Showing papers on "Longitudinal wave published in 1972"

Stability of periodic waves of finite amplitude on the surface of a deep fluid

Abstract: We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface η(r, t) and the hydrodynamic potential ψ(r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case.

Stationary solitary, snoidal and sinusoidal ion acoustic waves

Abstract: Stix's treatment of zero-damped electrostatic waves in a Maxwellian plasma is extended to the nonlinear regime. Stationary Bernstein-Greene-Krusk almodes which propagate with ion acoustic speed are constructed. This subclass consists of solitary, snoidal (=periodical waves, like ocean waves, which can be written in terms of Jacobian elliptic functions) and sinusoidal waves. A discrimination of those waves can be given by a single parameter, the steepness parameter, which contains nonlinearity, trapping of particles and dispersion. It turns out that Sadgeev's soliton represents a special case of the class solitons having the largest width and the lowest velocity. Hence a modified Korteweg-de-Vries equation must exist with a stronger nonlinearity.

Analysis of the resistance increase in waves of a fast cargo ship

Abstract: A new method to calculate the added resistance of a ship in longitudinal waves is discussed. For the particular case of a fast cargo-ship the calculated values are compared with experimental results, and a satisfactory agreement is shown. In addition the experiments with the considered shipform confirm that added resistance varies as the squared wave height for constant speed and wave length.

Colliding plane gravitational waves.

Abstract: The equations governing the collision of two plane gravitational waves are derived. The general exact solution representing this situation when both waves are linearly polarized are found, and some special solutions of possible physical interest are discussed in detail.

IMPROVED KORTEWEG--deVRIES EQUATION FOR ION-ACOUSTIC WAVES.

Abstract: The Korteweg‐deVries equation describing nonlinear ion‐acoustic waves in a plasma with finite ion temperature is derived and the temperature dependences of soliton width and speed are obtained.

On the interaction of surface and internal waves

Abstract: The steady-state interaction between surface waves and long internal waves is investigated theoretically using the radiation stress concepts derived by Longuet-Higgins & Stewart (1964) (or Phillips 1966). It is shown that, over internal wave crests, those surface waves for which cg0cosϕ0 > ci experience a change in direction of propagation towards the line of propagation of the internal waves and their amplitudes are increased. Here cg0 is the surface-wave group speed at U = 0, ϕ0 is the angle between the propagation direction of the surface waves at U = 0 and the propagation direction of the internal waves, and ci is the phase speed of the internal waves. If cg0cos ϕ0 < ci the direction of the surface waves is turned away and their amplitudes are decreased. Over troughs the opposite effects occur.At positions where the local velocity of surface-wave energy transmission measured relative to the internal wave phase velocity is zero, i.e. cg + U − ci = 0, there is a singularity in the energy of the surface waves with resulting infinite amplitudes. It is shown that at these critical positions two wavenumbers which were real and distinct on one side coalesce and become complex on the other. The critical positions are thus shown to be barriers to the propagation of those wave-numbers. It is also shown that there is a critical position representing the coalescence of three wavenumbers. Surface-wave crest configurations are shown for three numerical examples. The frequency and direction of propagation of surface waves that exhibit critical positions somewhere in an internal wave field are shown as a function of the maximum horizontal surface current. This is compared with measurements of wind waves that have been reported elsewhere.

Nonlinear waves in a Kelvin-Helmholtz flow

Abstract: Nonlinear waves on the interface of two incompressible in viscid fluids of different densities and arbitrary surface tension are analysed using the method of multiple scales. Third-order equations are presented for the space and time variation of the wavenumber, frequency, amplitude and phase of stable waves. A third-order expansion is also given for wavenumbers near the linear neutrally stable wave-numbers. A second-order expansion is presented for wavenumbers near the second harmonic resonant wavenumber, for which the fundamental and its second harmonic have the same phase velocity. This expansion shows that this resonance does not lead to instabilities.

The propagation of a weak nonlinear wave

Abstract: We consider the propagation of a weak nonlinear wave whose energy is concentrated in a narrow band of wavenumbers in a fluid which is both dispersive and dissipative. We use the small amplitude equations of Whitham's theory of slowly varying wave trains, modified slightly to include dissipation, to show that the modulation of the wave may be described by a nonlinear Schrodinger equation. For long waves which are purely dispersive we obtain the Kortewegde Vries equation, and for long waves which are dissipative we obtain Burgers’ equation by suitable transformations of the nonlinear Schrodinger equation. We mention the problem of Stokes waves in deep water and comment briefly upon invariant far-field theory.

