Showing papers on "Longitudinal wave published in 1968"

Interaction of Linear Waves with Oblique Shock Waves

Abstract: Linear wave interaction with oblique shock waves, noting dependence of transmission, reflection and generation coefficients on Mach number

Parametric excitation of coupled waves. ii. parametric plasmon--photon interaction.

Abstract: The general theory of the parametric excitation of coupled waves developed in the preceding paper is applied to the parametric interaction of the electrostatic waves in a plasma with radiation whose frequency is close to the electron plasma frequency. On the basis of the hydrodynamic equations, the coupled equations for the electron plasma wave and ion acoustic wave are derived. The effect of the Landau damping is also considered phenomenologically. Using the coupled wave equations, the expressions for the threshold radiation intensity, the frequency shift at the threshold and the growth rate above threshold are obtained. In particular, the dependence of these quantities on the wave-length of the electrostatic waves is discussed in detail. The results are compared with experiment of Stern and Tzoar.

The vibrational properties of an electron gas

Abstract: 1. Statement of the problem 721 2. Initial equations and their simplification 722 3. Solution of the linearized equations 724 4. Dispersion of longitudinal waves 728 5. Dispersion of longitudinal waves in an electron gas with a Fermi distribution function 730 6. Dispersion of transverse waves 732 7. Summary and conclusion 733

The physics of vibrations and waves

Abstract: Introduction to First Edition. Introduction to Second Edition. Introduction to Third Edition. Introduction to Fourth Edition. Introduction to Fifth Edition. Introduction to Sixth Edition. 1 Simple Harmonic Motion. 2 Damped Simple Harmonic Motion. 3 The Forced Oscillator. 4 Coupled Oscillations. 5 Transverse Wave Motion. 6 Longitudinal Waves. 7 Waves on Transmission Lines. 8 Electromagnetic Waves. 9 Waves in More than One Dimension. 10 Fourier Methods. 11 Waves in Optical Systems. 12 Interference and Diffraction. 13 Wave Mechanics. 14 Non-linear Oscillations and Chaos. 15 Non-linear Waves, Shocks and Solitons. Appendix 1: Normal Modes, Phase Space and Statistical Physics. Appendix 2: Kirchhoff's Integral Theorem. Appendix 3: Non-Linear Schro..dinger Equation. Index.

Longitudinal capillary waves. Part 1.—Theory

Abstract: The existence of hitherto unobserved surface waves with a longitudinal character is deduced theoretically and the properties of these waves are predicted. In contrast to capillary waves, the longitudinal waves are governed by the surface elastic modulus rather than by surface tension; they are accompanied by a horizontal rather than by a vertical surface movement and they are connected with a liquid flow of a highly dissipative character causing a rapid damping. Under normal experimental conditions capillary waves and longitudinal waves are superimposed, but the longitudinal wave is usually damped out much more rapidly, except at high elastic modulus.

On the shape of progressive internal waves

Abstract: An expansion technique, analogous to that of Stokes in the study of surface waves, is used to investigate the effects of finite amplitude on a progressive train of internal gravity waves. The paper is divided into two main parts, a study of interfacial waves in a two-fluid system and an examination of internal waves in a continuously stratified fluid. Experimental evidence is presented which confirms some of the theoretical predictions. The validity of the Boussinesq approximation is examined and particular examples are taken to illustrate the general results.

The propagation of continental shelf waves

Abstract: Taking the case of a continental shelf of exponential slope, and assuming zero horizontal divergence, the authors derive a simple theory of free shelf waves, which, however, is more general than previous theories in that shorter as well as longer waves (in comparison with previous work) are taken into account. The properties of the waves are discussed, and the dispersion curves for each mode are obtained. Although the phase velocities of shelf waves are always in the same sense as those of Kelvin waves, there is a negative group velocity for a range of wavelengths, indicating that energy can propagate in the opposite sense. A similar approach is used to derive a theory for free waves propagating on a shelf between two regions of constant depth. The limiting case of a shelf of zero width is also considered, and is compared with a limiting case of the ‘double-Kelvin’ waves discovered by Longuet-Higgins (1967).

Velocity of compressional waves in porous media at permafrost temperatures

Abstract: Measurements of velocity of compressional waves in consolidated porous media, conducted within a temperature range of 26 °C to −36 °C, indicate that: (1) compressional wave velocity in water‐saturated rocks increases with decreasing temperature whereas it is nearly independent of temperature in dry rocks; (2) the shapes of the velocity versus temperature curves are functions of lithology, pore structure, and the nature of the interstitial fluids. As a saturated rock sample is cooled below 0 °C, the liquid in pore spaces with smaller surface‐to‐volume ratios (larger pores) begins to freeze and the liquid salinity controls the freezing process. As the temperature is decreased further, a point is reached where the surface‐to‐volume ratio in the remaining pore spaces is large enough to affect the freezing process, which is completed at the cryohydric temperature of the salts‐water system. In the ice‐liquid‐rock matrix system, present during freezing, a three‐phase, time‐average equation may be used to estimat...

