Showing papers on "Wave propagation published in 1969"

Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma

Abstract: The hydromagnetic waves with small but finite amplitude in a cold collision-free plasma are investigated by using a nonlinear perturbation method. In the lowest order of perturbation, we can show that the system of equations for the magneto-acoustic wave propagating along a `critical' direction is reduced to a simple dispersive equation similar to the Korteweg-de Vries equation except that the third order derivative (the dispersion term) is replaced by the fifth order one. An extension of the problem to more general dispersive system is also made. On the other hand, the system of equations for the Alfven wave is reduced to a modified Korteweg-de Vries equation in the sense that the non-linear term f ∂ f /∂ξ in the Korteweg-de Vries equation is replaced by f 2 ∂/∂ξ. In the case of steady propagation this equation can be integrated to give a solution in closed form, which exhibits a solitary wave. Two kinds of solitary wave (both compressive and rarefied) are found to be possible.

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.

Group velocity of spiral waves in galactic disks.

Abstract: Group velocity of spiral density waves propagating in disk galaxies implying self destruction, discussing possible replenishment sources

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.

Electromagnetic Propagation through Materials Possessing Both Faraday Rotation and Birefringence: Experiments with Ytterbium Orthoferrite

Abstract: Electromagnetic wave propagation through materials that possess both Faraday rotation and birefringence is analyzed A matrix equation is developed which relates the amplitude and relative phase of the electric vectors between any two points along the propagation direction It is shown that the presence of birefringence can drastically affect the behavior of wave propagation and that it is considerably different from pure Faraday rotation Methods of measuring the material parameters are also described Criteria for viewing domains in this type of material are established It is shown that the thickness of the sample plays a great role in determining the contrast between domains and at some thicknesses no contrast at all can be obtained It is also shown that the method using elliptical analyzers gives greater contrast over the plane analyzers Photographs of domain patterns in a wedge of ytterbium orthoferrite are presented and they verify the calculated results

Recent Improvements in the Analysis of Surface Wave Observations

Abstract: Surface wave study techniques, discussing digital moving window analysis of group and phase velocity and use of time variable filters

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...

Reflection of Plane Waves from the Flat Boundary of a Micropolar Elastic Half‐Space

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

Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

Abstract: The propagation of transient waves in an elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are computed for all points of the half-space as well as for all load speeds. The disturbance is analyzed by using multi-integral transforms and an inversion scheme based on the well-known Cagniard technique. This reduces the displacements to single integral and algebraic contributions, each of which is identified as the disturbance behind a specific wave front. The same solution is valid for all load speeds, even though the wave front geometry varies greatly, depending on the speed of the load relative to the body wave speeds. Moreover, the surface displacements are obtained from the interior ones, but only after the Rayleigh waves are computed by a separate calculation. Then, by taking advantage of the form of the exact solution, wave front expansions and Rayleigh wave approximations are computed for all load speeds. Several other analytical results are obtained for restricted values of the load speed. In particular, when it exceeds both of the body wave speeds the steady-state displacement field is separated from the transient one and reduced to algebraic form. Also, for the limit case of zero load speed a new representation of the interior displacements for Lamb's point load problem is displayed in terms of single integrals.

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.

Free Vibrations of Circular Cylindrical Shells

Abstract: : Essentially a compilation of tables giving frequencies and mode shapes of vibrating cylindrical shells, the book is written for research and development engineers as well as for active scientists working in wave propagation and dynamics of thin shells. The text includes a self-contained treatment of the problem of propagation of plane harmonic waves along a hollow circular cylinder, within the framework of the three-dimensional theory of elasticity. The tables present frequencies of free vibrations of such cylinders for a wide range of axial and circumferential wave-lengths and of shell dimensions. In addition, graphs of frequency spectra and associated mode shapes are included. (Author)

Wave propagation in an elastic solid with a line of discontinuity or finite crack

