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Surface‐to‐borehole seismic waves have been used from the early days of seismic exploration [B. McCollum and W. W. LaRue, Bull. AAPG 15 (11), (12) (1931)]. As long as the arrival time of the most direct compressional wave was the only observation, a geophone hanging on a cable sufficed, and it did not matter that the ensuing signal was dominated by acoustic waves in the mud column. Mathematical modeling and field measurements have guided the development of geophones that are pressed against the hole wall, greatly reducing sensitivity to these mud‐borne waves and more faithfully measuring the three components of motion in the solid around the hole. Analysis and measurements also show that shear and compressional waves passing the borehole create pressure signals in the mud column. Pressure detectors have the advantage that they do not need to be clamped, but this means that suppression of the mud‐borne signals must be achieved in some other way. Borehole sources that act by way of volume change create stro...

The term underground sound is meant to encompass the instrumentation, observational methods, and interpretive techniques that have been developed for the utilization of seismic waves in the earth, quite analogous to the field of underwater sound. Wave propagation in the shallow crust of the earth is more complex than waves in water, simply because of the presence of shear waves in addition to compressional waves. Either type of wave is severely attenuated in traveling through earth materials, compared with similar distances of travel through the ocean. Finally, many rocks are anisotropic in average properties, resulting in further complexities. In exploration for petroleum, remarkably detailed images of geologic boundaries have been achieved by processing observed data so as to enhance reflected compressional waves and suppress shear waves and reverberations. Alternatively, shear‐wave reflections can be observed by using a vibratory source which radiates strong shear waves and processing the data so as to...

A treatment of waves in a viscoelastic gel filling a rigid cylindrical cavity shows that, when the wall of the cavity executes axial motion, resonances occur at certain frequencies characterized by large displacements and high energy loss. Application of these results to shear waves in a ‘‘porous solid’’ consisting of parallel tubes in an elastic skeleton produces attenuation that is orders of magnitude greater for gel saturation than for fluid saturation. The mechanism of energy loss is applied to a more realistic pore geometry by repeating the derivation of the Biot theory with addition of a small shear rigidity to the saturating phase. Calculations show that attenuation and dispersion are greatly increased for compressional waves, type‐II dilatational waves, and shear waves. Even so, the computed decrements are not substantially larger than the decrements measured for typical porous rocks.

The mechanical impedance of an electromagnetic speaker is affected by current flowing in the electrical elements, just as the electrical impedance is influenced by the motion of the mechanical system. This paper gives the relation between mechanical impedance and electrical load impedance, making use of the three complex speaker constants evaluated by the motional impedance method presented earlier by R. D. Fay. Measurements are given showing that the mechanical impedance of a WE555 horn driver can be varied over a fairly wide range.

This paper presents a theory describing the acoustic waves set up in a fluid‐filled borehole by the passage of plane elastic waves in the surrounding solid. It consists of a mathematical development of the following process. A stress in the solid around the borehole will in general cause the hole to contract or expand. A contraction compresses the fluid and causes pulses to radiate in both directions as tube waves. At some observation point, one of these pulses is observed after a delay due to the propagation as a tube wave. A complex wave in the solid is expressed as a combination of stresses distributed along elementary lengths of the borehole, and the total acoustic wave received at a point is expressed as the summation of elementary pulses arriving after suitable delays. The expressions derived give pressure or fluid particle velocity in terms of motion or stresses in the solid for plane compressional waves and plane shear waves arriving at any angle. The method can be extended to the case of a spherical wave cutting the borehole.

A simple analysis shows that, when a transient plane wave in an elastic medium is reflected at a plane boundary with a lossy medium, the transient waveform of the reflection is affected by the loss parameters of the second medium. If the attenuation in the second medium is small, and if the ρc products of the two media are matched, then the reflected waveform is the convolution of the incident waveform, with the integral of the Fourier transform of attenuation as a function of frequency. Thus, attenuation for a lossy solid or liquid can be obtained by this external‐pulse technique. Where attenuation is some simple function of frequency, its Fourier transform is some recognized generalized function. Sample waveforms have been observed using airborne sound in specially prepared tubes; good qualitative agreement with predicted waveform was obtained.

