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A generalized transverse wave is a propagating acceleration discontinuity on which the temperature and the entropy, together with their gradients, are continuous. In a heat-conducting elastic material the propagation and growth of such waves are uninfluenced by thermomechanical interaction. It is shown in this paper that in any given plane there is at least one direction in which a generalized transverse wave may propagate, and the existence is also proved of at least one direction in which a pair of generalized transverse waves may travel. Necessary and sufficient conditions are established for the speeds of propagation of these waves to be real. Relationships between transverse and generalized transverse waves are also studied, and in the last two sections of the paper the directions in which generalized transverse waves may propagate in an isotropic heat-conducting elastic material are systematically worked out and classified.

Using a Korteweg-de Vries equation the author studies the propagation of ion acoustic solitary waves in a two ion species plasma. As the dispersion relation presents two acoustic branches with quite different phase velocities, two types of soliton can propagate: the slow one which is subsonic with respect to one ion mode and supersonic to the other, has greater amplitude and smaller width than the fast one.

As motivated by the recent discovery of heat pulses propagating in dielectric crystals at low temperature, a continuum theory of thermoelasticity, which is modified to include the effect of thermal phonon relaxation, is applied to investigate the propagation of plane harmonic waves in unbounded anisotropic solids. Four characteristic wave speeds are found, three being analogous to those of isothermal or adiabatic elastic waves. The fourth wave, which is predominantly a temperature disturbance, corresponds to the heat pulses known also as the second sound. Velocity, slowness, and wave surfaces of the thermoelastic waves are analyzed and are illustrated with numerical and graphical results for NaF and solid helium crystals. A new definition of the group velocity for waves in a dissipative and dispersive anisotropic medium is proposed and is calculated and compared with the energy transport speed of thermoelastic waves.

The scattering of an SH-wave by a discontinuity in mass-loading on a semi-infinite elastic medium is investigated theoretically. The incident wave is either a plane body wave or a plane SH-surface wave. The problem is reduced to a Wiener-Hopf problem for the scattered wave. In this problem the amplitude spectral density of the particle displacement occurs as unknown function. Special attention is given to the numerical values of the surface wave contributions to the scattered field.

The propagation velocities and the variation of the amplitudes of thermo-acoustical waves in thermo-plastic materials are theoretically investigated. The constitutive equations of anisotropic thermo-plastic materials are derived from the concept of imaginary decomposition of the deformation rate tensor into the elastic and plastic contributions and from that of the plastic potential. From generalized Vernotte's heat conduction law the propagation condition of the jumps of the velocity gradients and of the temperature rate is obtained. In isotropic materials and in the case of a normal stress vector on the wave front we have two purely mechanical transverse waves and two thermo-longitudinal coupled waves. Formulae for the velocities and amplitudes are quite similar with those for thermo-elastic materials. The variation of the amplitude is discussed. There are, in general, three effects on the variation, that is, the non-planar, heat conduction and plastic flow effects. The transverse waves are subjected only to the non-planar effect, while the thermo-longitudinal waves may grow or decay according to the above three effects.

The propagation of elastic waves in a homogeneous solid is governed by a hyperbolic system of three linear second-order partial differential equations with constant coefficients. When the solid is also isotropic, the form of these equations is well known and provides the foundation of the conventional theory of elasticity (Love 1944). The explicit solution of the initial value, or Cauchy, problem for the isotropic case was found by Poisson, and in a different way by Stokes (1883). If the initial disturbance is sharp and concentrated, the resulting disturbance at a field point will consist of an initial sharp pressure wave, a continuous wave for a certain period, and a final sharp shear wave. The disturbance then ceases. Here we shall consider a medium which is homogeneous but not isotropic, and will describe, using Fourier transforms, the elastic waves produced by a local initial disturbance. The solution again consists of a continuous wave which lasts for a definite period of time, and a number of sharp waves, but the detailed nature of the waves may, in highly anisotropic media, be very different and much more complicated. The continuous wave may arrive at a field point in advance of the first sharp wave, though it will always terminate with the last sharp wave. The number of the sharp waves may not exceed 75. The solution appears as the sum of three modes, which correspond to the three sheets of a certain wave surface. The geometry of this surface, which may be quite complicated (Musgrave 1954 a ), qualitatively determines the nature of the solution. These calculations may serve as a foundation for the study of time-dependent elastic waves. There is also mathematical interest in this example of a hyperbolic system for which the wave surface may have certain types of singularities not usually considered in the existing general theory of hyperbolic differential equations.

