Showing papers on "Plane wave published in 1981"
TL;DR: In this article, the dispersion curves for the mode overtones and fundamental are directly observed in the transformed wave field, where the data wave field is linearly transformed from the time-distance domain into the slowness-time intercept (p − τ) domain.
Abstract: The dispersive waves in a common‐shot wave field can be transformed into images of the dispersion curves of each mode in the data. The procedure consists of two linear transformations: a slant stack of the data produces a wave field in the phase slowness‐time intercept (p — τ) plane in which phase velocities are separated. The spectral peak of the one‐dimensional (1-D) Fourier transform of the p — τ wave field then gives the frequency associated with each phase velocity. Thus, the data wave field is linearly transformed from the time‐distance domain into the slowness‐frequency (p — ω) domain, where dispersion curves are imaged. All the data are present throughout the transformations. Dispersion curves for the mode overtones as well as the fundamental are directly observed in the transformed wave field. In the p — ω domain, each mode is separated from the others even when its presence is not visually detectable in the untransformed data. The resolution achieved in the result is indicated in the p — ω wave ...
517 citations
TL;DR: In this article, the authors introduce the concept of an ideal (superconducting) current return plane placed below the ground surface at a complex distance p equal to the complex penetration depth for plane waves.
Abstract: For modelling current return in homogeneous ground, the paper introduces the concept of an ideal (superconducting) current return plane placed below the ground surface at a complex distance p equal to the complex penetration depth for plane waves. This "complex" plane appears as a mirroring surface, so that conductor images can be used to derive very simple formulae for self and mutual impedances under ground return conditions. Such equations, without proofs, were originally proposed by Dubanton and published by Gary.1 In this paper, plausibility arguments serve to initially justify the procedure, then the equations are analytically related to those of Carson and, finally, the errors, which in most cases are less than a few percent, are numerically evaluated. The ideal return plane at complex depth can also be used for multi-layer earth return.
378 citations
Book•
01 Jan 1981
TL;DR: In this article, the basic properties of acoustic wave properties are discussed, including linearization, uniformity, and Fourier analysis of wave properties, as well as the relationship between wave properties.
Abstract: CHAPTER TWO Basic Properties of Acoustic Waves 2.1 Ideal Fluids 2.2 Linearization 2.3 Uniform Fluids 2.4 One-Dimensional Plane Waves Speed of Sound in a Perfect Gas Speed of Sound in Other Fluids Relationships between Acoustic Quantities 2.5 Monochromatic Waves Plane, One-Dimensional Monochromatic Waves Plane, Monochromatic Waves in Three Dimensions Relation between Variables in a Monochromatic Wave Time Averages 2.6 Fourier Analysis Periodic Waveforms —Fourier Series Nonperiodic Functions—Fourier Transform 2.7 Acoustic Energy Energy Density Acoustic Intensity Reference Levels
287 citations
01 Jan 1981
TL;DR: The Numerical Electromagnetics Code (NEC-2) is a computer code for analyzing the electromagnetic response of an arbitrary structure consisting of wires and surfaces in free space or over a ground plane by the numerical solution of integral equations for induced currents.
Abstract: : The Numerical Electromagnetics Code (NEC-2) is a computer code for analyzing the electromagnetic response of an arbitrary structure consisting of wires and surfaces in free space or over a ground plane. The is accomplished by the numerical solution of integral equations for induced currents. The solution includes Numerical Green's Function for partitioned-matrix solution and a treatment for lossy grounds that is accurate for antennas very close to the ground surface. The excitation may be an incident plane wave or a voltage sour wire, while the output may include current and charge density, electric or magnetic field in the vicinity of structure, and radiated fields. Other options compute the maximum coupling between antennas and facilitate Numerical Electromagnetics Code (NEC-2) Numerical analysis Antenna response Electromagnetic radiation. structure input. Hence the code may be used for antenna analysis, EMP, or scattering studies. Part 1 of the document includes the equations on which the code is based and a discussion of the approximations and numerical methods used in the numerical solution. Some comparisons to demonstrate the range of accuracy of approximations are also included. Details of the coding and a User's Guide are provided as parts 11 and 111, respectively.