Measurements of pressure and particle velocity in fluid‐filled bore holes have yielded several unexpected results, some of which can be explained by very simple computations. One is an acoustic square wave, a flat‐topped pulse which is generated when a slug of water, broken off from the main column by cavitation, rejoins the column. In another instance, the geometry is such that a wave along a steel casing is the first signal to arrive at a pressure detector, and the initial pressure is negative, although the explosive pressure generating the transient is initially positive. The manner in which a steel casing causes reversal of pressure in the “casing break” is described. An expression is also presented which relates the negative pressure observed at the bottom of an uncased well to the elastic constants of the fluid and solid. As a fourth example, initial upward motion of a geophone hanging in a bore hole is shown to be consistent with the circumstances under which the observations were made.

A technique is described which makes use of Bragg reflection of x‐rays to give a “picture” of the vibration pattern on the face of a crystalline plate. The dependence of this pattern on the direction of the x‐ray beam is illustrated, and a comparison between dust patterns and x‐ray patterns for a vibrating bar fairly well establishes that the local curvature of the crystal is responsible for the observed increase in x‐ray reflection. Reference is made to quantitative measurements on statically bent quartz and to an approximate theory treating the effect.

If the active face, or acoustic output terminal, of a sinusoidal sound source moves as a plane piston, then the source can be characterized by a blocked pressure and an acoustic output impedance. If this piston is coupled to a microphone by means of a closed air column, the pressure at the microphone depends on the acoustic impedance of the microphone, on the impedance of the source, and on the air column. An expression for this pressure as a function of the length of the air column is developed, and data are presented which show how source impedance, tube attenuation and other quantities may be obtained.

The acoustic output impedance of a sound source can be determined from relative pressure measurements following a procedure which requires rather simple equipment and practically no time for processing data. The sound source must have a radiating area which is essentially a piston and which can be coupled to a closed tube of variable length. The tube is terminated by a rigid plug in which a pressure‐measuring microphone is mounted flush with the surface. Thus the microphone measures the pressure at the terminating plug as the length of the air column is varied. These pressure measurements are related to the acoustic impedance of the source by the expression (P0/Pl)2 = sin2kl[R2 + (X − cotkl)2], where p0 is the zero length or blocked pressure, pl is the pressure at length, k is 2π divided by the wave‐length, l is the length of the air column, R is the specific acoustic resistance ratio for the source, and X is the specific acoustic reactance ratio for the source.The principal value of the method lies in th...

Seismic waves fall within the general category of acoustical phenomena, and there is some logic to considering “underground sound” as an extension of underwater sound. Just add shear waves and proceed as in case one. Attenuation in earth materials disturbs the parallel and prevents the adoption of underwater sound techniques for underground applications. As examples in the petroleum industry, the Borehole Televiewer scans only the surface of the borehole wall; attempts to telemeter information along a drill string or a pipeline are foiled by the surrounding Iossy solid; and resolution of seismic prospecting is controlled by attenuation. Measurements of attenuation in earth materials describe a situation which has not been explained to this writer's satisfaction. Data strongly indicate an attenuation constant directly proportional to frequency and a velocity virtually independent of frequency. On the face of it, these two results contradict causality, which of course is not acceptable. The loss mechanism yielding these results is almost certainly nonlinear. The writer has shown that propagation in a lumped‐element model of a solid yields the above results, if the springs in the model exhibit hysteresis. It would be more satisfying to have a nonlinear wave equation which similarly yielded attenuation proportional to frequency and negligible dispersion over the frequency range for which data have been gathered on earth materials.