General variational methods for waves in elastic composites

Abstract: General variational theorems in which the displacement, the stress, and the strain in one case, and the displacement and the stress in another case, are given independent variations, and which include appropriate general bondary and discontinuity conditions, are developed with a view toward the application to harmonic waves in elastic composites with periodic structures. The one-dimensional case is first developed in detail, and in order to demonstrate the effectiveness of the results, especially their accuracy in providing the dispersion curve, waves propagating normal to layers in a layered composite are discussed, and numerical results are presented; see Tables I and II. Then the general three-dimensional case is considered, and the results are applied to waves propagating normal to the fibers in a fiber-reinforced composite.

Geometric Dispersion of Acoustic Waves by a Fibrous Composite

Abstract: This study was initiated to examine the transmission of acoustic waves through a fibrous composite whose only dispersive mechanism was geometric. The elastic-elastic composite chosen for study was composed of tungsten wires embedded in an aluminum matrix. This unidirectional com posite was manufactured in two constituent ratios, 2.2 and 22.1 percent by volume of tungsten. The dispersive characteristics of these composites were determined for harmonic waves propagating normal to the axes of the fibers by using standard water-bath techniques with wide-band transducers.The dispersion data obtained demonstrate that fibrous composites be have as wave filters which selectively transmit or reflect periodic waves. Further, this wave filtering is shown to be a boundary layer phenomenon which can and must be eliminated from dispersion data if it is to be meaningful.

Nonlinear dissipation of Alfvén waves

Abstract: Alfven waves are generated easily in many cosmic plasmas, but they possess no linear damping mechanism since they are not compressive. The most prominent nonlinear damping occurs when one Alfven wave decays into another plus a slow magnetosonic wave, or two Alfven waves combine into one fast magnetosonic wave; the resulting magnetosonic waves can then be dissipated. The nonlinear coupling rates are presented, with special emphasis on the astrophysically important case of sound speed ≪Alfven speed. Streaming cosmic rays generate Alfven waves moving in the direction of streaming, but they reabsorb the backward moving waves then produced by wave decay. The possible steady states for this system of cosmic rays and Alfven waves turn out to be highly restricted.

Study of waves in the earth's bow shock.

Abstract: Results of an examination of the perturbation vectors of waves upstream and downstream from the region of maximum compression in the bow shock on Ogo 5 under particularly steady solar-wind conditions. The polarization of the upstream waves was right-hand circular, and that of the downstream waves left-hand elliptical in the spacecraft frame. By observing that the polarization of the waves remained unchanged as the shock motion swept the wave structure back and forth across the satellite three times in eight minutes, it was found that the waves were not stationary in the shock frame. A study of the methods of determining the shock normal indicates that the normal estimated from a shock model should be superior to the normal based on magnetic coplanarity. The propagation vectors of the waves examined did not coincide with the shock-model normal, the average magnetic field, or the plasma-flow velocity. However, the major axis of the polarization ellipse of the downstream wave was nearly parallel to the upstream propagation vector.

On the stability of nonlinear cold plasma waves

Abstract: The exact form of a nonlinear, stationary wave, travelling against a background of cold plasma, is investigated for stability. The result is that it is marginally stable, and therefore can exist in practice. This is found without expanding in the wave amplitude. Another problem, that of a nonlinear wave at rest against a background of two counter-streaming cold electron beams, is also commented on.

Forced resonant second-order interaction between damped internal waves

Abstract: A theoretical and experimental study is made of the second-order resonant interaction between triads of linearly damped waves, one common member of which is continuously forced. In the case of a single triad, if the forced wave exceeds a critical amplitude defined by properties of the triad members, energy proceeds irreversibly to the other two waves. A stable limit state is reached where all power in excess of that required to sustain a critical amplitude in the forced wave is transferred to the other waves, which also reach steady terminal amplitudes.It is shown that when two or more triads are simultaneously at resonance the only stable limit state is one wherein the forced wave has fallen to the lowest critical amplitude, and the only other two waves remaining are those of the triad possessing this critical amplitude. Regardless of their initial amplitudes, all other waves not externally forced ultimately disappear.The theory is applied to the interaction of standing internal gravity waves in a linearly stratified liquid. The experiments described here quantitatively confirm the major predictions.