The wave equation is considered in media with variable velocity of propagation and a plane interface. The singular part of the Green’s function is constructed as a sum in which the $( {N + 1} )$st term is a degree less singular than the Nth with certain restrictions when a ray is tangent to the boundary. This expansion provides a uniformly valid approximation in a neighborhood of the arrival time at the observation point of the main reflected or transmitted wave surface, lateral waves, or surface waves. For first arrivals the result is asymptotic, whereas for subsequent wave fronts there is an error due to residual waves. One of the main results is that this method provides an estimate of the surface waves (if any). Also the method is directly applicable to more complicated hyperbolic equations or systems.

A mathematical expression of elastic waves reflected due to slight heterogeneity of the medium is presented explicitly to the first order approximation in the case of one-dimensional problems. The approximate solution of elastic waves propagated through a heterogeneous medium is obtained by regarding the effect of the heterogeneity as that of equivalent dynamical sources scattered within the medium and by applying Green's function to the first-order approximate wave equation.Comparison between the approximate reflected wave and the exact one shows the validity of the present theory.The applications of the solution are found in such problems as attenuation of elastic waves passing through a heterogeneous medium and interpretation of reflected waves from a certain layered structure.The finite difference method is applied to the wave equation with the aid of the present theory in order to demonstrate the nature of elastic waves propagated through a heterogeneous medium.

Using the principle of least action the equations of motion and momentum conditions for elastic disturbances in any continuous medium are derived. Consideration of energy flux defines the form of the wave surface as the first negative pedal of the surface of phase velocities for elastic plane waves in the medium . A general method for obtaining the forms of these surfaces and an associated inverse surface is given and general conclusions about the propagation of disturbances are drawn.

Summary The propagation of waves of Rayleigh type in a uniform semi-infinite elastic medium with two uniform surface layers is discussed. The waves are dispersive, and the wave velocity equation is obtained as a determinant of the tenth order. By making infinite the thickness of the surface sheet one can derive the velocity equation for waves in an internal stratum. By making the thickness of either sheet zero one can retrieve the known equation for the velocity of Rayleigh waves in a medium with a single surface layer. If the wave-length is very small the determinant reduces to the product of three determinants which, equated in turn to zero, are the velocity equations of very short Rayleigh waves at the free surface and of very short waves of Rayleigh type at the two interfaces. The results of a numerical solution of the wave equation and the application to the surface waves of earthquakes will be communicated in a later paper.

The reflection of time‐harmonic plane dilatational or shear waves from the rigid or stressfree boundary of an “arbitrary” linearly viscoelastic half‐space is studied. Properties of general plane waves—i.e., plane waves whose amplitudes vary across their wavefronts—are used to determine the reflected dilatational and shear waves. In general, the reflected waves are general plane waves and there is a phase shift at the boundary. When certain specified conditions prevail, the reflected waves are attenuated only in their direction of propagation. Necessary and sufficient conditions on the material properties are derived for the existence of surface waves. It is shown that no surface waves can exist for certain viscoelastic materials. In general, surface waves are possible only for discrete incidence angles. However, for a special class of viscoelastic materials, which we call elasticlike, reflected surface waves may exist for a range of incidence angles greater than a specific critical angle. For the elasticlike materials, the phase shift at the boundary is the same as in the analogous elastic problem. In general, the phase velocities and reflected angles are functions of the incidence angle, the frequency, and material properties.