278 citations
TL;DR: In this paper, the authors investigate the occurrence of magnetoacoustic surface waves at a single magnetic interface and consider in detail the case where one side of the interface is field-free.
Abstract: The solar atmosphere, from the photosphere to the corona, is structured by the presence of magnetic fields. We consider the nature of such inhomogeneity and emphasis that the usual picture of hydromagnetic wave propagation in a uniform medium may be misleading if applied to a structured field. We investigate the occurrence of magnetoacoustic surface waves at a single magnetic interface and consider in detail the case where one side of the interface is field-free. For such an interface, a slow surface wave can always propagate. In addition, a fast surface wave may propagate if the field-free medium is warmer than the magnetic atmosphere.
256 citations
TL;DR: In this article, a 3D volume integral equation algorithm was adapted to magnetotelluric (MT) modeling, where the two symmetry planes of a buried prismatic body and a normally incident plane wave source were used to reduce the computation time and storage.
Abstract: We have adapted a three‐dimensional (3-D) volume integral equation algorithm to magnetotelluric (MT) modeling. Incorporating an integro‐difference scheme increases accuracy somewhat. Utilizing the two symmetry planes of a buried prismatic body and a normally incident plane wave source greatly reduces required computation time and storage. Convergence checks and comparisons with one‐dimensional (1-D) and two‐dimensional (2-D) models indicate that our results are valid. We show theoretical surface anomalies due to a 3-D prismatic conductive body buried in a half‐space earth. Instead of studying the electric and magnetic fields, we have obtained impedance tensor and magnetic transfer functions by imposing two different source polarizations. Manipulation of the impedance tensor and magnetic transfer functions yields the following MT quantities: apparent resistivity and phase, impedance polar diagrams, tipper direction and magnitude, principal directions, skew, and ellipticity. With our preliminary analyses of...
183 citations
TL;DR: In this paper, the authors derived the angular distribution of light emitted into the half-spaces 1 and 2, and derived the dipole radiation patterns for the case in which any numbers of loss-free or absorbing, dielectric or metallic thin films are present between the loss-less layer 0 of arbitrary thickness containing the source and the halfspaces 0 and 2.
Abstract: The emission of light by sources, e.g., by luminescent centers, located in a thin nonabsorbing dielectric layer 0 between two half-spaces 1 and 2 is investigated theoretically. It is assumed that the light is emitted in electric or magnetic dipole transitions. But the theory is given in such a form that it can easily be extended to electric and magnetic quadrupole and higher-order multipole transitions. The electromagnetic boundary-value problem is solved rigorously for sources in layers 0 of arbitrary thickness. The radiation patterns, i.e., the angular distributions of light emitted into the half-spaces 1 and 2, are calculated. The theory takes into account the following effects that strongly influence the radiation patterns: (1) the wide-angle interferences that are a consequence of the coherence of the plane waves emitted into different directions, (2) the multiple-beam interferences that result from the multiple reflections of the plane waves between the interfaces 0/1 and 0/2, and (3) that evanescent waves present in the near field of the source radiate into media 1 and/or 2 if these media are denser than layer 0. This emission process is influenced by evanescent-wave effects analogous to the wide-angle interferences and the multiple-beam interferences of the plane waves. The limiting case of extremely thin layers 0 with optical thickness much smaller than the wavelength is also treated. Explicit analytical expressions are presented for the dipole radiation patterns in this case. Furthermore, the theory is generalized for sources in plane-stratified-layer systems. The dipole radiation patterns are derived for the case in which any numbers of loss-free or absorbing, dielectric or metallic thin films are present between the loss-free layer 0 of arbitrary thickness containing the source and the half-spaces 1 and 2.
120 citations
01 Sep 1981
TL;DR: In this article, the authors compared three analytical methods for a conducting sinusoidal surface in detail for both TE and TM polarized waves, and found the MMM method to be the most efficient one in terms of the rate and range of convergence.
Abstract: The scattering of waves from periodic surfaces is studied. We give a general review of this problem and compare three analytical methods for a conducting sinusoidal surface in detail for both TE and TM polarized waves. The three methods are: 1) the method developed by Masel, Merrill, and Miller (MMM); 2) the Modified Physical Optics (MPO) method; and 3) Waterman's Plane Harmonics (WPH) approach. We find the MMM method to be the most efficient one in terms of the rate and range of convergence. For dielectric media with periodic rough surfaces, an improved method is developed for calculating the reflected and transmitted powers. The results are used to compare with experimental data obtained at optical frequencies. It is shown that good agreement is achieved when the complex permittivity of the metal for the grating at the corresponding frequency is used.