Spectrum and Anomalous Resistivity for the Saturated Parametric Instability

Abstract: A wave spectrum peaked in angle and broadened in wave number is found from a non-linear saturation theory for the decay-type parametric instability in the case of nearly equal electron and ion temperatures. The dominant saturation mechanism is nonlinear damping of Langmuir waves by induced scattering from ions. The nonlinear resistivity for transverse waves, including the pump, is obtained from the related mechanism of conversion of transverse into longitudinal waves due to interaction with ions.

Reflection and amplification of acoustic-gravity waves at a density and velocity discontinuity

Abstract: The reflection and refraction of a plane acoustic-gravity wave at an interface separating two fluids in relative motion is calculated. It is found, as for purely acoustic waves, that the reflection coefficient for gravity waves can exceed unity (that is to say, the reflected wave can be amplified) if the shear flow speed exceeds the horizontal phase speed of the incident gravity wave. This result, which implies that gravity waves extract energy and momentum from the mean flow, is discussed, along with the idea of a critical layer at which the energy and momentum of gravity waves are absorbed into the mean flow.

Excitation of Parametric Instabilities by Radio Waves in the Ionosphere

Abstract: The excitation of parametric instabilities by radio waves in a magnetoplasma is discussed. A uniform medium is assumed and linear approximations are used. Excitation by a pump wave of ordinary polarization is hardly affected by the magnetic field. Low or zero frequency ion waves and high frequency Langmuir waves are excited simultaneously. For an extraordinary pump wave, the excited high frequency electrostatic waves are in the Bernstein mode. The threshold is slightly higher and excitation can occur only within certain 'allowed' frequency bands. A new type of parametric instability in which the excited waves are electromagnetic in nature and which is more strongly affected by the inhomogeneous nature of the medium is discussed qualitatively.

Propagation of a magnetospheric compressional wave to the ground

Abstract: A detailed analysis indicates that a pure compressional wave in the magnetosphere at the equator was observed on the ground near the northern conjugate area as an attenuated nearly pure transverse wave. The plane of polarization of the wave was tilted down ∼12° from the ground plane. Ground-based observations suggest that no proposed theory readily explains the occurrence of the magnetospheric wave.

Surface waves in hot plasmas

Abstract: Vlasov's equation and the full set of Maxwell's equations are solved as an initial value problem in a semi-infinite plasma. On specifying boundary conditions, a dispersion relation is obtained for surface waves in two situations, one without a wave incident on the boundary, the other with such a wave. The former case includes all previous results as special cases. In the latter case, it is found that surface waves cannot be excited by a wave incident on the boundary.

Channel and Crustal Rayleigh Waves

Abstract: Summary For a structure containing even a slight low-velocity channel in the upper mantle, the collection of higher mode Rayleigh waves decomposes naturally into a family of LVC channel waves and a family of crustal waves. Only the fundamental mode and the crustal waves need be considered as exciting Rayleigh waves significantly, since the channel waves do not generate significant amplitudes at the free surface. The broad properties of the interrelationship among the phase velocities of Rayleigh waves for the higher modes in the presence of a low-velocity channel (LVC) have been discussed a number of times heretofore. In this paper, we discuss the amplitude relationships among the eigenvectors for the first few higher modes without regard to the relative excitation of these eigenvectors by a particular source. We shall do so with the aid of an example taken from a particular model which has a LVC. The excitation of the modes will be treated separately. The example chosen is a shield structure taken from the literature (Harkrider 1970), to which two sedimentary layers have been added. The presence of sediments will be important even at long periods in the excitation functions; they are not overwhelmingly significant in this discussion. The structure is given in Table 1. The phase velocity of Rayleigh waves for the fundamental and the first three higher modes for this structure are shown in Fig. 1, for relatively short periods. The dispersion has been obtained using Knopoffs (1964) method as optimized by Schwab (1970); complete details are given by Schwab & Knopoff (1972). 2. Channel Rayleigh waves The apparent continuity of the phase velocities for adjacent modes is well known to be associated with the presence of a waveguide (Tolstoy & Usdin 1957; Tolstoy 1956; Mindlin & Deresiewicz 1955). Andrianova et al. (1967) have shown that such continuity from mode to mode occurs for Love waves in the presence of a lowvelocity channel. Below we show that this also occurs for Rayleigh waves in the presence of a low-velocity channel. Even with the poorly developed LVC present

Nonlinear theory of frequency shifts and broadening of plasma waves.

