Transient representation of the Sommerfeld‐Weyl integral with application to the point source response from a planar acoustic interface

Point source responses from a planar acoustic and/or elastic layer boundary (as well as from a stack of planar parallel layers) are generally obtained by using as a starting point the Sommerfeld‐Weyl integral, which can be viewed as decomposing a time‐harmonic spherical source into time‐harmonic homogeneous and inhomogeneous plane waves. This paper gives a powerful extension of this integral by providing a direct decomposition of an arbitrary transient spherical source into homogeneous and inhomogeneous transient plane waves. To demonstrate with an example the usefulness of this new point source integral representation, a transient solution is formulated for the reflected/transmitted response from a planar acoustic reflector. The result is obtained in the form of a relatively simple integral and essentially corresponds to the solution obtained by Bortfeld (1962). It, however, is arrived at in a physically more transparent way by strictly superimposing the reflected/transmitted transient waves leaving the interface in response to the incident transient homogeneous and inhomogeneous plane waves coming from the center of the point source.

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Transient response from a planar acoustic interface by a new point‐source decomposition into plane waves

Geophysics ◽

10.1190/1.1441951 ◽

1985 ◽

Vol 50 (5) ◽

pp. 766-774 ◽

Cited By ~ 5

Author(s):

Peter Hubral ◽

Martin Tygel

Keyword(s):

Point Source ◽

Wave Field ◽

Plane Waves ◽

Planar Interface ◽

Integration Limit ◽

Inhomogeneous Plane ◽

Homogeneous Plane ◽

Acoustic Interface ◽

New Formula ◽

Inhomogeneous Plane Waves

For the wave field of a point source in full space there currently exist two classical decompositions into plane waves. The wave field can be decomposed into either (a) homogeneous and horizontally propagating (vertically attenuated) inhomogeneous plane waves by using the so‐called Sommerfeld‐Weyl integral, or (b) upgoing and downgoing homogeneous plane waves only using the Whittaker integral. Transient representations of both integrals exist. We propose a new decomposition integral that has a greater flexibility than both classical decompositions. Solutions for the point‐source reflection/transmission response from a planar interface, if based on the Sommerfeld‐Weyl integral, have for instance inherently an infinite integration limit. With the new formula, by which the wave field of a transient point source is decomposed into upgoing and downgoing homogeneous as well as horizontally propagating inhomogeneous transient plane waves, the point‐source response is directly obtained in the form of an integral with a finite integration limit. It can also be interpreted in terms of certain plane waves by which the point source is simulated in a new manner. For that matter, solutions based on the new integral readily reveal the “evanescent” or “nonray” character of the point source. The new formula may be considered an extension of the Sommerfeld‐Weyl or Whittaker integral. It can be used to compute reflection/transmission responses in a compact form in situations where the Sommerfeld‐Weyl integral was hitherto considered fundamental.

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Static and propagating exponential solutions of Maxwell’s equations for crystals

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1992.0121 ◽

1992 ◽

Vol 438 (1904) ◽

pp. 485-510

Keyword(s):

Maxwell’S Equations ◽

Maxwell's Equations ◽

Plane Waves ◽

Electric Permittivity ◽

Complex Vector ◽

Time Harmonic ◽

Inhomogeneous Plane ◽

Isotropic Media ◽

Special Case ◽

Inhomogeneous Plane Waves

Solutions of Maxwell’s equations are considered for anisotropic media for which the electric permittivity κ and magnetic permeability µ are assumed to be arbitrary real positive definite symmetric second order tensors. The propagation of time-harmonic electromagnetic inhomogeneous plane waves or ‘propagating exponential solutions’ in such media has been presented previously. These solutions were systematically obtained by prescribing an ellipse - the directional ellipse associated with a bivector (complex vector) C - and finding the corresponding slowness bivectors. Here, it is shown that for some prescribed directional ellipses, not only propagating exponential solutions (PES), but also static exponential solutions (SES), may be obtained. There are a variety of possibilities. For example, for one choice of directional ellipse it is found that two SES and one PES are possible, whereas, for some other choices, only one SES or only one PES is possible. By a systematic use of bivectors and their associated ellipses, all the possible SES and PES are classified. To complete the classification it is necessary to examine special elliptical sections of the ellipsoids associated with the tensors κ , µ , κ -1 , µ -1 . In particular, sections by planes orthogonal to special directions called ‘generalized optic axes’ and ‘generalized ray axes’ play a major role. These axes reduce to the standard optic axes and ray axes in the special case of magnetically isotropic media.

