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1.
Summary The effects of a uniform external magnetic field on the propagation of waves in a homogeneous, infinitely conducting flat plate with free boundaries have been studied. It has been found that in general all the three types of waves —P, SV andSH waves—are coupled and the influence may be more pronounced in coupling the symmetric and antisymmetric types of motions in every mode.When the magnetic field is parallel to the plane faces and transverse to the direction of wave propagation, the shear wave polarized parallel to the field is purely elastic whereas the coupledP andS V waves are magnetoelastic and exhibit dispersion strikingly similar to the non-magnetic case, provided the electro-magnetic radiation into the surrounding free space is neglected.The results reported in an earlier communication [1]2) are also confirmed.  相似文献   

2.
Abstract

A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r 0, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance.

Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number Rc. The critical temperature gradient is smallest (Rc reaches a minimum) when the magnetic field strength parameter Λ ? 1. [Rc and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r 2 0 and its direction of propagation changes from eastward to westward as Λ increases through Λ c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λm ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.  相似文献   

3.
A time domain boundary element in a cylindrical co-ordinate system is developed for the analysis of wave propagation in a layered half-space. The field quantities (displacements and tractions) are expressed as products of Fourier series in the circumferential direction and as linear polynomials in the other spatial directions. An integral equation is written for each layer as an independent domain, and these equations are then assembled into a general equation by virtue of compatibility and equilibrium conditions between the interfaces. Examples of three-dimensional wave propagation in the layered half-spaces due to various forms of surface and inner-domain excitations are reported to demonstrate the accuracy and versatility of the method.  相似文献   

4.
The problem of radio wave propagation allowing for 3D localized lower ionosphere irregularity appears in accordance with the necessity of the theoretical interpretation of VLF remote sensing data. The various processes in the Earth's crust and in space (earthquakes, magnetic storms, sporadic E-layers, lightning induced electron precipitations, rocket launches, artificial ionosphere heating, nuclear explosions, etc.) may cause different power and size ionospheric disturbances. This paper presents a further development of the numerical–analytical method for 3D problem solving. We consider a vector problem of VLF vertical electric dipole field in a plane Earth-ionosphere waveguide with a localized anisotropic ionosphere irregularity. The possibility of lowering (elevating) of the local region of the upper waveguide wall is taken into account. The field components on the boundary surfaces obey the Leontovich impedance conditions. The problem is reduced to a system of 2D integral equations taking into account the depolarization of the field scattered by the irregularity. Using asymptotic (kr⪢1) integration along the direction perpendicular to the propagation path, we transform this system to a system of 1D integral equations. The system is solved in the diagonal approximation, combining direct inversion of the Volterra integral operator and the subsequent iterations. The proposed method is useful for study of both small-scale and large-scale irregularities. We obtained estimates of the TE field components that originate entirely from field scattering by a 3D irregularity.  相似文献   

5.
The transient response of a layered structure to plane wave excitation can be considered to be composed of a series of waves and a ground wave. For the case of a half-space of conductivity σ and permeability μ the maximum in the electric field is found at a depth z and time t when t=z2σμ/2. This formula can be used to estimate the depth to a buried horizontal conductor with an accuracy that depends upon the resistive contrast at the conductor's surface. The above ray type of solution can be converted to a solution composed of a number of modes by the use of a Poisson transform and the transformed solutions yield decay constants that are consistent with the previously reported results. In the case of a finite source, the maximum in the electric field is strongly directed. The direction depends upon the geometry of the source and the air-earth interface. Although the maximum varies with direction it can be shown that in some directions similar laws to that above are valid. The depth to a conductor can be estimated from the early part of the transients when the ground wave is removed. The removal of the ground wave from the transient is facilitated by the use of an apparent conductivity formula. Although these results were obtained under restrictive conditions they do provide some insight into the electrical transients that are encountered by studying more complex models.  相似文献   

6.
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7.
The capability of the numerical discontinuous deformation analysis (DDA) method to perform site response analysis is tested. We begin with modeling one‐dimensional shear wave propagation through a stack of horizontal layers and compare the obtained resonance frequency and amplification with results obtained with SHAKE. We use the algorithmic damping in DDA to condition the damping ratio in DDA by changing the time step size and use the same damping ratio in SHAKE to enable meaningful comparisons. We obtain a good agreement between DDA and SHAKE, even though DDA is used with first order approximation and with simply deformable blocks, proving that the original DDA formulation is suitable for modeling one‐dimensional wave propagation problems. The ability of DDA to simulate wave propagation through structures is tested by comparing the resonance frequency obtained for a multidrum column when modeling it with DDA and testing it in the field using geophysical site response survey. When the numerical control parameters are properly selected, we obtain a reasonable agreement between DDA and the site response experiment in the field. We find that the choice of the contact spring stiffness, or the numerical penalty parameter, is directly related to the obtained resonance frequency in DDA. The best agreement with the field experiment is obtained with a relatively soft contact spring stiffness of k = (1/25)(E × L) where E and L are the Young's modulus and mean diameter of the drums in the tested column. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