114 citations
TL;DR: In this article, a plane wave decomposition is used to transform an observed record section Φ(x,t) into its plane-wave decomposition Ψ(p,τ), which is an objective procedure which removes many of the interpreter's difficulties by casting the picking process into simpler variables.
Abstract: Slant stacking is used to transform an observed record section Φ(x,t) into its plane wave decomposition Ψ(p,τ). The invertible transformation untangles travel time triplications and removes focusing and defocusing effects, yielding signals with the plane wave amplitudes and phases. The principal arrival, composed of refraction and wide angle reflections, appears in these coordinates as a display of the intercept time function τ(p), which may be inverted by any of several currently popular methods. The transformation into the (p,τ) domain; is an objective procedure which removes many of the interpreter's difficulties by casting the picking process into much simpler variables. Moreover, the decomposition into plane waves directly yields the transient reflectivity matrix of the (one dimensional) medium. Several lines of argument lead us to conclude that the comparison of data and synthetic data should be done in the (p,τ) domain. The use of these techniques requires that the wave field be sampled with sufficient spatial density of receivers to avoid wave number aliasing. We show examples of plane wave decomposition for shallow data from the COCORP deep crustal reflection program.
106 citations
TL;DR: In this article, a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration, i.e., the Kirchhoff-summation approach, the plane-wave method (k-f method), and the finite-difference technique, is provided.
Abstract: The objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration, i.e., the Kirchhoff-summation approach, the plane-wave method (k-f method), and the finite-difference technique.Particular emphasis is given to the relationship between the different methods. By formulating the problem in the space-frequency domain (x, y, omega -domain), a systems approach can be adopted which results in simple and concise expressions. These expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure. In the situation of lateral velocity variations, the (de)convolution procedure becomes space-variant. The space-frequency domain is most suitable for recursive depth migration. In addition, frequency dependent properties such as absorption, dispersion, and spatial bandwidth can be handled easily.It is shown that all extrapolation methods are based on two equations: Taylor series and wave equation. In the Kirchhoff-summation approach all terms of the Taylor series are summed to an exact analytical expression--the Kirchhoff-integral for plane surfaces. It formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications. The Fourier-transformed version of the Kirchhoff-integral is used in the plane wave method (k-f method). This actually means that spatial (de)convolution in the x, y, omega -domain is translated into multiplication in the k x , k y , omega -domain. Of course, this is not allowed if the extrapolation operators are space-variant.In explicit finite-difference techniques a truncated version of the Taylor series is used with some optimum adjustments of the coefficients. For only one or two terms in the Taylor series, a spatial low-pass filter must be applied to compensate for the amplitude errors at high tilt angles. Explicit methods are simple and most suitable for three-dimensional (3-D) applications.In implicit finite-difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator. Properly designed implicit schemes do not show amplitude errors and, therefore, amplitude correction filters need not be applied. In comparison with explicit schemes, implicit schemes are more sensitive to improper boundary conditions at both ends of the data file.It is shown that the forward seismic model can be elegantly described by a matrix equation, using separate operators for downward and upward traveling waves. Using this model, inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to compensate for the upward propagation effects.
105 citations
TL;DR: In this article, a method is outlined for the calculation of synthetic seismograms which include the effects of absorption and dispersion, based on the communication theory approach and applicable to plane waves in a flat layered model.
Abstract: A method is outlined for the calculation of synthetic seismograms which include the effects of absorption and dispersion. The absorption model used is the usual model of exponential decay of amplitude with distance given by A=A0e-αz, where α is a linear function of frequency. This attenuation is accounted for mathematically by allowing the elastic modulus to be a complex function of frequency. This results in a complex velocity and wavenumber, and the reflection and transmission coefficients also become complex functions of frequency. The method is based upon the communication theory approach and is applicable to plane waves in a flat layered model. The source can be placed at an arbitrary depth. The equations are outlined in detail for a particular absorption‐dispersion pair taken from Futterman (1962). An example with a surface synthetic seismogram and synthetic traces at several depths is presented.
