Impulsive Plane Wave Research Articles

All-multiplicity amplitudes in impulsive PP-waves from the worldline formalism

We use the worldline formalism to derive Bern-Kosower type Master Formulae for the tree-level scattering of a charged particle and an arbitrary number of photons on impulsive PP-waves, where the coupling of the PP-wave to matter is treated fully non-perturbatively. We show that, in a certain kinematic regime characterised by a semi-classical positive energy condition, both off-shell currents and scattering amplitudes exhibit two novel factorisation structures. First, they may be written as currents in vacuum but with a single additional photon, averaged over the momentum of that photon. This converts the all-orders interaction with the PP-wave into a single effective interaction. Second, the currents and amplitudes may be written as a weighted average of the corresponding quantities in an impulsive plane wave background, with the average taken over all possible field strengths of the plane wave. This generalises a known single-photon result to arbitrary multiplicity.

Wightman function and stochastic gravity noise kernel in impulsive plane wave spacetimes

Wightman function and stochastic gravity noise kernel in impulsive plane wave spacetimes

Coupling of Impulsive EM Plane-Wave Fields to Narrow Conductive Strips: An Analysis Based on the Concept of External Impedance

The pulsed electromagnetic coupling to a narrow perfectly electrically conducting strip is analyzed using the concept of external impedance. For postulated spatial electric-current distributions, the problem is reduced to an integral equation of the time-convolution type that is, under the narrow-strip assumption, solved numerically via a marching-on-in-time scheme. Illustrative numerical examples concerning the impulsive plane-wave incidence demonstrate the space-time behavior of the induced electric-current density.

The pulsed EM plane-wave response of a thin planar antenna

Electromagnetic coupling of an impulsive plane wave to a thin planar antenna is described analytically using the reciprocity theorem of the time-convolution type. It is shown that the pulsed voltage response induced within a thin planar antenna can be expressed through a simple rim integration of either the “testing” voltage distribution or the generalized-ray representation of the time-domain Green’s function. Numerical experiments demonstrate the validity and high computational efficiency of the proposed closed-form coupling model.

Numerical evolution of plane gravitational waves in the Friedrich-Nagy gauge

The first proof of well-posedness of an initial boundary value problem for the Einstein equations was given in 1999 by Friedrich and Nagy. They used a frame formalism with a particular gauge for formulating the equations. This `Friedrich-Nagy' (FN) gauge has never been implemented for use in numerical simulations before because it was deemed too complicated. In this paper we present an implementation of the FN gauge for systems with two commuting space-like Killing vectors. We investigate the numerical performance of this formulation for plane wave space-times, reproducing the well-known Khan-Penrose solution for colliding impulsive plane waves and exhibiting a gravitational wave `ping-pong'.

Tutorial on seismic interferometry: Part 1 — Basic principles and applications

Seismic interferometry involves the crosscorrelation of responses at different receivers to obtain the Green’s function between these receivers. For the simple situation of an impulsive plane wave propagating along the [Formula: see text]-axis, the crosscorrelation of the responses at two receivers along the [Formula: see text]-axis gives the Green’s function of the direct wave between these receivers. When the source function of the plane wave is a transient (as in exploration seismology) or a noise signal (as in passive seismology), then the crosscorrelation gives the Green’s function, convolved with the autocorrelation of the source function. Direct-wave interferometry also holds for 2D and 3D situations, assuming the receivers are surrounded by a uniform distribution of sources. In this case, the main contributions to the retrieved direct wave between the receivers come from sources in Fresnel zones around stationary points. The main application of direct-wave interferometry is theretrieval of seismic surface-wave responses from ambient noise and the subsequent tomographic determination of the surface-wave velocity distribution of the subsurface. Seismic interferometry is not restricted to retrieving direct waves between receivers. In a classic paper, Claerbout shows that the autocorrelation of the transmission response of a layered medium gives the plane-wave reflection response of that medium. This is essentially 1D reflected-wave interferometry. Similarly, the crosscorrelation of the transmission responses, observed at two receivers, of an arbitrary inhomogeneous medium gives the 3D reflection response of that medium. One of the main applications of reflected-wave interferometry is retrieving the seismic reflection response from ambient noise and imaging of the reflectors in the subsurface. A common aspect of direct- and reflected-wave interferometry is that virtual sources are created at positions where there are only receivers without requiring knowledge of the subsurface medium parameters or of the positions of the actual sources.

