where k is the compressional or shear wave-vector amplitude, and a is the cylinder radius.

Mode conversion relation for elastic waves scattered by a cylindrical obstacle in a solid

A simple relation shown by White to exist between plane shear and compressional elastic waves undergoing mode conversion when normally incident upon an empty or fluid-filled cylindrical scatterer is extended to include the case of a solid scatterer. The obstacle is assumed to be an infinitely long cylinder embedded in an infinite isotropic solid medium. A sign error in White's result is corrected.

Response of a Rigid Spheroidal Inclusion to an Incident Plane Compressional Elastic Wave

The method of matched asymptotic expansions has been used to obtain a low-frequency solution for the diffraction of a plane compressional wave by a rigid spheroid embedded in an infinite homogeneous isotropic elastic medium. In contrast to standard techniques such as separation of variables which are not effective in this case, the present method gives, in a systematic manner, both the “inner” and “outer” fields. Expansions have been carried out which are correct to $O(\varepsilon ^2 ),\varepsilon $ being a suitably defined nondimensional wave number. These results are then used to determine the motion of the inclusion as well as the stress distribution on the spheroid and the far-field displacement components. Numerical studies for these properties are presented in graphical form and the results compared with known results in various limiting cases.