Showing papers on "Longitudinal wave published in 1973"

Structure-borne sound

Abstract: Although sound waves in structures cannot be heard directly, and only be felt at low frequencies, they play an important role in noise control, because many sound signals are generated or transmitted in structures before they are radiated into the surrounding medium. In several respects sound waves in structures and sound waves in gases or liquids are similar, there are, however, also fundamental differences, which are due to the fact that solids have a certain shear stiffness, wheras gases or liquids show practically none. As a consequence acoustic energy can be transported not only by the normal compressional waves but also by shear waves and many combinations of compressional (sometimes loosely called longitudinal) and shear waves . For noise control purposes bending waves (which are a special combination of compressional and shear waves) are of primary importance; for some special cases (quasi-) longitudinal waves and torsional waves also have to be considered.

Experiments on ion‐acoustic solitary waves

Abstract: Ion‐acoustic solitary wave (solitons) have been studied experimentally by employing a double‐plasma device. The solitary waves are found to be produced from both a single compressional pulse and a continuous wave. A rarefaction pulse also produces solitons if the pulse width is sufficiently wide. A theory based on the Schrodinger equation accounts for the number of solitons. Recurrence to the original state is observed when a continuous wave is launched. A simple wave‐wave coupling analysis for the recurrence of the original state is given.

Decay of MHD waves by phase mixing

Abstract: It is well known, [1–6], that the linearized equations of motion of ideal MHD possess a continuous spectrum which leads to damping of propagating waves through phase mixing. We show how this arises by examining the dispersion relation for plasmas with non-uniform profiles and comparing the results with those of a sharp boundary model. In this paper the special case of the non-uniform sheet-pinch is examined in order to present the mathematical details as clearly as possible. It is shown that as a result of the non-uniformity the frequency of the waves is a complex quantity having a real and imaginary part. The corresponding eigenfunctions and their mathematical pathology are discussed.