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Quality factor Q in dissipative anisotropic media

Geophysics ◽

10.1190/1.2937173 ◽

2008 ◽

Vol 73 (4) ◽

pp. T63-T75 ◽

Cited By ~ 23

Author(s):

Vlastislav Červený ◽

Ivan Pšenčík

Keyword(s):

Quality Factor ◽

Plane Waves ◽

Anisotropic Media ◽

Exact Expression ◽

Dissipated Energy ◽

S Wave ◽

Time Harmonic ◽

Inhomogeneous Plane ◽

Quality Factor Q ◽

Inhomogeneous Plane Waves

In an isotropic dissipative medium, the attenuation properties of rocks are usually specified by quality factor [Formula: see text], a positive, dimensionless, real-valued, scalar quantity, independent of the direction of wave propagation. We propose a similar, scalar, but direction-dependent quality [Formula: see text]-factor (also called [Formula: see text]) for time-harmonic, homogeneous or inhomogeneous plane waves propagating in unbounded homogeneous dissipative anisotropic media. We define the [Formula: see text]-factor, as in isotropic viscoelastic media, as the ratio of the time-averaged complete stored energy and the dissipated energy, per unit volume. A solution of an algebraic equation of the sixth degree with complex-valued coefficients is necessary for the exact determination of [Formula: see text]. For weakly inhomogeneous plane waves propagating in arbitrarily anisotropic, weakly dissipative media, we simplify the exact expression for [Formula: see text] con-siderably using the perturbation method. The solution of the equation of the sixth degree is no longer required. We obtain a simple, explicit perturbation expression for the quality factor, denoted as [Formula: see text]. We prove that the direction-dependent [Formula: see text] is related to the attenuation coefficient [Formula: see text] measured along a profile in the direction of the energy-velocity vector (ray direction). The quality factor [Formula: see text] does not depend on the inhomogeneity of the plane wave under consideration and thus is a convenient measure of the intrinsic dissipative properties of rocks in the ray direction. In all other directions, the quality factor is influenced by the inhomogeneity of the wave under consideration. We illustrate the peculiarities in the behavior of [Formula: see text] and its accuracy on a model of anisotropic, weakly dissipative sedimentary rock. Examples show interesting inner loops in polar diagrams of [Formula: see text] in regions of S-wave triplications.

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Scattering of inhomogeneous plane waves by a truncated cylindrical cap

Journal of Modern Optics ◽

10.1080/09500340.2015.1051149 ◽

2015 ◽

Vol 62 (19) ◽

pp. 1555-1560

Author(s):

Husnu Deniz Basdemir

Keyword(s):

Plane Waves ◽

Inhomogeneous Plane ◽

Inhomogeneous Plane Waves

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New capabilities of Chetaev´s model

Solar-Terrestrial Physics ◽

10.12737/21003 ◽

2016 ◽

Vol 2 (2) ◽

pp. 104-114

Author(s):

Михаил Савин ◽

Mihail Savin ◽

Юрий Израильский ◽

Yuriy Izrailsky

Keyword(s):

Experimental Data ◽

Plane Waves ◽

Reflection Coefficients ◽

Geomagnetic Pulsations ◽

Energy Fluxes ◽

Field Recordings ◽

Inhomogeneous Plane ◽

Inhomogeneous Plane Wave ◽

S Model ◽

Inhomogeneous Plane Waves

This paper considers anomalies in the magnetotelluric field in the Pc3 range of geomagnetic pulsations. We report experimental data on Pc3 field recordings which show negative (from Earth’s surface to air) energy fluxes Sz<0 and reflection coefficients |Q|>1. Using the model of inhomogeneous plane wave (Chetaev’s model), we try to analytically interpret anomalies of energy fluxes. We present two three-layer models with both electric and magnetic modes satisfying the condition |Qh|>1. Here we discuss a possibility of explaining observable effects by the resonance interaction between inhomogeneous plane waves and layered media.

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Energy flux and dissipation of inhomogeneous plane waves in hereditary viscoelasticity

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2019.0478 ◽

2019 ◽

Vol 475 (2231) ◽

pp. 20190478

Author(s):

N. H. Scott

Keyword(s):

Energy Density ◽

Energy Flux ◽

Plane Waves ◽

Time Averaging ◽

Complex Frequency ◽

Hereditary Viscoelasticity ◽

Inhomogeneous Plane ◽

Dissipative Material ◽

Comparable Magnitude ◽

Inhomogeneous Plane Waves

Inhomogeneous small-amplitude plane waves of (complex) frequency ω are propagated through a linear dissipative material which displays hereditary viscoelasticity. The energy density, energy flux and dissipation are quadratic in the small quantities, namely, the displacement gradient, velocity and velocity gradient, each harmonic with frequency ω , and so give rise to attenuated constant terms as well as to inhomogeneous plane waves of frequency 2 ω . The quadratic terms are usually removed by time averaging but we retain them here as they are of comparable magnitude with the time-averaged quantities of frequency ω . A new relationship is derived in hereditary viscoelasticity that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2 ω . It is shown that the complex group velocity is related to the amplitudes of the terms with frequency 2 ω rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials.

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