Abstract: With the aid of integral transforms, a method is presented for solving the problem of scattering of plane harmonic compression and shear waves by a line of discontinuity or crack of finite width embedded in an elastic medium of infinite extent. When the incoming waves are applied in an arbitrary direction, the scattered-wave field may be determined by separating the crack-surface boundary conditions into functions even and odd with respect to the variable along the line crack. The problem is reduced to the evaluation of a system of coupled Fredholm integral equations with special emphasis placed 011 finding the near-field solution which consists of a knowledge of the detailed structure of the displacements and stresses in a small region around the crack vertex. Dynamic stress-intensity factors, the critical values of which govern the condition of crack propagation, are defined and found to be dependent on the incident wave length and Poisson's ratio of the medium. At certain wave lengths, they are larger than those encountered under static loading. Such information is of particular importance in perdicting the fracture strength of structures subjected to oscillating loads. Introduction. Although the scattering of waves by obstacles of different shapes has been the subject of many past investigations in various branches of physics [1]—[3], to the authors' knowledge none of these investigations analyzed, in detail, the singular behavior of the stresses near a scatterer in the form of a line of discontinuity or finite crack. The main reason for this omission is the lack of an effective mathematical method for obtaining the near-field solution, which is of considerable theoretical interest and has innumerable applications in the field of fracture mechanics as well as in electromagnetic and acoustic theory. A popular approach to the diffraction of waves from obstacles has been that of separation of variables, where the formal solution of the wave equation is given by an infinite series of orthogonal functions. Such an approach, however, is effective only for obstacle shapes adapted to those coordinate systems in which the wave equation is separable. For this reason, the dynamic stress concentrations around circular and parabolic obstacles have received considerable attention in the past. A comprehensive survey of the literature in a field as wide and diversified as the propagation of elastic waves is clearly beyond the scope of this paper. In recent years, the Mow-Pao-Thau school [4]—[6] has * Received January 8, 1968; revised version received March 29, 1968. The research described in this paper was sponsored by the U. S. Navy under Contract Nonr-610(06) with the Office of Naval Research, Washington, D. C. 194 G. C. SIH AND J. F. LOEBER [Vol. XXVII, No. 2 published a number of papers on this subject. References to other work can be found in [4]-[6], It is well known that problems involving diffraction of plane harmonic, horizontally polarized shear waves (SH-waves) by a semi-infinite crack can be formulated in terms of integral equations, and solved by the Wiener-IIopf technique [7]. As pointed out by Sih [8], however, since the static limit of the semi-infinite crack solution is zero, it is not possible to estimate the precise magnification of the stresses due to dynamic effects. To overcome this shortcoming, Loeber and Sih [9] proposed to add another characteristic dimension into the problem, namely the crack width, and managed to obtain the exact behavior of the crack-front displacement and stress fields for the case of SH-waves diffracted by a finite or internal crack. Ang and Knopoff [10] have attempted to solve the internal crack problem earlier but their method yields results which are restricted to low frequencies and to distances far away from the crack. In elastodynamics, the farfield crack solution is not useful in the sense that it offers no information to the development of the theories of crack propagation. Generally speaking, the far-field solution can always be determined by the standard method of Wiener-Hopf [7] in a straightforward manner. On the other hand, considerable difficulty is encountered when the WienerHopf method is applied to find the near-field solution. One of the difficulties arises from the factorization of certain functions into functions analytic in the upper and lower half planes. The problem of the diffraction of electromagnetic waves' incident upon a slit has also been treated by Schmeltzer and Lewin [11] using the function-theoretic approach. Their results are left in terms of several complicated integrals the evaluation of which becomes a problem in itself, particularly in seeking the analytical form of the solution in the vicinity of the slit. Having discussed the previous work related to crack problems of SH-waves, it is natural to follow the discussion with a few remarks concerning the diffractions of plane harmonic compression waves (P-waves) and vertically polarized shear waves (SY-waves) by a line crack. Although both Miles [12] and Papadopoulos [13] have investigated crack problems of this type, their work discusses only the qualitative character of the displacement potentials without any explicit information given as to the nature of the local stress distribution. The mathematical description of these problems is somewhat complex because the scattered waves, caused by the line crack, are composed of both compression and shear waves even though the input wave may be of one type, either the Por SVwaves. For this and other reasons, the near-field solution of waves scattered by a crack with finite width is yet to be found. The purpose of this paper, aside from obtaining the stress solution close to the crack point, is to offer a method of solution for solving diffraction problems involving Pand SY-waves incident upon a line of discontinuity. The method can handle different types of boundary conditions2 on the line of discontinuity. For illustration, only the case of a traction-free crack will be considered. An important conclusion is that within certain ranges of wave lengths the dynamic stress distribution around the crack is quite sensitive to changes in the wave number. This is displayed graphically for different values of the irThe scattering of plane-polarized electromagnetic waves by a screen in a fluid medium is mathematically analogous to the SH-wave crack problem in elastodynamics. 2By following the steps outlined in this paper, it is clear that the problems of a rigid and rigidsmooth strip can be solved in the same way. 1969] WAVE PROPAGATION IN AN ELASTIC SOLID 195 Poisson's ratio. The knowledge gained in this investigation is believed to add further impetus to the understanding of the propagation of cracks under fluctuating loads. Field equations and input waves. Consider the propagation of elastic waves, produced by the action of oscillating compressional and shear forces, which vary harmonically in time and are applied in the .xy-plane containing a through crack. In the plane, there arise both compressional and shear waves, and the resulting displacements can be expressed in terms of two scalar functions and *p each of which depend upon x, y, and t. The rectangular components of the displacement vector are ux = d/dx + Sip/dy, uy = d/dy — d\\p/dx, (1) Substituting Eq. (1) into the equations of motion under the conditions of plane strain, the following wave equations on (/> and