8.
Abstract

We introduce a general expansion approach to obtain a fully consistent closed set of magnetohydrodynamic equations in two independent variables, which is particularly useful to describe axially symmetric, time-dependent problems with weak variation of all quantities in the radial direction. This is done by considering the hierarchy of expanded magnetofluid equations in cylindrical coordinates and equating terms with equal powers in the radial coordinate r. From geometrical considerations it is shown that the radial expansions of the pertaining physical quantities are either even series or odd series in r; this introduces a significant reduction in the number of variables and equations. The closure of the system is provided by appropriate boundary conditions. Among other possible applications, the method is relevant for the analysis of structure and dynamics of magnetic field concentrations in stellar atmospheres.  相似文献   

9.
The Cagniard-de Hoop method is ideally suited to the analysis of wave propagation problems in stratified media. The method applies to the integral transform representation of the solution in the transform variables (s, p) dual of the time and transverse distance. The objective of the method is to make the p-integral take the form of a forward Laplace transform, so that the cascade of the two integrals can be identified as a forward and inverse transform, thereby making the actual integration unnecessary. That is, the exponent (–sw(p)) is set equal to –sτ, with τ varying from some (real) finite time to infinity. As usually presented, the p-integral is deformed onto a contour on which the exponent is real and decreases to –∞ as p tends to infinity. We have found that it is often easier to introduce a complex variable τ for the exponent and carry out the deformation of contour in the complex τ-domain. In the τ-domain the deformation amounts to ‘closing down’ the contour of integration around the real axis while taking due account of singularities off this axis. Typically, the method is applied to an integral that represents one body wave plus other types of waves. In this approach, the saddle point of w(p) that produces the body wave plays a crucial role because it is always a branch point of the integrand in the τ-domain integral. Furthermore, the paths of steepest ascent from the saddle point are always the tails of the Cagniard path along which w(p) →∞. That is, the image of the pair of steepest ascent paths in the p-domain is a double covering of a segment of the Re τ-axis in the τ-domain. The deformed contour in the p-domain will be the only pair of steepest ascent paths unless the original integrand had other singularities in the p-domain between the imaginary axis and this pair of contours. This motivates the definition of a primary p-domain, i.e. the domain between the imaginary axis and the steepest descent paths, and its image in the τ-domain, the primary τ-domain. In terms of these regions, singularities in the primary p-domain have images in the primary τ-domain and the deformation of contour on to the real axis in the τ-domain must include contributions from these singularities. This approach to the Cagniard-de Hoop method represents a return from de Hoop's modification to Cagniard's original method, but with simplifications that make the original method more tractable and straightforward. This approach is also reminiscent of van der Waerden's approach to the method of steepest descents, which starts exactly the same way. Indeed, after the deformation of contour in the τ-domain, the user has the choice of applying asymptotic analysis to the resulting ‘loop’ integrals (Watson's lemma) or continuing to obtain the exact, although usually implicit, time-domain solution by completing the Cagniard-de Hoop analysis. In developing the method we examine the transformation from a frequency-domain representation of the solution (ω) to a Laplace representation (s). Many users start from the frequency-domain representation of solutions of wave propagation problems. In this case issues arising from the movement of singularities under the transformation from ω to s must be considered. We discuss this extension in the context of the Sommerfeld half-plane problem.  相似文献   

10.
Abstract

We discuss the propagation of internal waves in a rotating stratified unbounded fluid with randomly varying stability frequency, N. The first order smoothing approximation is used to derive the dispersion relation for the mean wave field when N is of the form N 2 = N o 2(1 + ?μ), where μ is a centered stationary random function of either depth (z) or time (t), N o = constant and O < ?2 ≦ 1. Expressions are then derived for the change in phase speed and growth rate due to the random fluctuations μ; in particular, attention is focused on the behaviour of these expressions for short and long correlation lengths (case μ = μ(z)) and times (case μ = μ(t)). For the case μ = μ(z), which represents a model for the temperature and salinity fine-structure in the ocean, the appropriate statistics of the fluctuations observed at station P (50°N, 145°W) have been incorporated into the theory to estimate the actual importance of the effects due to these random fluctuations. It is found that the phase speed of the mean wave decreases significantly if (i) the wavelength is short compared to g/No 2 or (ii) the wave number vector is essentially horizontal and the wave frequency is very close to N o. Also, the random fluctuations cause a significant growth (decay) in the amplitude of a wave propagating upwards (downwards) through a depth of a few kilometers. However, in the direction of energy propagation, the kinetic energy is conserved. Finally, it is shown that the average effect of the depth dependent fluctuations at station P is to slightly decrease the stability frequency and the magnitude of the group velocity.  相似文献   

