Showing papers on "Longitudinal wave published in 1969"

Stress-induced velocity anisotropy in rock: An experimental study

Abstract: Application of uniaxial stress to a sample of granite causes elastic wave velocity anisotropy. Compressional waves travel fastest in the direction of the applied stress. Two shear waves travel with generally different speeds in any direction, exhibiting acoustic double refraction which increases with increasing stress.

Bow shock associated waves observed in the far upstream interplanetary medium

Abstract: Fifty orbits of Explorer 34 data have been used to study 0.01–0.05 Hz transverse waves in the interplanetary medium region between the bow shock and the spacecraft apogee of 34 RE. It is concluded that the waves are associated with the earth's bow shock since they only occur when projection of the interplanetary field observed at the spacecraft intersects the shock. The waves are observed 18.5% of the time when a total of 134 days of interplanetary data is considered, but more than 90% of the time when the field has the proper orientation with respect to the bow shock. On the basis of this result it is suggested that these waves with 20–100 second periods are a permanent feature of the solar wind-earth interaction. The transverse component of the waves is typically several gammas in amplitude in 4–8 gamma fields. The disturbance vector in the XY plane generally exhibits the same sense of rotation in a coordinate system where the field is oriented along the positive z axis. Attenuation of wave amplitudes with distance from the bow shock is estimated to be only a factor of 2 when the spacecraft is 15 RE from the bow shock. The absence of waves at particular field orientations, even though the field line intersects the shock, is interpreted as a propagation effect. This observation is the basis for calculations that yield an average velocity in the plasma frame of 2.7 ± 0.4 times the solar wind velocity. Whistler propagation and local generation by two-stream instability are discussed as alternate theoretical explanations for the presence of the waves. It is suggested that the data favor the latter mechanism.

Momentum transport by gravity waves

Abstract: SUMMARY A review of theories concerning the propagation of internal gravity waves in a horizontally uniform shear flow concludes that an upward transport of horizontal momentum inevitably accompanies the generation of such waves in the atmosphere, the mean flow being affected only at precisely those levels where the waves are dissipated. If the mean wind depends on horizontal position there may be a continuous transfer of momentum from the waves to the mean flow during propagation. In the absence of intense clear air turbulence or a critical level where the intrinsic frequency vanishes, many waves propagate upwards to great heights, even if the mean wind is time dependent. Computations of the wave drag in a 19 m s-I gradient wind over hilly terrain in north Wales show that on that occasion it amounted to 4 dyne cm-*, of which 3 dyne cm-* probably acted on the atmosphere above 20 km.

A New Class of Nonlinear Waves in Parallel Flows

Abstract: Waves in parallel shear flows are found to have different characteristics depending on whether nonlinear or viscous effects dominate near the critical layer. In this paper a nonlinear theory is developed which gives rise to a class of disturbances not found in the classical viscous theory. It is suggested that the modes found from such an analysis may be of importance in the breakdown of laminar flow due to free stream disturbances.

The Transformation of a Solitary Wave over an Uneven Bottom

Abstract: Based on a set of approximate equations for long waves over an uneven bottom, numerical results show that as a solitary wave climbs a slope the rate of amplitude increase depends on the initial amplitude as well as on the slope. Results are also obtained for a solitary wave progressing over a slope onto a shelf. On the shelf a disintegration of the initial wave into a train of solitary waves of decreasing amplitude is found. Experimental evidence is also presented.

Character of pseudo surface waves on anisotropic crystals

Abstract: When the free surface is anisotropic, mode of elastic surface‐wave propagation can arise that has many of the properties of a normal surface wave but has a phase velocity higher than that of the transverse bulk waves in the corresponding direction. The pseudo surface wave is a coupled mode involving terms decaying rapidly beneath the free surface and a term representing a bulk wave radiating into the solid. For many choices of crystal and plane of propagation, the contribution of the bulk term over a range of directions is small enough that the energy of the wave is essentially concentrated within a few wavelengths of the free surface and flows parallel to the surface as with the normal elastic surface waves. Moreover, in certain specific directions, the bulk term disappears completely and the pseudo‐surface wave has all the properties of a normal surface wave. The method of computation of the characteristics of the pseudo surface waves is outlined here and typical results of velocity, displacements and e...

Progressive internal waves on slopes

Abstract: The refraction of progressive internal waves on sloping bottoms is treated for the case of constant Brunt—Vaisala frequency. In two dimensions simple, explicit expressions for the changing wavelengths and amplitudes are found. For small slopes, the solutions reduce to simple propagating waves at infinity.The singularity along a characteristic is shown to be removable, though the solutions are now inhomogeneous waves. The viscous boundary layers of the wedge geometry are briefly considered with the inviscid solutions remaining as interior solutions.A theory valid for small slopes is obtained for three-dimensional waves. The waves are refracted in the usual manner, turning parallel to the beach in shallow water.