TL;DR: In this paper, the diffraction of externally generated sound in a uniformly moving flow at the trailing edge of a semi-infinite flat plate is studied, and the coupling of the sound field to the hydrodynamic field by way of vortex shedding from the edge is considered.
Abstract: The diffraction of externally generated sound in a uniformly moving flow at the trailing edge of a semi-infinite flat plate is studied. In particular, the coupling of the sound field to the hydrodynamic field by way of vortex shedding from the edge is considered in detail, both in inviscid and in viscous flow.In the inviscid model the (two-dimensional) diffracted fields of a cylindrical pulse wave, a plane harmonic wave and a plane pulse wave are calculated. The viscous proess of vortex shedding is represented by an appropriate trailing-edge condition. Two specific cases are compared, in one of which the full Kutta condition is applied, and in the other no vortex shedding is permitted. The results show good agreement with Heavens’ (1978) observations from his schlieren photographs, and confirm his conclusions. It is further demonstrated, by an explicit expression, that the sound power absorbed by the wake may be positive or negative, depending on Mach number and source position. So the process of vortex shedding does not necessarily imply an attenuation of the sound.In the viscous model a high-Reynolds-number approximation is constructed, based on a triple-deck boundary-layer structure, matching the harmonic plane wave outer solution to a known incompressible inner solution near the edge, to obtain the viscous correction to the Kutta condition.
TL;DR: In this article, two different methods are presented for efficient computation of two-dimensional wave fields in focal regions, one based on impulse-response integral and the other based on angular-spectrum representation.
Abstract: Two different methods are presented for efficient computation of two-dimensional wave fields in focal regions. Both methods are valid for arbitrarily large relative apertures. One method is based on the impulse-response integral and the other on the angular-spectrum representation. The latter method is used to analyze the discrepancy between applying the Kirchhoff or the Debye assumption to obtain an approximation for the field in the aperture. Two cases of idealized incident waves are analyzed in detail. First, we treat the case of a perfect incident wave, i.e., a wave that, in the limit of an infinitely large aperture, would produce a δ-function field distribution on the focal line if account were taken of evanescent waves. Second, the incident wave is taken to be the field radiated by a point source and subsequently focused by a lens that delays the phase of the incoming wave in a perfect manner without influencing its amplitude. The latter wave has the same phase distribution over the aperture as the perfect wave, but a different amplitude distribution.
TL;DR: In this article, the authors extended the exact analysis for plane wave scattering from a sinusoidal surface to include the case of an arbitrary periodic surface and showed that the field amplitudes of the upgoing scattered (or evanescent) waves (valid above the highest surface excursion) are expressed as summation over these expansion coefficients multiplied by integrals over the surface function.
Abstract: We extend our exact analysis for plane wave scattering from a sinusoidal surface to include the case of an arbitrary periodic surface. Both Dirichlet (transverse electric polarization) and Neumann (transverse magnetic polarization) boundary conditions are considered. The method uses Green's theorem and our previous idea of expanding the surface fields in Fourier series multiplied by the Kirchhoff or physical optics approximation. The expansion coefficients solve a set of linear equations. The field amplitudes of the upgoing scattered (or evanescent) waves (valid above the highest surface excursion) are expressed as a summation over these expansion coefficients multiplied by integrals over the surface function. The Rayleigh hypothesis is not invoked. Some examples are presented. For an analytic surface, steepest descent methods yield the asymptotic values of the amplitudes. Using this and other asymptotic results, the convergence of the scattered wave expansion is studied as it is analytically continued into the surface wells, and a simple and explicit confirmation of the conditions under which the Rayleigh hypothesis is valid is presented as well as new results for other examples. The periodic surface examples include a sinusoid, an echelette, a quadratic surface, a log cosine surface, a vortexlike surface, a cycloid, a trapezoid, a full-wave rectified surface, and a surface of semicircular cylinders (bosses). The method is general and applies to a very broad class of physical problems.
TL;DR: In this paper, a simple problem arising in the electromagnetic study of gratings is addressed, namely, whether plane wave field representations are efficient or even correct, for a class of groove profiles.