The mechanical impedance of an electro‐mechanical device is affected by current flowing in the electrical elements, just as the electrical impedance is influenced by the motion of the mechanical system. The purpose of this paper is to determine over what limits the mechanical impedance of electro‐magnetic loudspeakers and horn driver units can be varied by varying the load impedance across the terminals. A simple extension of the motional impedance analysis presented by R. D. Fay provides a means of calculating the mechanical impedance of an electro‐magnetic speaker as a function of electrical load. The speaker when used in this way may be considered a continuously variable acoustic impedance. It is a secondary standard, since electrical measurements provide a means of calculating the impedance in terms of the surge impedance of a column of air. The calculated impedances were checked by direct measurement using the standing wave, or hyperbolic tangent, method. By means of a horn driver, the impedance of an area equal to approximately one square inch can be varied over fairly wide limits. The range of variation for cone speakers, with correspondingly larger areas, is much more restricted.

Because of fine layering, many sedimentary rocks can be characterized as transversely isotropic, possessing an axis of symmetry perpendicular to the layering. The many kinds of measurements made in fluid‐filled holes drilled parallel to this axis prompted this study of axisymmetric cylindrical waves. The literature shows how such waves in an isotropic solid can be expressed in terms of a compressional potential or a shear potential. This paper shows that in a transversely isotropic medium, particular combinations of these potentials are needed simultaneously, yielding either a quasi‐compressional wave or a quasi‐shear wave. The expressions are used to compute the transient response of an acoustic logging tool. The speeds of refracted compressional and shear waves agree with the speeds of plane waves traveling along the axis of symmetry. Dispersive waves in the fluid annulus yield further information about the elastic constants of the solid. The potentials also portray the radiation of elastic waves from a point force or from a transducer on the surface.

Expressions are derived for speed and attenuation of compressional waves travelling normal to the boundaries of a laminated medium which consists of alternating layers of two fluid‐saturated porous solids. Fluid flow across boundaries is taken into account through approximate expressions, applicable at low frequencies, derived through extension of the approach used by Gassmann. Calculations show that when one of the saturating fluids is gas, substantial dispersion and extreme attenuation may result.

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.

Geoacoustics, the application of acoustical theory and techniques to wave propagation in the earth, is centrally involved in two current developments in earthquake seismology and geophysical prospecting. The same theory which describes sound absorption by acoustical panels also describes wave propagation in fluid‐saturated rocks. The effect of fluid on the speeds of shear and compressional waves in rocks may soon provide reliable forecasts of imminent earthquakes. Because of the profound effects of gas saturation, compared to liquid saturation, gas reservoirs can be mapped directly by seismic reflection profiling. Both of these significant developments are covered by a theory which shows how speeds and density depend on static stresses as well as on the intrinsic properties of the fluid and of the skeleton. When gas saturation occurs in discrete pockets or in thin layers, substantial dispersion of compressional waves is indicated by the theory, with attenuation coefficients 10 to 30 times larger than norm...

The receiver analysis developed by R. D. Fay gives a method for calculating the complete performance of an electromagnetic speaker from a plot of electrical impedance measured when the speaker is loaded by a closed tube of variable length. This of course includes the performance of the speaker as a microphone, although the statement is not explicitly made. The present article applies the above analysis to a dynamic microphone and compares the sensitivity thus obtained with an independent calibration at lower audiofrequencies. This method yields a primary calibration, in that neither calibrated source nor comparison microphone is required. The high mechanical resistance of a dynamic microphone and its low coupling constant, as compared with an electromagnetic speaker, result in a much smaller variation in electrical impedance, with changing air load. The variation is of the order of 5 percent of the minimum impedance, as compared with perhaps 80 percent change in speaker impedance. The cross‐sectional area...

Measurements of sound waves at any considerable depth in the earth must almost of necessity be made in fluid‐filled boreholes, although it is a knowledge of the sound field in the absence of the borehole which is desired. This paper describes a method for computing the wave form to be expected at any point in the borehole due to a spherical wave in the earth. The same method is applied to sound in a gas‐filled inner cylinder due to traveling waves in a concentric outer cylinder. It is assumed that sound pressure in the outer medium acting on an incremental length of the cylinder sets up a wave which travels away in both directions with little dispersion or attenuation. Its effect is observed at all other points along the cylinder at later times. The total pressure at any point is made up of the sum of such contributions from earlier disturbances up and down the cylinder. This is expressed as an integral, which permits easy computation of received signals in terms of wave forms in the outer medium. In general, a transient in the outer medium is measured as two arrivals, one of which may be reversed in sign. Computed arrival times and magnitudes are compared with experiment for the case of concentric cylinders.