Mixed region collapse with internal wave generation in a density-stratified medium

Abstract: An experiment has been conducted modelling a two-dimensional mixed region collapsing in a continuously density-stratified medium. The process of collapse can be divided into three stages. Empirical formulae have been derived to describe the process of the first two stages, during which gravitational effects determine the modelling criterion. The collapse in the final stage is complicated by viscous effects; a ratio between the Froude number and the Reynolds number seems to provide a probable modification of the time scale.The pattern of internal waves generated by the initial impulsive collapse of this mixed region can be represented by moving rays connecting either wave crests or troughs. These rays move away from the collapse centre and at the same time decrease their slopes from the horizontal. A simpler steady-state wave pattern generated by an oscillating plunger has also been studied. Taken together, these experimental results are interpreted to show that the energy density of the collapse is skewed toward high frequencies and is peaked at 8/10 of the Brunt-Vaisala frequency of the medium.

A Nonlinear Mechanism for the Generation of Sea Waves

Abstract: Recent observations of the growth of sea waves under the action of wind have established that the rate of growth is several times greater than has yet been accounted for. In this paper a new mechanism of wave generation is proposed, based on the idea of a maser-like action of the short waves on the longer waves. It is shown that when surface waves decay they impart their momentum to the surrounding fluid. Short waves are readily regenerated by shear instability. But a longer wave passing through shorter waves causes the short waves to steepen on the long-wave crests. Hence the short waves impart more of their momentum to the crests of the long waves, where the orbital motion of the long waves is in the direction of wave propagation. If the short waves are decaying only weakly (under the action of viscosity), the effect on the long waves is slight. But when the short waves are forced to decay strongly by breaking on the forward slopes of the long waves the gain of energy by the latter is greatly increased. Calculations suggest that the mechanism is capable of imparting energy to sea waves at the rate observed.

A variational method for weak resonant wave interactions

Abstract: Whitham’s variational method is formulated so as to apply to weak second-order resonant interactions among waves whose amplitudes and phase angles vary slowly with position and time. The method is applied in detail to capillary-gravity wave interactions. An internal gravity waves problem is also discussed briefly. The method leads to new and substantial simplifications of the interaction equations. This makes possible the proof of local conservation of total mean wave energy and momentum laws. These, together with another integral of the motion, are found to be of central importance in classifying and characterizing the slow modulations of planewave-like form. Such a classification is given in detail for all initial values of phase angles and relative amplitudes. All progressive uniform waves in the capillary range are found to be unstable with perturbation growth rates which can be of first order in the wave slopes. In this formulation amplitude dependent first-order corrections of classical frequency and/or wave-number arise for all waves participating in a resonance. A few predictions which could be verified by simple experiments are made.

Sound and Shock Waves in Liquids Containing Bubbles

Abstract: The propagation of infinitesimal sound waves in a liquid containing gas bubbles is considered. Relative motion of gas bubbles and liquid is explicitly allowed for, and it is shown that a significant error in the speed of waves may arise if the relative motion and fluctuations of mass fraction are neglected. The structure of steady shock waves is also derived.

A Quantitative Evaluation of Seismic Signals at Teleseismic Distances—II: Body Waves and Surface Waves from an Extended Source

Abstract: Summary Expressions are derived for the body wave and surface wave displacement at epicentral distances of between 30" and 100" from an extended or moving source. The source is assumed to lie entirely within a finite region on a plane. Otherwise it can be quite general. The effects of layering at the source and receiver are taken into account. Attenuation due to linear anelasticity is allowed for by an empirical factor. Propagation through the mantle is assumed to follow ray theory and the sphericity of the Earth is taken into account by the use of geometrical spreading factors. Expressions for the surface waves generated by a point source in a layered halfspace have been given by both Haskell (1964) and Harkrider (1964) using essentially the same method (i.e. the Thomson-Haskell matrix theory) but with different notations. Later on, Fuchs (1966) derived similar formulae in Harkrider's notation for the body waves radiating into the lower half-space. These correspond to the waves from a seismic source which travel through the mantle before being refracted back to the surface by the velocity gradient. A scheme by which the body wave pulse from a seismic source may be calculated, allowing for the effects of transmission through the mantle and crust, was given by Carpenter (1966). The analysis applies to the waves recorded at epicentral distances between 30" and 100"; i.e. waves travelling along a ray path which lies partly in the mantle and is unaffected by the core. Kogeus (1968) applied Fuchs's results to Carpenter's theory to allow for the effects of the layered crust. He derived teleseismic waveforms due to an explosive source near the surface. A method for extending these results to sources of finite extent was indicated by Harkrider (1964) who derived expressions for the surface waves radiated from a source consisting of a horizontal point force moving with finite speed along a line. More realistic models of explosive and earthquake sources and their integration into the Thomson-Haskell theory are given by Hudson (1969) (Part I of the present Methods are, therefore, available for constructing waveforms of body waves and surface waves at distances in the range 30"-100" from a wide range of source models. We shall begin by deriving expressions for the Fourier time transforms (with transform variable o) of the body waves and surface waves from a point source of