Abstract: It is shown that the linear dispersion relation for plasma waves is converted into a nonlinear one by making the replacement ω→ω+δω+idw. Here, ω is the wave frequency, and δω and dw are real nonlinear wave quantities that are explicitly calculated. The wave damping decrement dw has been determined in previous investigations, and shown to increase wave damping. Here, the nonlinear frequency shift δω is determined for a wide class of conditions. It is shown to have a direct and calculable influence on the frequencies of general electrostatic waves in magnetoplasmas. It is further shown that δω implies a splitting and broadening of frequency spectra of low‐frequency waves. The calculated value of such broadening is in good agreement with observations by Sheffield, and Halseth and Pyle. The present method is based on the use of an averaging operator to express the nonlinear dispersion relation in terms of perturbed orbit function. Cumulant expansions are then used to express the orbit function in terms of δω,...

The propagation of non-axisymmetric Alfven waves in an argon plasma

Abstract: The propagation of non-axisymmetric waves in a uniform, cylindrical argon plasma was investigated; the frequency range and the plasma conditions were chosen so that both the slow and the fast waves were observed. Measurements dealt mainly with the dispersion relations and the radial variation of the wave fields which were compared with predictions from magnetohydrodynamic theory assuming that the boundary condition to be applied was that for an infinitely conducting boundary, viz., that the radial magnetic field becomes zero at the plasma boundary. The propagation of the m=-1 slow wave and the m=+1 fast wave was consistent with theory but there was a discrepancy for the m=+1 slow wave. The m=-1 fast wave was not observed to propagate.

On the momentum of quasi-monochromatic waves in a plasma

Abstract: The formulae for the momentum of quasi-monochromatic wave packets of transverse and longitudinal waves in a plasma without a magnetic field are derived including the terms of the second order in the amplitude of the electromagnetic field. The well-known increase of the momentum of the transverse wave penetrating into the plasma is given by the momentum (transported with the group velocity of the wave) of the averaged motion of the plasma. The laws of energy and momentum conservation lead simply to some results of the theory of the wave decay.

A theory of the low current ionization waves (striations) in inert gases

Abstract: The solution of the Boltzmann equation found in the foregoing paper was utilized to calculate the dispersion of the fast as well as slow ionization waves (moving striations) in the positive column of a low-current neon discharge. Two balance equations of heavy particles were used: atomic ions guiding the fast waves and metastable atoms guiding the slow waves. The theory yields basically two kinds of waves: hydrodynamic r-variety and those waves that occur due to the spatial resonances of the electron gas: s-, s′- and p-variety, which are all experimentally known. Moreover, the theory predicts a new “p′”-variety as a fast wave with the same characteristic potential as the p-wave. Fixation of the characteristic potentials of the low-current ionization waves as well as high ratio of the group to phase velocity naturally follows from the theory as is shown by means of an approximative method. Numerical solutions are also presented giving full agreement for the optimum wavelength forE/p 0≲3 V/cm. torr, but for higherE/p 0 having by a few per cent lower values. The phase-shifts between various perturbed plasma parameters are also computed showing certain differences if compared with those expected from a hydrodynamic description but having in all cases the production term for the particles leading the wave in the right place demanded by the ionization mechanism of amplification and propagation of the wave.

Propagation of Plane Waves in Biaxially Anisotropic Layered Media

Abstract: This paper is concerned with the general theory of wave propagation in layered biaxially anisotropic media. Details are presented for the calculation of the induced waves that are due to an arbitrarily polarized and obliquely incident wave impinging on a three-layer structure. The total numbers of partial waves with their respective phase velocities, direction of phase propagation, and polarization are determined by the use of the Fresnel equation and Snell’s law applied to each layer. The vector amplitudes of the partial waves are found by proper matching of the field components at the interfaces. The expressions thus found are shown to reduce to known results for a uniaxial three-layer structure. An extension of this theory to an arbitrary number of layers is also presented.