Excitation of Surface Elastic Waves by Transient Surface Heating

Abstract: The generation of surface elastic waves by the transient surface heating of piezoelectric and nonpiezoelectric solids is described. A Q‐switched ruby laser produces the surface heating; the frequencies of the resultant surface waves are Fourier components of the laser waveform. The use of a spatially periodic illumination is shown to increase the effectiveness of generation at a selected frequency. This method of generating surface waves appears suitable for microwave frequency operation as well as operation at high wave amplitudes at low frequencies.

Longitudinal capillary waves. Part 2.—Experiments

Abstract: It has been shown theoretically that longitudinal waves can easily be studied at an interface with a high elastic modulus. Measurements on mixed monolayers of cholesterol and dimyristoyl lecithin show that the behaviour of longitudinal waves is in accordance with predictions from theory. The elastic modulus at this interface as obtained from the wavelength of longitudinal waves at 200 c/sec, is several times the “equilibrium” modulus obtained from the slope of force-area curves.

Granites: Relation of Properties in situ to Laboratory Measurements.

Abstract: Granite compressional wave velocity and electrical resistivity variations with depth, explaining discrepancy between in situ and laboratory measurements on dry samples

Dispersion of free harmonic waves in fiber- reinforced composites.

Abstract: : A set of displacement equations of motion is proposed that is suitable for the dynamic analysis of fiber-reinforced composites. In deriving the equations, representative elastic moduli are used for the binder, and the elastic and geometric properties of the fibers are combined into effective stiffnesses. With the aid of certain assumptions regarding the deformation of the fibers, and by employing a smoothing operation, approximate kinetic and strain energy densities for the fiber-reinforced composite are obtained. Application of Hamilton's principle yields the displacement equations of motion. The proposed set of equations is employed to study the propagation of plane harmonic waves propagating in the direction of the fibers and normal to the fiber direction. Plane transverse waves propagating in the direction of the fibers are dispersive, and dispersion curves are shown. By proper choice of the representative elastic moduli of the binder, the phase velocity at infinite wave length for transverse waves propagating in the direction of the fibers, and the constant phase velocities for longitudinal waves and waves propagating in the other directions, agree with the values predicted by the effective modulus theory. (Author)

Temporal and Spatial Plasma Wave Echoes

Abstract: It is shown that, if a longitudinal wave is excited in a collision‐free plasma and Landau‐damps away, and later a second wave is excited and also damps away, then a third wave will spontaneously appear in the plasma. This wave appears long after the first two waves have damped away at a time proportional to the interval between the first two waves, and is in that sense an echo. It is also shown that, if a wave is continuously excited at one point in a plasma and a second wave is continuously excited many Landau damping lengths from the first point, then a third wave will spontaneously appear many Landau damping lengths from the second point. Fundamentally, plasma wave echoes are possible because of the reversible nature of Landau damping. However, small‐angle Coulomb collisions are very effective in destroying the echo.

A note on “kelvin” waves in the atmosphere1

Abstract: A special solution of the linearized equations for an equatorial β-plane is described. The meridional velocity is set identically zero and it is shown that the solution is a vertically propagating gravity wave in the x-z plane which is in geostrophic balance in the meridional plane. These waves are shown to be similar to Kelvin waves in a bounded ocean. Some implications of the atmospheric “Kelvin” waves for equatorial dynamics are discussed.

Frequency response of an interdigital transducer for excitation of surface elastic waves

Abstract: Excitation of elastic waves in a piezoelectric solid by an interdigital transducer, in terms of a simplified model, has been analyzed using Lamb's integral solution. The transducer generates three types of disturbances propagating at the velocities of irrotational, equivoluminal, and Rayleigh waves. While the amplitudes of the former two waves diminish as the three-half power of the distance from the transducer, that of the latter does not decay with distance on the surface. Dependence of particle displacement of the waves on electric fields, number of electrodes, electrode width, and wave numbers has been derived. It has been shown that a conventional interdigital transducer, in which all the electrode widths and all the spacings are respectively equal among themselves, can respond to fundamental and odd harmonic excitations, but not to the even harmonics. Means to improve the insertion loss and the bandwidth of the transducer are discussed. A flat overall frequency response can be synthesized by parallel operation of two or more transducers, the fundamental frequencies of which are properly staggered.

A Note on the Breaking of Waves

Abstract: A model equation for water waves has been suggested by Whitham (1967) to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and it is of independent mathematical interest. For an approximate kernel of the form e$^{-b|x|}$, it is shown first that solitary or periodic waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into `bores'. The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root infinity and the present argument does not go through. Nevertheless, the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.

Seismic Waves near 110°: Is Structure in Core or Upper Mantle Responsible?