Analytic Analysis of Transient Scattering From a Finite Second-Order Surface Illuminated by an Incident Plane Wave

A closed-form analytic solution based on a time-domain (TD) physical optics (PO) approximation is developed for the scattering from a finite second-order surface when illuminated by a transient impulsive plane wave. This TD-PO solution can be applied via a convolution to derive the early time transient fields scattered from the same scatterer that is illuminated by a realistic astigmatic finite-energy pulse. The closed form TD-PO solution is obtained by inverting the corresponding frequency-domain PO solution into TD. This solution can be expressed in terms of reflection and diffraction components of the scattering mechanisms as in other conventional high-frequency asymptotic solutions. Numerical examples are presented to demonstrate its physical phenomenon of the scattering mechanisms

Reconstruction of multilayered lossy dielectrics from one-sided plane wave impulse reflection responses: the bistatic case

Motivated by numerous scattering problems where the media is lossy and yet can only be investigated from a single side, the authors discuss algorithms for reconstructing a lossy, stratified dielectric from one-sided plane wave impulse reflection responses. Novel features of these algorithms include: 1) solving the lossy media problem using only one-sided reflection response data; 2) exploiting oblique probing angles; 3) having origins in digital signal processing (DSP) theory; 4) employing fast algorithms that have been modified to solve the forward scattering problem; and 5) solving the scattering problems exactly, including accounting for multiple reflections. A new algorithm is introduced that reconstructs such a media from one-sided transverse electric (TE) and transverse magnetic (TM) impulsive plane wave reflection responses from a single oblique angle. The oblique-probing angle necessitates a bistatic measurement configuration. New probing constraints are developed for a previously presented algorithm based on probing at two angles of incidence. Algorithm stability and data correction are addressed. Numerical examples illustrate the new algorithm in synthesizing the media transient response and reconstructing the media.

Diffraction of an Impulsive Plane Wave at a Conducting Wedge

Diffraction of an Impulsive Plane Wave at a Conducting Wedge

Plane light-like shells and impulsive gravitational waves in scalar-tensor theories of gravity

We study gravitational plane impulsive waves and electromagnetic shock waves in a scalar-tensor theory of gravity of the Brans-Dicke type. In vacuum, we present an exact solution of the Brans-Dicke field equations and give an example in which a plane impulsive gravitational wave and a null shell of matter coexist on the same hypersurface. In the homogenous case, we characterize them by their surface energy density and wave amplitude and discuss the inhomogenous case. We also give an exact solution of the Brans-Dicke field equations in the electrovacuum case which admits a true curvature singularity and use it to built an example where a plane impulsive gravitational wave and an electromagnetic shock wave have the same null hypersurface as history of their wavefronts and propagate independently and decoupled from a null shell of matter. This last solution is shown to correspond to the spacetime describing the interaction region resulting from the collision of two electromagnetic shock waves leading to the formation of two gravitational impulsive waves. The properties of this solution are discussed and compared to those of the Bell-Szekeres solution of general relativity.

Inverse scattering for time-varying one-dimensional layered media: algorithms and applications

We formulate and present algorithms and applications for the inverse scattering problem for a one-dimensional layered medium whose reflection coefficients vary with time. We show that given the reflection responses to impulsive plane waves incident on the medium at different times, this problem can be solved either by solving a series of nested non-Toeplitz systems of equations, or by propagating a set of coupled layer-stripping algorithms with reflection coefficients varying in time. All multiple scattering effects are included here. We show that our results reduce to previous results in the special case of time-invariant media. Applications include the two-dimensional (2D) inverse resistivity problem of reconstructing the 2D resistivity of a 2D medium from surface measurements, and the reconstruction of rapidly curing layered media. We also present a time-varying discrete Miura transform linking the time-varying discrete wavesystem inverse scattering problem to a time-varying discrete Schrödinger equation, and a new important feasibility condition on the reflection responses, which seems to be new in the context of the 2D inverse resistivity problem. A numerical example is provided.

Singular hypersurfaces in scalar - tensor theories of gravity

We study singular hypersurfaces in tensor multi-scalar theories of gravity. We derive in a distributional and then in an intrinsic way, the general equations of junction valid for all types of hypersurfaces, in particular for lightlike shells and write the general equations of evolution for these objects. We apply this formalism to various examples in static spherically symmetric spacetimes, and to the study of planar domain walls and plane impulsive waves.

A Priori Estimates and Convergence for the Discrete Forward and Inverse Problems of Reflection Seismology

The authors study numerical methods for the forward and inverse problems of one-dimensional reflection seismology. An impulsive plane wave in pressure is applied to the surface of a stratified elastic half space and the resulting particle velocity is measured at the surface. The forward problem is to determine the particle velocity response at the surface from the characteristic impedance of the medium. The inverse problem is to recover the characteristic impedance of the medium as a function of travel time from the measured particle velocity response at the surface. Assuming the impedance profile is in the Sobolev space $H^3 $, the authors prove the second-order convergence of the forward problem in the discrete $H^1 $ -norm using energy estimates designed to handle the propagating singularity in the solution. In the discrete problem, a priori estimates for the impedance and its differences that depend only on the measured particle velocity at the surface are proved. Finally, these a priori estimates and the convergence of the forward problem are used to prove second-order convergence of the inverse problem in the discrete $H^1 $-norm, assuming the impedance profile is in $H^3 $. The convergence proof requires weaker hypotheses (by more than one derivative) and has stronger conclusions (convergence in a stronger norm) than earlier proofs of convergence.