It is shown that Truesdell's theorem on the existence of waves in an inhomogeneous elastic material can be readily carried over to the case of an elastic non-conductor of heat, provided that the isentropic elasticity tensor is strongly elliptic. That is, there is at least one direction in which one longitudinal wave and two transverse waves with orthogonal amplitudes may exist and propagate. Further, if one considers a situation for which the waves are necessarily plane, then one finds that the amplitudes of these waves depend on the mechanical properties of the material alone, independent of its thermal properties. In other words, one would not be able to ascertain the effects of the thermal properties of the material on the behaviour of these waves.

In non-isotropic single crystals the normals to the wavefronts of elastic waves are not colinear with the vectors representing either the energy flow or the particle displacement. Calculations have been carried out on the propagation characteristics of sound waves in two particular trigonal crystals, α-quartz and sapphire.The development of the eigenvalue equation for the velocity and the formulae for the components of the displacement and energy-flow vectors are summarized. The assumption that the wave has a plane wavefront normal to a given direction leads to three solutions, one representing a quasi-longitudinal wave and the other two representing quasi-transverse waves. The velocities of propagation, directions of displacement, and directions of energy flow for the three waves have been calculated for many orientations of the wave normal. Detailed results for propagation near one of the pure-mode axes are presented.

The equations for the speed of waves in the following cases are derived in an elementary way: transverse waves along a stretched wire and through an elastic solid; longitudinal waves through a fluid in a uniform tube, through a uniform rod, and along a stretched spring; long gravity waves through a shallow canal. Several errors in previously published derivations are pointed out.

Exact steady‐state solutions for both longitudinal waves and shear waves traveling in isotropic rods of circular cross section are analyzed. Numerical examples are described for rods of large diameter to wavelength ratio.The effect of mode conversion on the propagation of short duration longitudinal wave trains is considered; and an approximate expression for the resulting loss is developed.Experimental results for 10–25 Mc/sec longitudinal and shear waves in fused silica rods of 1.13 cm radius are described.

A model of a simple one-dimensional medium has been developed which allows most of the important features of traveling waves to be quantitatively demonstrated. Air-suspended disks are connected by light springs and the resulting “lumped-parameter” line is easily analyzed. Phase and group velocity, dispersion, the cutoff frequency, and other more subtle features of wave motion can be computed and measured in class. The device is also useful for studying standing waves.

The mathematical formalism for obtaining dispersion relations for acoustic waves in plates of arbitrary anisotropy is outlined, and dispersion curves for propagation in a (001)‐cut cubic plate are presented. These results are compared to the uncoupled SV and P modes which, in turn, are related to the slowness curves for bulk waves. This approach provides an explanation for the behavior of the computed dispersion curves, and it also provides a means of approximating plate wave dispersion curves from the behavior of the slowness curves. The relationship of plate waves to surface waves is also explored for directions in which pseudosurface waves are known to propagate.

Effects of a spherical cavity in an infinite, homogeneous, isotropic elastic solid, containing non-viscous compressible liquid, on the propagation of elastic waves are investigated mathematically. The waves emitted by a simple harmonic point source in the solid are of the types known as SH and P in seismology. The discussion is restricted to the case ka » 1 (ka = 2 π cavity radius/wave length). Series solutions are transformed into contour integrals by Watson's method. Evaluation of these by the method of residues results in expressions describing the P and S components of the diffracted waves.

The propagation of plane waves in a piezoelectric medium has been studied. A simultaneous solution of Maxwell's field equations and Newton's law of force gives the necessary conditions for wave propagation in the infinite medium. By piezoelectric coupling, every electromagnetic wave has associated with it an elastic wave. Nonpiezoelectric crystals are, at best, birefringent electromagnetically and trirefringent acoustically. In piezoelectric media, the electromagnetic and elastic modes are coupled, thus giving five phase velocities for both the electromagnetic and acoustic waves. These media are therefore electromagnetically and acoustically pentrefringent. The slower electromagnetic waves are the field variations accompanying the elastic waves, while the fast elastic waves are the mechanical deformations accompanying the electromagnetic wave.The displacement in the elastic wave will differ in phase by ninety degrees from the electromagnetic wave, its amplitude being proportional to the electric field str...