Elastic waves in rotating media

Abstract: 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. Introduction. In this paper plane wave propagation in a linear, homogeneous, isotropic elastic medium will be considered, with the assumption that the entire elastic medium is rotating with a uniform angular velocity. If the coordinate system is taken as fixed in the rotating medium, this introduces additional terms in the equations of motion: a centripetal and a Coriolis acceleration. We consider small-amplitude waves propagating in the medium and exclude any discussion of the time-independent stresses and displacements that are caused by centrifugal forces and other possible body forces. In the following section the equations governing plane-wave solutions in an infinite rotating medium are formulated and it is shown that there are three real slowness surfaces, each corresponding to the root of a cubic characteristic equation. It is shown that the phase speed in all cases depends on the ratio of the wave frequency to the rotational frequency, thus making it clear that the rotation causes the material to be dispersive. The actual slowness surfaces are given for various values of Poisson's ratio and the frequency ratio. In the next section the energy flux for plane waves is discussed and it is proved, for any admissible plane wave, to be perpendicular to the slowness surface at the point indicated by the slowness vector (essentially the wave number vector) of the wave. Actual displacements that occur are discussed qualitatively in the subsequent section. It is seen that, in general, the various modes are neither shear nor compressional, but combinations of both. All exceptional cases of pure shear or pure compressional modes are discussed. In the last section free surface phenomena are discussed. To describe the reflection of plane waves from a plane free surface, use is made of the slowness surfaces. Much qualitative information on types of reflected waves can be brought out even without the use of the explicit expressions for the slowness vector, such as under what circumstances one or two of the reflected waves will be surface waves, i.e. have a complex * Received December 30, 1971; revised version received March 28, 1972. 116 MICHAEL SCHOENBERG AND DAN CENSOR slowness vector. As an extension of this, a solution which consists only of surface waves is discussed. This can be thought of as a generalized Rayleigh wave. The analysis involved in the derivations is roughly similar to that used for wave propagation in anisotropic media by Synge [1], [2] and Musgrave [3], among others. A rotating medium can be thought of as a type of transverse isotropic medium; i.e., all directions orthogonal to some direction, in this case the axis of rotation, are equivalent. However, there are basic differences between rotational and material anisotropy. For a rotating medium, substitution of a plane wave solution into the equations of motion leads to a homogeneous set of linear equations for which the matrix of the coefficients is Hermitian, instead of symmetric, as is the case for non-rotating media. Hence the eigenvectors, i.e. the displacement vectors, even for real slowness vectors, are complex instead of real, implying that the particle trajectories are elliptical. Further, as mentioned above, the solutions for a rotating medium are frequency-dependent. In addition, there is no easily perceived eigenvector that can be used to find one root of the cubic characteristic equation, thus leaving only a quadratic equation to be solved, as for a nonrotating transverse isotropic medium (see [1, p. 331]). The standard index notation is used throughout, e.g. for the position vector x = Xi&i = x1e1 + x2e2 + x3es . The cap will always denote unit vectors. Use is made both of vector and indicial notation. The rotating elastic medium. Consider an infinite homogeneous, isotropic, linear elastic medium characterized by a density p, a shear modulus and a bulk modulus. These quantities determine, in the usual fashion, a shear wave speed C, and a pressure wave speed Cv . The medium is rotating uniformly with respect to an inertial frame, and the constant rotation vector in an Xi , x2 , rectangular Cartesian frame rotating with the medium is ii = S!w. The unit vector w will denote the direction of the axis of rotation (according to the right-hand rule) throughout. The displacement equation of motion in such a rotating frame has two terms that do not appear in the non-rotating situation. As we are looking for time-varying dynamic solutions, the time-independent part of the centripetal acceleration X (£} X x), as well as all body forces, will be neglected. Thus our dynamic displacement u is actually measured from a steady-state deformed position, the deformation of which, however, is assumed small. The equation which governs the dynamic displacement u is (iCl Cl)V(V-u) + C2,V2u = u + a X (n X u) + 2£2 X u, (1) where the dot denotes time differentiation. The term Q X (H X u) is the additional centripetal acceleration due to the time-varying motion only, and the term 2£i X u is the Coriolis acceleration. All other terms are as usual for a linear elastic medium under the assumptions of smajl strains and displacements. We look for plane wave solutions of the form u = (R U exp iw ^ -')\" = (R U exp ~ 1 (2) where (R denotes the real part, U is a constant vector, in general complex, a; is the angular frequency, C is the phase speed, n is a unit vector in the direction of propagation, and S is the non-dimensional slowness vector with amplitude S = CJC in the direction of fi. ELASTIC WAVES IN ROTATING MEDIA 117 Substitution of (2) into (1) gives (1 0)(S-U)S + /3S2U = U (1/oj2)£1 X (Q X U) + (2i/u)Q X U, /3 = (C./C,,)2, or, by making use of Si = S2w and the vector identity w X (w X U) = (wU)w — U (i /s)(s-u)s + /3S2u = (i + r2)u r2w-Uw + 2irw x u, r = a/«. (4) This vector equation stands for three scalar equations on the three components U, , and they can be written as Mull, = [(1 + T2) 5,,— T3wtu— 2iTeiikwk — (1 JS)sis/ — fisksk 8u]Ui = 0, (5) where Sif is the Kronecker delta and eijk is the permutation tensor. Note that, because of the Coriolis term, the matrix Mu is no longer symmetric but Hermitian. The necessary and sufficient condition for the existence of non-trivial eigenvectors, [/,•, is det Mu = 0. In general, there may be real and complex vectors, , which satisfy the condition det M a = 0. However, complex

Elastic waves in free anisotropic plates

Abstract: 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.

Surface Waves in Acoustics

Abstract: : The article surveys the essential results obtained concerning acoustic surface waves on curved surfaces, and establishes their relationships with the corresponding plane surface waves. In view of the limited generality of the existing literature, the excitation mechanisms considered here for the generation of such surface waves have mostly been taken as those provided by a plane incident acoustic wave; more general mechanisms are briefly mentioned. A classification of surface and lateral waves on flat surfaces, and with a description of their properties is given. Surface waves generated by sound scattering on surfaces of simple or arbitrary curvature, both penetrable and impenetrable is discussed. Theoretical results are considered mainly, but selected illustrative examples of the recent experiments are given also. (Author)