Reconstruction of multilayered lossy dielectrics from plane wave impulse responses at two angles of incidence

Motivated by the radioglaciology inverse problem, we present new algorithms for reconstructing a lossy, stratified dielectric from its impulsive plane wave reflection responses at two different angles of incidence. Novel features of these algorithms include: 1) a digital signal processing formulation that does not require discretization of continuous equations; 2) use of the asymmetric Levinson algorithm for rapid solution of the forward and inverse problems; 3) a novel use of layer stripping ideas, featuring iteration between the forward and inverse problems, with each iteration recursively determining another layer of the medium; and 4) another recursive algorithm for determining the bottom lossy half-space from reflection data only. Numerical examples illustrate the new algorithms on the reconstruction of a synthetic but realistic layered ice shelf from both noiseless and noisy radar reflection data.

Long Time Evolution of Wavefronts in Random Media

This work studies the evolution of the front of an impulsive plane wave in a layered random medium. A general second-order hyperbolic equation with an inhomogeneous plane wave solution in a frame moving with random propagation speed is considered. In this frame it takes the form of a coupled system of ordinary differential equations along characteristics for the propagation modes. The author shows that in the asymptotic limit in which the amplitude of random inhomogeneities decreases but the distance at which the pulse is observed increases the leading part of the pulse, the wavefront, stabilizes to a deterministic waveform. This waveform depends on the statistical properties of the medium and the initial pulse only, and it is explicitly calculable. The theory is applied to the elastic wave equations in a two-dimensional medium.

Erratum: Gravitational Faraday rotation induced from interacting gravitational plane waves

The polarization of gravitational plane waves is studied. In particular, it is found that each interacting gravitational plane wave can be split into two parts: the shock part and the impulsive part. The plane of the polarization for the plane gravitational shock waves usually gets rotated due to two kinds of interaction: one is the nonlinear interacton between the oppositely moving plane gravitational shock waves, and the other is the interaction between the gravitational shock wave and the matter fields which are present. The change of polarization due to the nonlinear interaction is exactly a gravitational analogue of Faraday rotation, but with the oppositely moving gravitational shock wave as the magnetic field and medium. The effect of the above two kinds of interaction on the impulsive gravitational plane waves is different from that on the shock waves. Only the interaction between the impulsive gravitational wave and the matter fields can change the polarization of the impulsive wave.

A fast algorithm for the reconstruction of a layered acoustic medium from its transmission response

We derive a layer‐recursive algorithm for reconstructing an equal‐travel‐time layered acoustic medium from its transmission response to an impulsive plane wave incident from below. The medium is bounded below by an infinite half‐space from which it is probed, and it is bounded above by a free surface at which its transmission response is measured. This problem arises in the nondestructive evaluation of layered composite materials, in which the transmission response is measured. The algorithm is similar in form to the Levinson algorithm, which reconstructs a layered medium from its reflection response; it is closely related to the lattice algorithm for computing the QR factorization of a matrix. We relate the new algorithm to the Levinson, fast Cholesky, and lattice algorithms, all of which are well‐known fast algorithms for signal processing.

Wave propagation with tunneling in a highly discontinuous layered medium

An impulsive plane wave traverses a stratified medium consisting of a large number N of homogeneous isotropic perfectly elastic layers. The directly transmitted wave is greatly reduced by the cumulative effect of scattering loss at each of the many interfaces. However, close to the arrival of the direct wave is a broad pulse, arising from multiple scattering; this pulse does not decay as rapidly as the direct wave and ultimately appears to diffuse about a moving center. The latter process, which is determined by the medium statistics, leads to time delays, effective anisotropy, and apparent attenuation. The present work may be regarded as an extension of that described by Burridge, White and Papanicolaou (1988) and Burridge and Chang (1989) to allow for tunneling P waves for S-wave incidence beyond the critical angle. When the reflection coefficients at the interfaces are scaled as 1/√ N while N → ∞, and when time is measured in units of vertical travel time across an average layer, numerical solutions of the exact problem show that the shape of the broad transmitted pulse approaches the limiting form given as the solution of a certain integrodifferential equation in accordance with our asymptotic theory.

Early-time currents induced on a cylinder by a cylindrical electromagnetic wave

An analytical study of the early-time current response of a perfectly conducting circular cylinder to a cylindrical electromagnetic wave, produced by a parallel filament carrying a unit-step current, is presented. Surface current density expressions for both the shadow and illuminated regions are obtained by applying the Watson transformation to the frequency-domain solution and then evaluating the corresponding integrals separately, in the shadow region by a residue series and in the illuminated region by using the saddle point technique. Analytical results obtained by using a double Laplace transform approach for the shadow region and the Luneberg-Kline expansion approach for the illuminated region are given. Numerical results illustrate the variation of the distribution of current density with time for different locations of the line current with respect to the cylinder, and also allows a comparison of the alternate methods applied by the authors, showing their corresponding range of validity. The response to an impulsive plane wave is derived as a special case.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Emission and absorption of plane impulsive gravitational waves

Emission and absorption of plane impulsive waves of radiation by a thin shell of dust are studied. At the moment of interaction local conservation laws apply. An impulsive gravitational wave does not participate in the energy balance. Rather, it is associated with a change in isotropy of the source (absorber). Changes in the kinetic energy of the source exactly match the changes in its rest mass.