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 solution is given for the problem of elastic wave propagation from a finite line source. The line source is considered as the limit of an elongated ellipsoid of revolution, the length of the minor axis of which tends to zero. A pressure that is a given function of time is applied to the inner surface of the ellipsoid, whereas this surface is free from shearing stress. It is shown that, in addition to a pressure wave, a shear wave is generated. This shear wave is relatively unimportant for values of Poisson’s constant around 0.25, but becomes appreciable for values larger than 0.4.

Reflection of plane waves from stress free flat surface of micropolar elastic half space, presenting reflection laws and amplitude ratios

Goodier and Bishop established that there is a “critical” incident angle that a train of Sv waves makes with the traction‐free surface of an elastic half‐space and that as the incident angle passes through this critical value from above, the reflected P wave passes through grazing incidence and emerges as a surface wave. In this paper, study is made of the fate of a train of Sv waves on striking the plane interface between two elastic materials. It is found that there are five critical angles associated with such a wave and the emerging wave which passes through grazing incidence corresponding to each angle is transformed into a surface wave. The wave is like a Rayleigh wave in that the magnitude of motions accompanying it decay exponentially away from the interface, particle motions have an elliptical orbit, and the wave travels with a constant phase velocity. The wave is unlike a Rayleigh wave in that its velocity is not the velocity of Rayleigh waves and, in fact, depends on the incident angle. The met...

In this paper, the propagation of harmonic waves with an arbitrary number of circumferential nodes in an infinitely long two‐layered composite circular elastic rod is investigated. The composite rod is made of a circular solid rod encased by a circular shell having different material properties. The frequency equation derived on the basis of the three‐dimensional linear isotropic elastic theory is presented. The reduction of this equation to the frequency equations for some special problems, such as longitudinal wave propagations, torsional wave propagations, flexural wave propagations, axial‐shear vibrations, and plane‐strain vibrations is discussed. Simplified equations for phase velocities of longitudinal and torsional waves at very long wavelength are obtained. Numerical results, in terms of frequency and real wavenumber, are given for a composite rod made of a soft core with a stiff casing.

Plane harmonic waves in a rotating elastic medium are considered. The inclusion of centripetal and Coriolis accelerations in the equations of motion with respect to a rotating frame of reference leads to the result that the medium behaves as if it were dispersive and anisotropic. The general techniques of treating anistropic media are used with some necessary modifications. Results concerning slowness surfaces, energy flux and mode shapes are derived. These concepts are applied in a discussion of the behavior of harmonic waves at a free surface.

In this paper are given the results of an experimental study of the interaction of an explosion wave with a water‐backed steel plate. Data are given showing the dependence of the transmitted and reflected wave on the angle of incidence, and of the diffracted wave on position behind the plate. These data are illustrative of the use of the explosion wave in acoustical research. The plate acts as a filter, removing the high frequencies from the transmitted wave and the low frequencies from the reflected wave. The reflected wave is approximately constant in shape, or time scale, with angle of incidence. Its amplitude has a broad maximum at normal incidence. In addition to the reflected, diffracted, and transmitted wave, waves can travel along the plate, in which case the plate can act as a wave guide. As a consequence of dispersion in the guided waves a precursor precedes the explosion wave as it travels along the plate. The dependence of the frequency, length, and amplitude of this precursor upon orientation...