Interactions of Solitary Waves —A Perturbation Approach to Nonlinear Systems—

Abstract: An approximate method of studying interactions between two solitary waves which propagate in opposite directions is presented In the first approximation, the solution is described by s superposition of two solitary waves which are governed by their respective Korteveg-de Vries equation The second order approximation gives a small correction where the two waves overlap one another The method is extended to the system, in which there exist n “quasi-simple” waves (the simple waves under the effects of higher derivative terms, such as dispersions of dissipations) The possibility that n “quasi-simple” waves can be superposed to describe nonlinear systems is studied Applications to ion acoustic waves in collisionless plasmas and shallow water waves are discussed

Alfvén waves in the solar wind: Wave pressure, poynting flux, and angular momentum

Abstract: We consider three effects of Alfven waves propagating in the solar wind. (1) Modification of the angular momentum balance of the solar wind by Alfven waves in the presence of thermal anisotropy is considered. The Alfven waves are found to reduce the azimuthal velocity υϕ at 1 AU. This effect occurs because the Alfven waves are transverse and represent an additional component of the pressure perpendicular to the magnetic field. The effect is large if /B0² ≳ ⅓, and it is concluded that thermal anisotropy cannot be invoked to explain the large azimuthal velocity of the solar wind. (2) Modification of the angular momentum balance of the solar wind by Alfven waves by finite-wavelength (non-WKB) effects is considered. The Alfven waves reduce υϕ at 1 AU by reducing the heliocentric distance of the critical point that appears in the equation for υϕ. This effect occurs because the waves act like a Reynold's ‘viscosity,’ but the sign is such that the viscosity is negative, leading to antirotation of the solar wind. This effect is only important for waves with ω−1 ≳ 10 hours. (3) Finite-wavelength modifications of the wave pressure are considered. It is found that the wave pressure is reduced close to the sun. This effect is important near 2 RE for waves with ω−1 ≳ 2 hours.

Damping and nonlinear wave-particle interactions of Alfvén-waves in the solar wind

Abstract: Since most Alfven-waves in the solar wind are observed to come from the Sun, nonlinear wave-particle interactions can be expected to constitute their dominant dissipation process. The growth or damping of two circularly-polarized Alfven-waves with wave vectors parallel to the ambient magnetic field is calculated using kinetic theory. If the waves are oppositely polarized they both damp proportional to their frequency. If the waves are of the same polarization, both the lower frequency wave and the plasma particles gain energy at the expense of the higher frequency wave. Thus, with increasing distance from the Sun, a steepening of the power spectrum is expected. For waves propagating in the same direction, the interaction is negligible for small β, while it becomes appreciable for β≥10−1. For conditions typical of the solar wind near 1 AU an observed half-hour linearly-polarized wave, for example, with δB=0(B0) has a damping time of about 10 h.

Diffraction of Elastic Waves in Multiply Connected Bodies

Abstract: : ;Contents: Linear problems of wave dynamics of isotropic deformable bodies; Representation of the solution of a wave equation in certain coordinate systems; Diffraction of waves in bodies limited by circular cylindrical surfaces; Wave diffraction in bodies with spherical cavities; Solution to problems of the diffraction of elastic waves on cylindrical surfaces; Wave diffraction in semi-bounded bodies with cavities; Principles of the theory of wave propagation in fibrous and granular media.

On Waves in Composite Materials with Periodic Structure

Abstract: The propagation of waves through nonhomogeneous elastic material, with a periodic structure of elastic constants and density variation, can be conveniently treated in terms of Floquet waves. These ...

Plane-Wave Representations for Scalar Wave Fields

Abstract: The theory of the representation of wave fields in terms of superpositions of monochromatic plane waves is presented for fields satisfying the inhomogeneous scalar wave equation The discussion includes expansions of the type originally used by E T Whittaker involving only homogeneous plane waves, and of the type introduced by H Weyl involving both homogeneous and inhomogeneous plane waves Expressions for the plane-wave amplitudes for both types of representations are obtained in terms of the source function, and precise conditions under which each expansion is valid are given It is shown that when both types of expansions are valid, the superposition of inhomogeneous plane waves in the Weyl-type representation is equal to the superposition of the homogeneous plane waves that propagate into a specific half-space in the Whittaker-type representation It is shown also that in restricted space-time regions only a certain subset of the plane waves in the Whittaker-type expansion contribute to the field This result leads to a simple expression for the field valid at large distances from the source

Transverse surface waves on a piezoelectric material carrying a metal layer of finite thickness

Abstract: Theory of the propagation of a surface wave related both to Bleustein‐Gulyaev waves and Love waves is developed. The wave has unidirectional particle motion perpendicular to the direction of propagation and parallel to the surface of a piezoelectric material which is covered with a finite‐thickness layer of an isotropic conducting material. An equation relating phase velocity to material costants is solved in closed form for a piezoelectric material of class 6mm, and conditions for the existence of various modes are presented. Piezoelectricity allows a nonleaky but dispersive wave to exist under conditions is which no love wave is possible, namely, when the shear wave velocity in the layer is greater than that in the substrate. Numerical results are presented for aluminium, gold, and zinc layers on PZT 4 ceramic.