Full wave calculations of gravity wave propagation through the thermosphere

Abstract: Full wave calculations have been performed within the frequency range of gravity waves (10−3 ≤ ω ≤ 10−2 sec−1) for a thermospheric model between 150 and 500 km altitude. In this altitude range gravity waves are coupled with heat conduction waves. Reflection, transmission, conversion, and coupling from one wave type into the other one is described by the elements of the scattering matrix. The dependence of these elements on height and angle of incidence is discussed. The transmission coefficients of gravity waves calculated by full wave theory are compared with simple ray calculations and show that ray treatment is a sufficient approximation for obliquely upward propagating gravity waves and that gravity waves predominate throughout the thermosphere. The thermosphere reacts like a selective filter with respect to upward propagating gravity waves with optimal transmission at kx ∼ ω/C (ω = angular frequency; C = velocity of sound; kx = horizontal wave number).

The attenuation of compressional waves in marine sediments

Abstract: A number of mechanisms have been proposed to account for the attenuation of compressional waves in water‐saturated sediments. These include viscous losses between the particles and the fluid, and “solid friction” losses between the particles. The mechanisms are discussed and it is shown how the low values of attenuation observed in pure clays arise from the electrical interaction forces between the surface‐active particles. It is proposed that the attenuation in clay‐ and silt‐size sediments (up to 6 phi mean diameter) arises from viscous interaction between the clay‐water “fluid” and the non‐surface‐active particles. Both new and published experimental measurements indicate that the proposed mechanism is valid, at least in a frequency range 30 to 370 kHz. For sediments of mean particle diameter greater than 6 phi, both new and published experimental results are presented to show that, although under the circumstances of a very well sorted sediment under zero overburden pressure a viscous dissipation mech...

Small amplitude waves in high β plasmas

Abstract: We present a brief survey of the phase velocities and polarizations of linear waves in an infinite, uniform, high β plasma using the two-fluid approximation. Questions of wave-particle damping are deferred. Several features of high β plasma waves differ from the familiar low β case.

Three-Dimensional and Shell-Theory Analysis of Elastic Waves in a Hollow Sphere: Part 1—Analytical Foundation

Abstract: In Part 1 of this paper an exact analysis of the nonaxisymmetric wave propagation in a hollow elastic sphere is presented. It is found that the characteristic frequency equation is independent of the longitudinal wave number. Approximate equations for thin shells and membranes are derived by way of asymptotic expansions. In general, the vibrations fall into two distinct classes, one of which is equivoluminal. Also included in the paper is a six-mode shell theory in which the effects of transverse normal strain are included. A technique due to van der Neut is used to separate the governing partial differential equations whereby two frequency equations corresponding to the two classes of vibrations are obtained.

On the Transmission of Plane Waves through Layered Linear Viscoelastic Media

Abstract: 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.

Non-linear coupling of waves in a magnetized plasma with particle drift motions

Abstract: We study non-linear interaction between three monochromatic waves which propagate parallel to the direction of a magnetic field in a plasma. The approach to the problem is hydromagnetic, including temperature effects, and the method of solution is that of coupled mode theory. In particular drift motions of the particles along the magnetic field are considered taking into account relativistic effects. The interaction of two transverse waves and one longitudinal wave is treated as well as that of three longitudinal waves. Besides, the case of two longitudinal waves and one perpendicular wave has been studied in some detail assuming the latter to have a long wavelength.The present paper represents a generalization of the problem of three-wave interaction in a plasma to situations more complex than have been treated before. The most essential limitation consists in the assumptions made for the directions of propagation of the waves.

The scattering of surface waves by a vertical plane barrier

Abstract: In this paper we use a method due to Williams(1) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.

Surface waves in hot plasmas.

Abstract: The solution of the initial value problem for a semi‐infinite Vlasov plasma, with no applied fields, is given. In the long‐wavelength limit, in addition to the usual plasma waves, terms corresponding to surface plasma waves, for which the dispersion relation is given, are found.

Rayleigh and Lamb Waves on Cylinders

Abstract: Acoustic waves propagating on the surface of solid and hollow elastic cylinders immersed in water were observed and their characteristics shown to be consistent with theory These waves were excited by an 11‐μsec pulse, projected onto the cylindrical surface at an excitation angle given by sin 1(co/cs), where co is the velocity of sound in water and cs is the velocity of the surface wave The waves may be observed, since they continually radiate energy back into the water at the excitation angle Surface waves were observed to exist on pure (98%) aluminum, 304 stainless steel, and various aluminum‐alloy cylinders Circumference‐to‐wavelength ratios were varied from 10 to 400 to discern the dispersive effects owing to the curvature of the target It was also found that inhomogeneities correlate with pure propagation characteristics in some of the targets The waves observed on air‐ and water‐filled cylinders immersed in water are shown to correspond to the symmetric and antisymmetric vibrations in the shell