Abstract: The properties of plane waves are well known, and it is probably because they are that we often try to represent any unknown field as a combination of such waves. Is such a representation always efficient or even correct? We try to answer this question for a simple problem arising in the electromagnetic study of gratings. We first summarize in a didactic way some important theoretical results that are probably not well known to those working in optics. Thereafter we report on recently performed numerical experiments. Plane-wave field representations indeed permit us to obtain quickly reliable results for some particular profiles. Nevertheless, the methods leading to the solution of an integral equation are the most useful because they are applicable to a larger class of groove profiles.
TL;DR: In this paper, a solution for electromagnetic wave scattering from an infinitely long circular cylinder with a possibly radially stratified internal structure and which is buried in a lossy half space is given.
Abstract: A solution is given for electromagnetic wave scattering from an infinitely long circular cylinder with a possibly radially stratified internal structure and which is buried in a lossy half space. The fields of the primary source are allowed to vary along the cylinder's axis. The scattered fields result from induced multipole line sources of electric and magnetic types located at the cylinder's axis. Explicit expressions are obtained for the source amplitudes up to the dipole terms that take into account the presence of the air-earth interface. The dipole sources may contribute significantly to the scattered fields even when the cylinder's radius is small relative to the wavelength in the surrounding medium. Some specific results are given for a buried tunnel and a conducting cylinder surrounded by an air gap and for an incident plane wave from the air region.
TL;DR: In this paper, the precursors of plate impact experiments on single crystals are re-examined from the viewpoint of the elastodynamics of moving dislocations, and it is shown that dislocation drag forces due to continuum elasticity effects are small for dislocation velocities less than, say, 80% of the elastic shear wave speed.
Abstract: P recursor decay in plate impact experiments on single crystals is re-examined from the viewpoint of the elastodynamics of moving dislocations. Superposition of solutions for many dislocations set in motion by an incident plane wave is used to relate the decay of the wave amplitude at the front of the plane wave to the density and velocity of dislocations at the wavefront. The resulting precursor decay relation is the same as the one derived from an elastic/visco-plastic model of the material, except for a small correction due to differences between the effects of forward and backward propagating dislocations. Motivated by this added support for the validity of the precursor decay equation, the values used for the quantities in this equation are re-examined. Recent experimental results and the elastodynamics analysis are interpreted as indicating that the commonly-used values of dislocation velocity are probably satisfactory, but that the values used for dislocation density are several orders of magnitude too small near the lapped surfaces of the crystal. These large dislocation densities are identified as the probable dominant cause of the lower-than-predicted precursor amplitudes that are recorded in experiments. More accurate experimental data and inclusion of the non-linear elasticity effects are essential in determining whether or not the observed precursor decay in the bulk of the specimen can be explained by the motion of dislocations present initially. Calculations of energy radiated from screw and edge dislocations that start from rest and move thereafter at constant velocity confirm that dislocation drag forces due to continuum elasticity effects are small for dislocation velocities less than, say, 80% of the elastic shear wave speed. At supersonic speeds the continuum drag effects become so large that sustained supersonic motion of dislocations appears unlikely.
Book•
30 Jun 1981
TL;DR: This chapter discusses the applications of geometrical diffraction theory to antennas in the field of aperture theory, and some examples are given of diffraction by conductors with sharp edges.
Abstract: Chapter 1: Introduction Chapter 2: Plane waves from apertures Chapter 3: Fourier transform representation of aperture patterns Chapter 4: Near-field radiation patterns Chapter 5: Aperture gain Chapter 6: Applications of aperture theory to antennas Chapter 7: Diffraction by conductors with sharp edges Chapter 8: Geometrical theory of diffraction by edges Chapter 9: Applications of geometrical diffraction theory to antennas Appendixes
TL;DR: In this paper, the authors provide a theoretical framework for understanding the local phase structure and the motion of any dislocation in a scalar wave, where even the most general type of dislocation may be considered to be carried, locally, by a plane wave whose orientation relative to the trajectory determines the climb of the dislocation.