The propagation of two‐dimensional time harmonic waves through a plane layered viscoelastic medium is considered in terms of the equivalent elastic plane strain problem with modified Lame constants that are complex and frequency dependent. The problem is formulated directly in terms of stresses and displacements rather than potentials and is solved by matrix methods. If the incident wave is not attenuated in the direction parallel to the layering, as would occur if the incident wave were to travel through a semi‐infinite elastic half‐space before striking the viscoelastic layers, interface waves could be generated only if one of the layers is “pseudoelastic,” i.e., has at least one real wave speed. In this case, interface waves may be generated in the same manner as in the purely elastic case. If the incident wave is attenuated in this direction, however, interface waves could be generated for specific angles of incidence and material properties.

Summary The reflected and transmitted waves due to an elastic plane sinusoidal P or SV wave impinging on the plane interface between an elastic and a linearly viscoelastic medium are found analytically for any type of viscoelastic behaviour. The properties of these waves depend both on the frequency of the incident wave and the angle of incidence of the impinging wave. Some general properties of the transmitted waves are that both the dilatational and equivoluminal waves in the viscoelastic media have refraction angles less than 90°, the displacement trajectories of material points in the viscoelastic media are ellipses, and the waves attenuate with increasing distance from the interface.

We are interested in the manner of propagation of hydromagnetic waves in the Earth's upper atmosphere. The density changes continuously in the atmosphere and the steady magnetic field is a dipole field, but we discuss here the laws of reflection and refraction of hydromagnetic waves at a plane boundary between two semi-infinite homogeneous media of anisotropic electrical conductivity under a uniform steady magnetic field. When the steady magnetic field is perpendicular to the boundary plane, the laws of reflection and refraction are simple, but when the steady magnetic field makes an angle with the plane, they are complicated. The incident ordinary (or extraordinary) waves may be reflected or refracted as the ordinary or extraordinary waves and propagate anisotropically.

In this paper some special types of finite amplitude wave motions are considered, for which kinematical non-linearities do not arise in the equations of motion of an elastic solid. Consequently, only constitutive non-linearities occur and, for special classes of materials, solutions may be read off from corresponding solutions in the linear theory. These include SH-waves1 and Love waves in layered or inhomogeneous media. Finite amplitude transverse circularly-polarized harmonic progressive waves are shown to propagate in any compressible or incompressible isotropic elastic material. Some effects of homogeneous pre-stress are also investigated.

The problem of wave propagation in linear elastic membranes is treated using the method of Hadamard. Speeds of propagation, wave shape and decay, as well as coupling effects, are obtained for longitudinal and transverse waves. Examples are considered which illustrate the features of the theory.

An expression for effective elastic constants which control the behavior of acoustic waves interacting with polarization waves in ferroelectrics is derived using the linear response theory. If interactions due to the piezoelectric effect only are taken into account, then the frequency and space dispersion of effective elastic constants are essentially determined by those of the linear dielectric susceptibility. Some consequences of this direct relation between the elastic and dielectric properties of ferroelectrics are discussed for the ultrasonic attenuation and the Brillouin or neutron scattering by acoustic waves.

The possibility of obtaining true microminiaturization by the use of elastic wave circuitry on solids is hampered by the lack of knowledge regarding the behavior of the constituents of those circuits. Since boundary value problems involving elastic waves in solids are generally very intricate and difficult to solve, a direct frontal attack on those problems will in many cases lead to frustration. In this paper, a series of steps is outlined which avoids the frontal attack and lends itself to a systematic procedure for achieving the understanding sought. It involves the application of concepts and techniques of proven value in electromagnetic microwaves to corresponding categories of problems in elastic guided waves. To demonstrate the value of this approach, it is used to derive the properties of several well-known types of elastic wave on layered media, such as Rayleigh surface waves, leaky Rayleigh waves, Lamb waves, and Love waves. In the building-block approach employed, the results derived separately include transmission-line models for body waves in fluids and isotropic solids, with expressions for the characteristic impedances and the velocity and stress vector mode functions, and equivalent networks for several types of interface which are constituents of the layered media mentioned above. The propagation properties of the guided waves are then obtained by the use of the transverse resonance procedure in a systematic, simple, and direct fashion.