11.
A method of wave mode determination, which was announced in Balikhin and Gedalin, is applied to AMPTE UKS and AMPTE IRM magnetic field measurements downstream of supercritical quasiperpendicular shock. The method is based on the fact that the relation between phase difference of the waves measured by two satellites, Doppler shift equation, the direction of the wave propagation are enough to obtain the dispersion equation of the observed waves. It is shown that the low frequency turbulence mainly consists of waves observed below 1 Hz with a linear dependence between the absolute value of wave vector |k| and the plasma frame wave frequency. The phase velocity of these waves is close to the phase velocity of intermediate waves Vint = Vacos().  相似文献   

12.
Abstract

Two open curves with fixed endpoints on a boundary surface can be topologically linked. However, the Gauss linkage integral applies only to closed curves and cannot measure their linkage. Here we employ the concept of relative helicity in order to define a linkage for open curves. For a magnetic field consisting of closed field lines, the magnetic helicity integral can be expressed as the sum of Gauss linkage integrals over pairs of lines. Relative helicity extends the helicity integral to volumes where field lines may cross the boundary surface. By analogy, linkages can be defined for open lines by requiring that their sum equal the relative helicity.

With this definition, the linkage of two lines which extend between two parallel planes simply equals the number of turns the lines take about each other. We obtain this result by first defining a gauge-invariant, one-dimensional helicity density, i.e. the relative helicity of an infinitesimally thin plane slab. This quantity has a physical interpretation in terms of the rate at which field lines lines wind about each other in the direction normal to the plane. A different method is employed for lines with both endpoints on one plane; this method expresses linkages in terms of a certain Gauss linkage integral plus a correction term. In general, the linkage number of two curves can be put in the form L=r + n, |r|≦1J2, where r depends only on the positions of the endpoints, and n is an integer which reflects the order of braiding of the curves.

Given fixed endpoints, the linkage numbers of a magnetic field are ideal magneto-hydrodynamic invariants. These numbers may be useful in the analysis of magnetic structures not bounded by magnetic surfaces, for example solar coronal fields rooted in the photosphere. Unfortunately, the set of linkage numbers for a field does not uniquely determine the field line topology. We briefly discuss the problem of providing a complete and economical classification of field topologies, using concepts from the theory of braid equivalence classes.  相似文献   

13.
This paper introduces the conversion of Euler's equation from a Cartesian coordinate system to a radial coordinate system, and then demonstrates that for sources of the type 1/rN (where r is the distance to the source, and N is the structural index) it can be solved at each point in space without the need for inversion, for a known structural index. It is shown that although the distance to the source that is obtained from Euler's equation depends on the structural index used, the direction to the source does not. For some models, such as the gravity and magnetic response of a contact, calculation of the analytic signal amplitude of the data is necessary prior to the application of the method. Effective noise attenuation strategies, such as the use of moving windows of data points, are also discussed. The method is applied to gravity and magnetic data from South Africa, and yields plausible results.  相似文献   

14.
Two seismic wave attenuation factors, scatteringattenuation Q s -1 and intrinsicabsorption Q i -1 are measured using theMultiple Lapse Time Window (MLTW) analysis method forthree different frequency bands, 1–2, 2–4, and 4–8 Hz.Data from 54 temporally deployed seismic stationslocated in northern Chile are used. This methodcompares time integrated seismic wave energies withsynthetic coda wave envelopes for a multiple isotropicscattering model. In the present analysis, the waveenergy is assumed to decay with distance in proportionto1/GSF·exp(- (Q s -1+Q i -1)· r/v), where r, and v are the propagationdistance, angular frequency and S wave velocity,respectively, and GSF is the geometricalspreading factor. When spatial uniformity of Q s -1, Q i -1 and v isassumed, i.e. GSF = 4r 2, theestimates of the reciprocal of the extinction length,L e -1 (= (Q s -1+Q i -1)·/v), are 0.017,0.012 and 0.010 km-1, and those of the seismicalbedo, B 0 (= Q s -1/ (Q s -1+Q i -1)), are 0.48, 0.40and 0.34 for 1–2, 2–4 and 4–8 Hz, respectively, whichindicates that scattering attenuation is comparable toor smaller than intrinsic absorption. When we assumea depth dependent velocity structure, we also findthat scattering attenuation is comparable to orsmaller than intrinsic absorption. However, since thequantitative estimates of scattering attenuationdepend on the assumed velocity structure (strength ofvelocity discontinuity and/or Moho depth), it isimportant to consider differences in velocitystructure models when comparing attenuation estimates.  相似文献   