Abstract: Scattered scalar wavefields contain line singularities where the phase of the wave is indeterminate and the amplitude is zero. Unless the wave is monochromatic, these dislocation lines, which are analogous to crystal dislocations, move along trajectory surfaces, changing their positions relative to the wave by glide and climb. The edge-screw character of a given dislocation varies along its length and as it moves. When it has no close neighbours its glide and screwness, and the way they change, are completely determined by the distribution over the trajectory surface of two scalar quantities: the phase of the dislocation and its time of arrival. It is shown how even the most general type of dislocation may be considered to be carried, locally, by a plane wave, whose orientation relative to the trajectory determines the climb of the dislocation. Around a general isolated dislocation the equiphase surfaces form a helicoid; they are equally spaced along any radial line, but with a discontinuity of $\pi$ across the dislocation line itself. The paper provides a theoretical framework for understanding the local phase structure and the motion of any dislocation in a scalar wave.
TL;DR: In this article, the relationship between these far-field representations and Kostrov's representation of radiated energy in terms of fault surface traction and particle velocity was established, and the authors showed that the excess of the static strain energy change over the work done by the fault surface tractions, called Wo by Kanamori, is a good approximation to the radiated energies when fault propagation speed is near the Rayleigh wave velocity and the time rate of change of fault surfaces traction is small.
Abstract: This note clarifies the relationships among various expressions for the energy radiated by elastodynamic seismic sources. The radiated energy can be expressed in terms of the far-field particle velocities provided that the stressparticle velocity relationship asymptotically approaches the plane wave relationship with increasing distance from the source. This condition is satisfied for all sources that can be characterized by a moment density tensor. For the case in which the source can be characterized as a point, i.e., the wavelengths of all emitted radiation are much greater than source dimensions, the radiated energy is expressed in terms of the moment tensor. The relation between these far-field representations and Kostrov's representation of radiated energy in terms of fault surface traction and particle velocity is established. Kostrov's representation is arranged in various forms to reveal the source of radiated energy as the deviations of the fault surface tractions and particle velocities from the values which would occur during quasi-static fault motion between the same end states. Moreover, the excess of the static strain energy change over the work done by the fault surface tractions, called Wo by Kanamori, is shown to be a good approximation to the radiated energy when fault propagation speed is near the Rayleigh wave velocity and the time rate of change of fault surface traction is small.
TL;DR: In this article, a new strong nongeometrical arrival, denoted by S *, was detected in the synthetic seismograms computed for a P point source located in the proximity of the free surface.
Abstract: When the Alekseev-Mikhailenko method was used for the exact numerical solution of Lamb's problem, a new strong nongeometrical arrival, denoted by S *, was detected in the synthetic seismograms computed for a P point source located in the proximity of the free surface. Strong dependence of the amplitude of the S * arrival on the depth of the source is seen in computed seismograms. It is shown that under favorable circumstances, i.e., when the source is less than one wavelength from the free surface, the S * arrival may be stronger than ordinary body waves at the same depth. This may be particularly important for studies of the seismic wave fields in oil exploration, where explosive sources are close to the surface. We show that the S * arrival, which features a linear polarization and propagates with the shear wave velocity, may be interpreted as a result of interaction between inhomogeneous plane waves in the integral representation of the P point source, and the free surface. Mathematically, the S * arrival corresponds to the saddle point contribution of the integral along the branch cut originating at the horizontal slowness p = α/β ( α, β indicate P - and S -phase velocities, respectively). This branch cut must be considered when the saddle point approximation to the Weyl-Sommerfeld integral for a shallow P point source is used. Existence of another nongeometrical effect, namely the nonzero vertical component of the converted PS wave reflected from the free surface at normal incidence, is also shown. In our opinion, this arrival can be easily explained by higher order terms in the corresponding ray series. The prominence of both nongeometrical effects suggests they should be incorporated into any synthetic seismogram computations carried out for shallow explosive sources.
TL;DR: In this paper, two methods to determine the electromagnetic wave distribution function are presented: the first is based on the use of the dirichlet kernels and provides us with a local average.