This paper discusses axially symmetric solutions for elastic waves propagating along an infinite cylinder, namely those which can be expressed as F(r) eikz e−iwt. M. A. Biot [J. Appl. Phys. 23, 997–1005 (1952)] presented results for phase velo|cities along the axis less than shear velocity in the solid and indicated that for higher phase velocities, the waves cannot propagate without attenuation. The present paper gives solutions for higher phase velocities, showing that no attenuation in the direction of the axis is required. Special attention is given to the limiting case of phase velocity equal to the speed of compressional waves in the solid, which yields the coupling of a fluid‐filled borehole to a plane compressional wave in the surrounding solid. In the low‐frequency limit, this coupling agrees with an approximate expression published earlier [J. Acoust. Soc. Am. 25, 906–915 (1953)].

Many measurements of the earth's elastic properties are best made through the use of a fluid‐filled bore hole. A theory describing the mechanism by which elastic waves in a solid generate bore‐hole signals was published by White [J. Acoust. Soc. Am. 25, 906 (1953)]. This paper is concerned with the experimental verification of the bore‐hole coupling theory. A model consisting of a metal rod loosely coupled to a metal pipe was used. An elastic pulse was generated at one end of the rod and permitted to travel along the rod. During the course of travel down the rod, energy from the pulse is transferred to the pipe and can be measured. The rod corresponds to the earth and the pipe to the fluid in the bore hole. Since the same mechanism for transfer of energy applies to the rod‐pipe model as to the earth‐bore‐hole system, the bore‐hole coupling theory is applied to the model and the computed wave forms compared to the experimentally observed signals.

As part of a program of fundamental research on seismic waves, a generator was built for applying a transient horizontal force at the surface of the ground and the resulting seismic waves were observed in some detail. The force is applied when a mass swinging through an arc strikes a target anchored to the earth. Surface geophones along a line in the direction of the force register vertically polarized shear waves refracted back up to the surface, whereas geophones on a line perpendicular to the force register horizontally polarized shear waves. The speeds of the two types of shear waves are often different, indicating anisotropy. Geophones buried below the target show a down‐going shear wave. Variation of amplitude with angle, and other features, are in qualitative agreement with the results given by Rayleigh and others for the waves due to a force at a point in an infinite solid. Love waves and other surface waves were observed, which of course would not be expected from an nterior force.

The velocity of elastic waves traveling parallel and perpendicular to the layering in a laminated medium has been determined by means of static elasticity. The procedure is to assume the medium to be in static equilibrium and exposed to certain stresses. The stresses are of such a nature as to generate strains similar to those which would exist during the passage of an elastic wave through the medium. From the application of Hooke's law, an effective stiffness constant for the medium is obtained. The ratio of this effective stiffness to the effective density is the square of the velocity of the elastic wave. For a medium consisting of layers of two materials with the same density but with a velocity contrast of two, the velocity of compressional waves traveling parallel to the layering is 20 percent higher than the velocity of the same wave traveling perpendicular to the layers. The SH shear wave has a velocity which is 25 percent higher than the SV shear wave for the same laminated medium.

If the electrical impedance of an electromagnetic speaker is plotted as a vector when the acoustic loading consists of an air column of variable length terminated by a rigid plug, a circle diagram results. If the air column is terminated instead by an acoustic sample, a second circle diagram results. Various relationships between the two circles have been worked out which yield the acoustic impedance of the sample with considerable precision. The method inherently averages many individual readings and automatically checks the self‐consistency of the data. It is applicable to samples with little absorption since full advantage is taken of the possibility of coupling such a sample most effectively to the speaker by means of the air column. The method has been used to determine the acoustic impedance of commercial materials at low audiofrequencies, where the absorption is quite small.