d2 2(d~\\p d2\\p\\ d2 \\p .. C\\^? + W2) = ^• c*\\d? + w) = d?\" () In Eq. (2), cl and c2 stand, respectively, for the velocities of compression (irrotational) and shear (equivoluminal) waves in an infinitely extended elastic medium; they are given by c, = [(X + 2m)/p]1/2, c2 = (m/p)1/2 (3) with p being the mass density. As usual, in the case of generalized plane stress the Lame constant X in Eq. (3) is to be replaced by 2\\/i/(X + 2/x), while the shear modulus of elasticity u remains unchanged. From the stress and displacement relations, it is found that (d^ ay \\ \\5x'2 dx dy/ ' vxx = XV + 2fx +2\" @ £i) ■ « = (o _ ?Jk i

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.

Low-frequency waves in the magnetosphere

Abstract: LF wave propagation and emission in magnetosphere, discussing steady noise and discrete emissions

Some Simple Modes of Wave Propagation in an Infinite Piezoelectric Plate

Abstract: Exact solutions of the equations of the fully coupled linear theory of piezoelectricity are obtained for some simple types of two‐dimensional waves in an infinite plate. It is shown that the coupling of the mechanical and the electrical fields can give rise to dispersion curves with complex branches and to waves that are largely confined to the region near the major surfaces of the plate.

Collisionless heating of the solar-wind plasma. II.

Abstract: Collisionless plasma heating by damping hydromagnetic waves applied to solar wind qualitative model, discussing magnetoacoustic wave energy

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.

Shock‐Induced Dynamic Yielding in Copper Single Crystals

Abstract: Three groups of single‐crystal disks approximately 5‐mm thick with surface normals along [100], [110], and [111] crystallographic directions were prepared from 99.99+ at.% pure copper. These specimens were shock loaded to about 50 kbar in a state of uniaxial strain by nitroguanidine explosive plane‐wave generators, and the propagated wave profiles were measured with quartz gauges. Elastic wavefronts for the single crystals exhibited sharp risetimes (of the order of 10 nsec) to dynamic yield points, and subsequent stress relaxations preceding the plastic wavefronts. For the propagation distances of about 5 mm, the measured yield point normal stresses were about 2.0, 1.3, and 1.3 kbar, respectively, for wave propagation in the [100], [110], and [111] directions. Although the principal stress states at the yield points differed, analysis reveals that the shear stresses on {111} 〈110〉 slip systems were about the same for all orientations. Single‐crystal disks prestrained by about 3½% exhibited essentially zero yield stresses and ramp‐like elastic waves. Similar behavior observed for polycrystalline specimens indicates the importance of initial dislocation density on dynamic yielding. In all cases the plastic wave velocities were the same. Constitutive relations derived on the basis of dislocation dynamics are given for the three single crystal orientations. From these relations the decay of the respective dynamic yield points with increasing propagation distance can be predicted as a function of the dislocation mobility and the initial mobile dislocation density. Within the framework of this theory it is shown that simple dislocation damping models for the mobility are not consistent with the experimental results.

Magnetic emissions in the magnetosheath at frequencies near 100 Hz

Abstract: Noise signals in earth magnetosheath interpreted as electromagnetic waves propagating in whistler mode

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).

Experimental Observation of Sommerfeld and Brillouin Precursors in the Microwave Domain

Abstract: An experimental investigation of transient electromagnetic wave propagation in the microwave frequency range related to the theoretical investigation of Sommerfeld and Brillouin is reported. Both the Sommerfeld and Brillouin precursors are shown experimentally.