Abstract: Two methods to determine the electromagnetic wave distribution function are presented. The first is based on the use of the dirichlet kernels and provides us with a local average. It has the disadvantage, however, of a nonsystematic approach to positive solutions. The second uses the maximum entropy concept. It leads to particular solutions that are smooth and positive everywhere. The two methods are shown to be complementary. Applications to VLF electromagnetic waves observed onboard Geos 1 are discussed. One of the most striking results is that the wave energy of the natural VLF emissions is generally concentrated within two wave packets whose wave normals are approximately in the same off-meridian plane and oriented in the same way relative to the direction of the earth's magnetic field. It is suggested that those two wave packets have a common source.
TL;DR: In this paper, a hybrid ray-mode field representation was developed whereby the truncation remainder of a mode series can be expressed in terms of ray fields or, conversely, the truncated remainder of the ray series can also be expressed as a guided mode series.
Abstract: A line-source excited parallel plane waveguide with perfectly conducting or surface impedance walls is investigated in the high frequency range where alternative field representations involve many rays or many modes. It is shown that a hybrid ray-mode field representation can be developed whereby the truncation remainder of a mode series can be expressed in terms of ray fields or, conversely, the truncation remainder of a ray series can be expressed in terms of guided modes. A great variety of appropriate ray-mode combinations, each determined from a physically well-founded criterion, is possible: with respect to propagation angles measured at the source, the retained modes are those whose characteristic plane wave propagation angles fill the void left by the truncated ray series and vice versa. The analysis thus clarifies and quantifies the intimate relation between a bundle of rays and a corresponding bundle of modes. The accuracy of various ray-mode mixtures, and computational simplifications gained thereby, are illustrated in several numerical examples covering different parameter ranges, with the conventional guided mode series serving as a reference solution.
TL;DR: In this paper, the dispersion relation for a Rayleigh surface wave propagating normal to the grooves of a diffraction grating has been obtained by two different methods: the first is based on the use of the Rayleigh hypothesis, and the second uses the extinction theorem form of Green's theorem to obtain an exact pair of homogeneous linear integral equations for the boundary values of the elastic displacement field, from the solvability condition for which the rayleigh-wave dispersion relations is obtained.
Abstract: The dispersion relation for a Rayleigh surface wave propagating normal to the grooves of a diffraction grating has been obtained by two different methods. The first is based on the use of the Rayleigh hypothesis. The second uses the extinction theorem form of Green's theorem to obtain an exact pair of homogeneous linear integral equations for the boundary values of the elastic displacement field, from the solvability condition for which the Rayleigh-wave dispersion relation is obtained. When the latter pair of equations are solved by Fourier-series expansions, the dispersion relation obtained is identical with the one obtained on the basis of the Rayleigh hypothesis, in the case that the grating profile function is an even function of its argument. Numerical solutions of the dispersion relation are obtained for a sinusoidal profile and for a symmetric sawtooth profile. For the former profile, solutions are found to be convergent for corrugation strengths far beyond the value at which the Rayleigh hypothesis is known to become invalid for the scattering of a scalar plane wave from a hard sinusoidal surface. For the sawtooth profile (for which in the aforementioned problem the Rayleigh hypothesis is invalid due to the nonanalyticity of the profile) convergent results are here obtained for appreciable values of the corrugation strength. These results indicate that the Rayleigh hypothesis can be used for the calculation of the dispersion curve for a Rayleigh wave propagating across a large-amplitude grating, even when the grating profile function is not an analytic function of its argument. The dispersion curves obtained possess gaps for values of the one-dimensional wave vector, characterizing the propagation of the Rayleigh wave, corresponding to the boundaries of the one-dimensional Brillouin zones defined by the period of the grating. An analytic expression is obtained for the form of the dispersion curve in the vicinity of each gap, in the small corrugation limit, for an arbitrary grating profile. Comparison of the predictions of this analytic result with the exact numerical results allows the limits of validity of the small-amplitude approximation to be determined.
TL;DR: Two methods, a ray optic approximation and a coupled beam method, are proposed for the numerical solution of the coupled wave equations, which are illustrated by a number of calculated examples.
Abstract: The problem of a guided wave obliquely incident on a grating etched in a thin film guide is considered. Two-dimensional coupled wave equations for the incident and reflected beams are derived for the cases of TE–TE, TE–TM, and TM–TM coupling. Two methods, a ray optic approximation and a coupled beam method, are proposed for the numerical solution of the coupled wave equations. Both methods are illustrated by a number of calculated examples.