Propagation of Solitary Pulses in Interactions of Plasma Waves

Abstract: New solutions of solitary pulses in three-wave interactions are obtained, which were not given in the previous paper of this series. In those solutions, the wave of lower frequency has a shock-type envelope and behaves as a “pump” while the other two waves have the pulse-type ones. In addition, by considering two effects, linear translations of the waves and nonlinear interactions, a physical mechanism of formation of the solitary pulses is found. The theory is applied to the interaction among two Alfven waves and an ion-acoustic wave, which propagate along the external magnetic field,

Steady‐state solutions for relativistically strong electromagnetic waves in plasmas

Abstract: New steady-state solutions are derived which describe electromagnetic waves strong enough to make plasma ions and electrons relativistic. A two-fluid model is used throughout. The following solutions are studied: (1) linearly polarized waves with phase velocity much greater than c; (2) arbitrarily polarized waves with phase velocity near c, in a cold uniform plasma; (3) circularly polarized waves in a uniform plasma characterized by a scalar pressure tensor. All of these waves are capable of propagating in normally overdense plasmas, due to nonlinearities introduced by relativistic effects. The propagation of relativistically strong waves in a density gradient is examined, for the example of a circularly polarized wave strong enough to make electrons but not ions relativistic. It is shown that such a wave propagates at constant energy flux despite the nonlinearity of the system.

Amplification and decay of long nonlinear waves.

Abstract: The interaction of weakly nonlinear waves with slowly varying boundaries is considered. Special emphasis is given to rotating fluids, but the analysis applies with minor modifications to waves in stratified fluids and shallow-water waves. An asymptotic solution of a variant of the Korteweg-de Vries equation with variable coefficients is developed that produces a 'Green's law' for the amplification of waves of finite amplitude. For shallow-water waves in water of variable depth, the result predicts wave growth proportional to the -1/3 power of the depth.

Some Results for Surface Gravity Waves on Shear Flows

Abstract: This work attempts to fill some gaps in the subject of steady surface gravity waves on two-dimensional flows in which the velocity varies with depth, as is the case for waves propagating on a flowing stream. Following most previous work the theory is basically inviscid, for the shear is assumed to be produced by external effects: the theory examines the non-viscous interaction between wave disturbances and the shear flow. In particular, some results are obtained for the dispersion relationship for small waves on a flow of arbitrary velocity distribution, and this is generalized to include the decay from finite disturbances into supercritical flows. An exact operator equation is developed for all surface gravity waves for the particular case of flow with constant vorticity; this is solved to give first-order equations for solitary and cnoidal waves in terms of channel flow invariants. Exact numerical solutions are obtained for small waves on some typical shear flows, and it is shown how the theory can predict the growth of periodic waves upon a stream by the development of a fully-turbulent velocity profile in flow which was originally irrotational and supercritical. Results from all sections of this work show that shear is an important quantity in determining the propagation behaviour of waves and disturbances. Small changes in the primary flow may alter the nature of the surface waves considerably. They may in fact transform the waves from one type to another, corresponding to changes in the flow between super- and sub-critical states directly caused by changes in the velocity profile.