Abstract: Summary An analysis is given of longitudinal waves which are observed at distances between 105° and 115° some minutes after the arrival of the P waves diffracted by the mantle-core boundary. A number of seismograms which clearly show these wave trains is examined; recordings at WWSS stations from the Chilean shock of 1965 March 28 are specially studied. The observations show (i) that discernible PKiKP waves with wavelengths of order 10 km are reflected from the boundary of the Earth'sinner core back to distances of at least 105° and (ii) that many longitudinal wave onsets (the PdP phase) having travel-times up to 60 s before PKiKP and 90 s before PP arrive near 110° by means of reflection from the lower side of physical discontinuities in the upper mantle of the Earth. The first result is consistent with a relatively sharp increase in P velocity between the transition zone and the inner core at a radius of about 1220 km. The second suggests the existence of a number of discrete shells of different elastic properties in the Earth'supper mantle above 400 km; in particular, one prominent group of PdP waves of order 2 s period is consistent with reflection from a discontinuity near 385 km. This result provides confirmation of the overall high velocity gradient near this depth inferred by L. Johnson; there is an indication, however, of first-order discontinuities in the velocity function assumed to be smooth by Johnson.

Magneto--gravity waves and the heating of the solar corona.

Abstract: It is generally believed that the heating of the solar corona is caused by waves originating in the photosphere and propagating into the corona where their energy is dissipated. The medium through which these waves propagate is in general permeated by magnetic fields complicating the behaviour of this propagation considerably. We have therefore analysed the wave motions in a plasma permeated by constant magnetic and gravitational fields. In general, three waves modes were found, which we called the + mode, −mode, and the Alfven mode. Each mode was found to be strongly coupled to each of the three kinds of motion; acoustic, gravity, and hydromagnetic. However, the Alfven mode was found to be separable from the dispersion relation, and therefore independent of compressibility and gravity. The local dispersion relation is derived and expressed in nondimensional form independent of the constants that describe a particular atmosphere. From the dispersion relation one can show that rising waves propagate either with a constant or a growing wave amplitude depending on the magnitudes and directions of the gravitational field, magnetic field, and the wave vector. The variation of the density with height is taken into account by a generalized W.K.B. method. Equations are found which give the height at which wave reflection occurs, giving the upper bound for possible wave propagation.

Scattering of Lattice Waves by Dislocations

Abstract: The problem of the scattering of lattice waves by the static strain field of dislocations has been treated by means of the Green's function method. The calculation starts from the force constant model like that of Carruthers, the relaxation time obtained is, however, about one hundredth of that evaluated by him. For the case of longitudinal waves scattered by edge dislocations in alkali halides, MgO, Cu and Ge, the scattering cross sections calculated by the present theory exhibit fairly better agreement with the experimental results as compared with those calculated by Klemens and Carruthers.

Nonlinear Interaction of Cyclotron Harmonic Waves

Abstract: This paper presents an analysis of nonlinear interaction between cyclotron harmonic waves propagating perpendicular to the static magnetic field in a warm magnetoplasma. It is demonstrated that the synchronism conditions for traveling wave parametric amplification, and passive mode conversion can easily be satisfied for such waves, and an analysis is carried out for a large‐signal pump wave interacting with two small‐signal waves. The theory employs the Vlasov equation and Maxwell's equations, and provides expressions suitable for numerical computation. Growth rates and corresponding pump powers are computed, and it is concluded that the effect should be observable in typical laboratory plasmas. The paper concludes with a discussion of possible device applications.

The effect of wave energy spectra on wave run-up

Abstract: Previous investigations carried out "by the Delft Hydraulics laboratory have shown the necessity of applying irregular waves m studies on wave run-up. The installation of a wave generator driven "by hydraulic actuators has created the possibility of producing irregular waves with arbitrary wave spectra. Investigations performed with this type of wave generator show the influence of the shape of the energy spectrum on the wave run-up on smooth straight slopes of 1:4 and 1:6. The results are compared with run-up figures derived from experiments with wind generated waves and with monochromatic waves.

Excitation of Longitudinal Waves near Electron‐Cyclotron Harmonics

Abstract: Longitudinal waves were excited radially across a plasma column subjected to an axial magnetic field. A strip line placed externally to the column was the source of excitation. The wavelength and propagation direction were measured by a movable antenna immersed in the plasma. The measurements are compared with theory and with observations made with an externally situated receiving antenna. The following conclusions are drawn. (a) Contrary to the commonly accepted notion, the longitudinal waves are propagating rather than standing waves. The propagation is radial towards the axis of the plasma column. Since the waves are backward waves, the energy propagates radially outwards. (b) The internal and external measurements are related to good accuracy through the geometrical optics, “phase quantization” condition, ∫ k dr ≈ 2mπ, where k is the propagation constant, r is the radial coordinate, and m is an integer. The left‐hand side of the equation is deduced from measurements of k made with the movable antenna ...