TL;DR: In this article, a generalized acousto-elastic theory is presented for the determination of propagation speeds of plane waves in bodies experiencing elastic-plastic deformation, and specific expressions for the velocities are generated by assuming an isotropic form of the free energy which includes cubic terms in the elastic strain.
Abstract: A generalized acoustoelastic theory is presented for the determination of propagation speeds of plane waves in bodies experiencing elastic–plastic deformation. The development is carried out in the framework of an isothermal theory for the response of an elastic–plastic continuum by considering the small deformations of a propagating wave superimposed on a finite homogeneous deformation. Specific expressions for the velocities are generated by assuming an isotropic form of the free energy which includes cubic terms in the elastic strain. The velocities are first presented for a finite strain theory and then reduced to linear form for the case of infinitesimal strain.
TL;DR: This study predicts the steady state temperature rise in a homogeneous tissue sphere exposed to plane wave electromagnetic energy, assuming heat transfer mechanisms of convection, conduction, and evapotranspiration from the surface.
Abstract: This study predicts the steady state temperature rise in a homogeneous tissue sphere exposed to plane wave electromagnetic energy. The differential equation for heat transfer is solved, assuming heat transfer mechanisms of convection (due to blood flow), conduction, and evapotranspiration from the surface. The tissue heating is described by the specific absorption rate (SAR), which is a known function of the size and dielectric properties of the sphere and the frequency of the incident electro-magnetic energy. We consider the limiting cases of irradiation at very low frequencies, very high distance in the SAR is effectively averaged to produce a smoothly varying temperature increase. The results of this study are used to predict the maximum temperature rise in the human head, produced by an incident electromagnetic plane wave.
TL;DR: In this paper, a recursive procedure is introduced for the calculation of the properties of an earth model consisting of regions with smoothly varying properties separated by discontinuities in seismic parameters or parameter gradients.
Abstract: Summary. A recursive procedure is introduced for the calculation of the properties of an earth model consisting of regions with smoothly varying properties separated by discontinuities in seismic parameters or parameter gradients. Within each region the solution is constructed from a Langer uniform approximation in terms of Airy functions supplemented by a series of terms representing multifold wave interaction with the parameter gradients. This procedure allows an efficient treatment of turning point problems and in general it is an adequate approximation to retain only the leading order term in the interaction series. At an interface between varying media the solution may be expressed in terms of generalized interface coefficients or alternatively recast into a form which separates the effects of gradients and the interface itself. In the latter case the plane wave reflection and transmission coefficients are to be used. The resulting calculation scheme for the reflection matrices is an extension of the recursive scheme for uniform layers. The simple phase delay transmission effect of a uniform region is replaced by the reflection and transmission from a gradient zone sandwiched between two uniform media. This recursive scheme gives good results for crustal and upper mantle models, and only about a dozen subdivisions of the stratification are required down to 950 km. Checks on the accuracy of the computation from unitarity relations between reflection coefficients show that for periods less than 20 s, in the upper mantle, the error associated with neglect of the interaction series is less than 0.1 per cent except for a few cases. When a turning point is just at a structural boundary the error can increase to 2 per cent. For a low-velocity zone where close turning points occur for slownesses within the range of the inversion, rather larger errors may occur but these do not affect waves which turn well above or well below the zone. The leading order approximation allows no conversion of wave types except at a discontinuity in elastic properties and the error introduced by neglecting conversions at steep gradient zones of limited vertical extent (as at upper mantle discontinuities) may reach a few per cent.
TL;DR: In this paper, a plane wave is incident onto a finite plate set in a rigid baffle, and the scattered field is examined in the limit when fluid-plate coupling effects are large.
Abstract: A plane wave is incident onto a finite plate set in a rigid baffle, and the scattered field is examined in the limit when fluid-plate coupling effects are large. An asymptotic solution is obtained, matching an outer region with inner regions at either edge of the plate. Waves are found to be present on the flexible surface, and resonance is shown to occur for particular values of the plate half-length, a. Away from a resonance, the leading term in the expansion of the outer potential is the solution of the boundary value problem in the absence of the plate. As a resonance is approached, however, eigensolutions, with singularities at the plate edges, also become present at this order.