15.
Acoustic plane wave scattering at a vertical fault structure represents the simplest two-dimensional model of geophysical exploration that can be investigated by analytical techniques. The exact and complete solution, in the time domain, for the scattering of the pressure field of an acoustic plane wave normally incident on a vertical fault structure is determined adapting previous results given for the frequency domain. The wave form of the pressure field of the incident plane wave is expressed by a causal time function that decays exponentially with time at every point above the fault (z<0). The zero-order term of the scattered pressure field has been computed above the fault. This zero-order term consists of an inverse Fourier transform which reduces to a closed expression forx=0, and contains an integral of a Hankel function forx#0. The high frequency part of the inverse Fourier transform forx#0 is computed employing asymptotic expressions for the Hankel function. The integral of the asymptotic expression of the Hankel function reduces to: (i) a Fresnel integral which contains a plane wave term for |x||z|; and (ii) a stationary point plane wave term plus an upper limit term for |x|=O(|z|). For the latter case the plane wave term cancels, leaving a cylindrical wave emanated from the edge of the fault. The wave front is well defined in shape, in phase and in amplitude. The amplitude of the scattered field is discontinuous atx=0, presents a jump and is well defined for |x| small and is rather smooth for |x| large.  相似文献   

16.
A Bremmer Series decomposition of the solution y(t) to the lossless wave equation in layered media is where the yj(t) are physically meaningful constituents (i.e., y1(t) are primaries, y2(t) are secondaries, etc.). This paper reviews Mendel's state space models for generating the constituents; reviews Bremmer's integral equation models for generating the constituents; and demonstrates how Mendel's state space models can be obtained by a careful decomposition of Bremmer's integral equation models. It shows that Mendel's equations can be viewed as approximate numerical solutions of Bremmer's integral equations. In a lossless homogeneous medium, the approximations become exact.  相似文献   

17.
Numerical modelling techniques are now becoming common for understanding the complicated nature of seismic wave propagation in fractured rock. Here the Indirect Boundary Element Method (IBEM) is applied to study scattering of elastic waves by cracks. The problem addressed in this paper is the diffraction of P and S waves by open 3-D cracks of arbitrary shape embedded in a homogeneous isotropic medium. The IBEM yields the value of the jump of displacements between opposite surfaces of the crack, often called Crack Opening Displacement (COD). This is used to evaluate the solution away from the crack. We use a multi-regional approach which consists of splitting a surface S into two identical surfaces S+ and S− chosen such that the crack lies at the interface. The resulting integral equations are not hyper-singular and wave propagation within media that contain open cracks can be rigorously solved. In order to validate the method, we compare results of displacements of a penny-shaped crack for a vertical incident P-wave with the classic results by Mal (1970) obtaining excellent agreement. This comparison gives us confidence to study cases where no analytic solutions exist. Some examples of incidence of P or S waves upon cracks with various shapes are depicted and the salient aspects of the method are also discussed. Both frequency and time-domain results are included.  相似文献   

18.
To better understand (and correct for) the factors affecting the estimation of attenuation (Q), we simulate subsurface wave propagation with the Weyl/Sommerfeld integral. The complete spherical wavefield emanating from a P‐wave point source surrounded by a homogeneous, isotropic and attenuative medium is thus computed. In a resulting synthetic vertical seismic profile, we observe near‐field and far‐field responses and a 90° phase rotation between them. Depth dependence of the magnitude spectra in these two depth regions is distinctly different. The logarithm of the magnitude spectra shows a linear dependence on frequency in the far‐field but not in those depth regions where the near‐field becomes significant. Near‐field effects are one possible explanation for large positive and even negative Q‐factors in the shallow section that may be estimated from real vertical seismic profile data when applying the spectral ratio method. We outline a near‐field compensation technique that can reduce errors in the resultant Q estimates.  相似文献   

19.
This paper collects certain results concerning wave propagation in two-and-one-half dimensions, i.e., three-dimensional (3-D) wave propagation in a medium that has variations in two dimensions only. The results of interest are for sources and receivers in the plane determined by the two directions of parameter variation. The objective of this work is to reduce the analysis of the in-plane propagation to 2-D analysis while retaining–at least asymptotically–the proper 3-D geometrical spreading. We do this for the free space Green's function and for the Kirchhoff approximate upward scattered field from a single reflector. In both cases the derivation is carried out under the assumption of a background velocity c(x, z) with the special cases c = c0 and c = c(z).  相似文献   

20.
Atmospheric gravity waves, with small to medium scales, prevail in the atmosphere and have global ef- fects. Many researches show that gravity waves are the main source that causes the variation of wind and temperature field in the stratosphere, and that the break-up of upward propagating gravity waves is the dominant sources of small scale turbulent and mixing processes in the middle atmosphere. Theories and ob- servations indicate that the redistribution of momen- tum, caused by the generati…  相似文献   

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