On the existence of critical levels with applications to hydromagnetic waves

Abstract: It is proved that a critical level, at which a wave packet is neither reflected nor transmitted, can exist only if the wave normal curve, which is formed by taking the cross-section through the wave normal surface in the plane of propagation, possesses an asymptote which is parallel to the direction of variation of the properties of the medium through which the wave packet moves. This condition, when applied to various types of hydromagnetic waves (such as hydromagnetic waves of the inertial or gravity type, or slow magnetoacoustic waves), shows that critical levels for such waves can exist only if the direction of spatial variations of the medium is perpendicular to the ambient magnetic field. Provided that the angle between the gravitational acceleration, or the rotation axis, and the magnetic field is not zero, hydromagnetic critical levels, characteristic of the gravity or inertial type, act like ‘valves’ in the sense that the wave packet can pierce the critical level from one side and is captured from the other side. It is also pointed out that critical-level behaviour is to some extent a consequence of the WKBJ approximation since the other limit, namely when the waves feel an almost discontinuous behaviour in the properties of the medium, gives markedly different results, particularly in the presence of streaming, which can give rise to the phenomenon of wave amplification.

Water wave models and wave forces with shear currents

Abstract: Water wave modeIs, incorporating shear currents, are developed for linear and nonlinear waves. The first model assumes a constant vorticity over the depth of the fluid; the case for a wave propagating on a linear shear current. For greater generality, a second model is presented which assumes that the fluid is composed of two layers, each with a different, but constant, vorticity. The nonlinear solutions require the use of a numerical perturbation procedure. The last wave model, using a finite difference approach, generates waves propagating on arbitrary vorticity distributions. Examples of the effect of the vorticity on the waves are presented. Further, the use of two of these models in the analysis of actual measured wave data is shown. Wave force measurement programs conducted for the purposes of obtaining drag and inertia coefficients on structures are affected by the presence of currents, and the biases introduced into these coefficients by neglecting the currents are investigated via small amplitude wave theory. Further, an approximate procedure for de termining currents for measured wave force data is presented, as weIl as some results from Wave Project 11 data, which were obtained during Hurricane Carla in the Gulf of Mexico.

Ultrasonic Velocities in BaTiO3

Abstract: The sound velocities in BaTiO 3 has been observed near its paraelectric to ferroelectric phase transition point. The anomalous part of the sound velocity of the longitudinal wave propagating along [100] direction depends on temperature as A [( T - T 0 )/ T 0 ] -ζ with ζ=0.41 and A =1.79×10 4 cm·sec -1 where T 0 is the paraelectric temperature. The critical index ζ is close to the theoretical value of 1/2 as predicted by Dvořak. On the other hand, the velocity of the transverse wave does not show any appreciable critical behavior. The value of A is also compared with the theory given by Dvořak. The observed curve for the longitudinal wave is in good agreement with the theoretical curve, while that for the transverse wave shows a notable discrepancy.

Critically Refracted Waves in a Spherically Symmetric Radially Heterogeneous Earth Model

Abstract: Summary A theoretical analysis of acoustic waves refracted by a spherical boundary across which velocity and density increase abruptly and below which velocity and density may either increase or decrease continuously with depth is formulated in terms of waves generated at a harmonic point source and scattered by a radially heterogeneous spherical body. Through the application of an Earth-flattening transformation on the radial solution and the Watson transform on the sum over eigenfunctions, the solution to the spherical problem for high frequencies is expressed as an integral for the corresponding half-space problem in which the effect of boundary curvature maps into an effective positive velocity gradient with depth. The results of both analytical and numerical evaluation of this integral can be summarized as follows for body waves in the crust and upper mantle: (1) In the special case of a critical velocity gradient (a gradient equal and opposite to the effective curvature gradient), waves interacting with the boundary at the critical angle of incidence have the same form as the classical head wave for flat, homogeneous layers. (2) For gradients more negative than critical, the amplitude of waves incident at the critical angle decay more rapidly with distance than the classical head wave. (3) For gradients that are positive, null, and less negative than critical, the amplitude of waves near the critical angle decays less rapidly with distance than the classical head wave, and at sufficiently large distances, the refracted wave field can be adequately described in terms of ray-theoretical diving waves. At intermediate distances from the critical point, the spectral amplitude of the refracted wave is scalloped due to multiple diving wave interference.

Stokes' expansion of internal deep water waves to the fifth order

Abstract: Stokes' expansion is applied to the internal waves of finite amplitude, which propagate on the interface between two layers of infinite thickness. Stream function, wave profile, phase velocity and mass transport velocity are given in the fifth order approximation. It is shown that (a) phase velocity increases with increase of wave steepness, (b) mass transport appears in the direction of the wave propagation in both layers as in the case of the surface waves, and (c) when the density difference is very small, the wave profile is flattened not only at the troughs but also at the crests.