On the high-frequency limit of the second-order small-slope approximation

Abstract

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

On the high-frequency limit of the second-order small-slope approximation

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

Inverse problem of the wave equation and the Schwinger approximation

A new method of profile inversion for acoustic waves propagating in a medium with spherical inhomogeneity based on the Schwinger variational method is presented. The wave equation of interest is transformed into a Schrodinger equation, so that the Born approximation and the new method could also be applied at high frequencies. It is shown that the new method is stable and is more accurate than the Born approximation. To illustrate the method, an exactly solvable analytical example is presented. Also numerical examples using synthetic data, with and without additive noise, are given and the corresponding inversion results and the stability of the method are studied.     

On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders

The aim of this paper is to propose a numerical strategy for computing the solution of two-dimensional time-harmonic acoustic multiple scattering problems at high-frequency. The scatterers are assumed to be circular, leading therefore to semi-analytical representation formulae of the scattered field through the solution of a large linear system of equations. Taking advantage of the special block Toeplitz structure of the matrix of the linear system, a fast iterative and preconditioned numerical method yielding large memory savings is proposed. Several numerical experiments for general configurations are presented to show the efficiency of the numerical method.     

Using simulated Tianqin gravitational wave data and electromagnetic wave data to study the coincidence problem and Hubble tension problem

In this study, we used electromagnetic wave data (H0LiCOW, \begin{document}$ H(z) $\end{document}, SNe) and gravitational wave data (Tianqin) to constrain the interacting dark energy (IDE) model and investigate the Hubble tension and coincidence problems. By combining these four types of data (Tianqin+H0LiCOW+SNe+\begin{document}$ H(z) $\end{document}), we obtained the following parameter values with a confidence interval of \begin{document}$ 1\sigma $\end{document}: \begin{document}$ \Omega_m=0.36\pm0.18 $\end{document}, \begin{document}$ \omega_x=-1.29^{+0.61}_{-0.23} $\end{document}, \begin{document}$ \xi=3.15^{+0.36}_{-1.1} $\end{document}, and \begin{document}$H_0=70.04\pm $\end{document}\begin{document}$ 0.42~ {\rm kms}^{-1}{\rm Mpc}^{-1}$\end{document}. According to our results, the best value of \begin{document}$ H_0 $\end{document} shows that the Hubble tension problem can be alleviated to some extent. In addition, the center value of \begin{document}$ \xi+3\omega_x = -0.72^{+2.19}_{-1.19}(1\sigma) $\end{document} indicates that the coincidence problem is slightly alleviated. However, \begin{document}$ \xi+3\omega_x = 0 $\end{document} is still within the \begin{document}$ 1\sigma $\end{document} error range, which indicates that the ΛCDM model is still the model in best agreement with the observational data at present. Finally, we compared the constraint results of the electromagnetic and gravitational waves on the model parameters and found that the constraint effect of electromagnetic wave data on model parameters is better than that of simulated Tianqin gravitational wave data.     

On the gravitational wave induced oscillations of viscoelastic bodies

An equation for the propagation of oscillations in a viscoelastic solid, induced by gravitational waves, is derived here. A linearized version of a relativistically invariant constitutive equation of integral type is employed in connection with the appropriately linearized field equations of general relativity. This theory could be applied for a more realistic design of gravitational wave detectors.     

On the velocity of electromagnetic and gravitational wave fronts

It is shown that, in the space of the general theory of relativity, the velocity of an electromagnetic wave front in a non-inertial frame of reference, and of electromagnetic and gravitational wave fronts in a synchronous frame of reference, is constant and equal to the fundamental velocity. It is noted that the concept of a wave front is meaningful in synchronous frames of reference, of which inertial systems in a Minkowski world are a special case.     

Signal photon flux generated by high-frequency relic gravitational waves

The power spectrum of primordial tensor perturbations Vt increases rapidly in the high frequency region if the spectral index n_t 0.It is shown that the amplitude of relic gravitational waves ht(5×10~9 Hz) varies from 10~(-36)to 10~(-25) while n_t varies from-6.25×10~(-3) to 0.87.A high frequency gravitational wave detector proposed by F.-Y.Li detects gravitational waves through observing the perturbed photon flux that is generated by interaction between relic gravitational waves and electromagnetic field.It is shown that the perturbative photon flux N_x~1(5×10~9 Hz)varies from 1.40×10~(-4) s~(-1) to 2.85×10~7 s~(-1) while n_t varies from-6.25×10~(-3) to 0.87.Correspondingly,the ratio of the transverse perturbative photon flux N_x~1 to the background photon flux varies from 10~(-28) to 10~(-16).     

Test of a model coupling of electromagnetic and gravitational fields by using high-frequency gravitational waves

Models of the coupling of electromagnetic and gravitational fields have been studied extensively for many years. In this paper,we consider the coupling between the Maxwell field and the Weyl tensor of the gravitational field to study how the wavevector of the electromagnetic wave is affected by a plane gravitational wave. We find that the wavevector depends upon the frequency and direction of polarization of the electromagnetic waves, the parameter that couples the Maxwell field and the Weyl tensor, and the angle between the direction of propagation of the electromagnetic wave and the coordinate axis. The results show that this coupling model can be tested by the detection of high-frequency gravitational waves.     

On the weak-field approximation in the two-body problem of general relativity

We consider the two-body problem of general relativity taking into account the retardation of interactions. The equations of motion are shown to be simplified, and this enables one to neglect the effects of heredity. The problem of validity of the approximations involved is considered in the extended-particle formalism. It is shown that under the correct treatment the self-interaction terms do not lead to unphysical solutions.     

Methods of spline approximation in the problem of amplitude-time analysis of a pulse wave

We analyze the applicability of the methods for restoration of signal derivatives by cubic and B-splines in the problem of amplitude-time analysis of a pulse wave. It is found that interpolating splines are sufficiently sensitive to the presence of noises in the signal processed. We propose a pre-processing procedure that is based on the use of a specially constructed low-frequency filter and that allows us to improve significantly the efficiency of the methods. The limiting levels of the error in finding the ordinates of characteristic points of a pulse wave are determined and their dependences on both the noise level and amplitude of the input signal are revealed. The results obtained allow us to update the requirements on the quality of the recording channel of the Automated Pulse Diagnostics Complex (APDC). An application package for the amplitude-time analysis of the pulse wave, which is a component of the APDC, is developed on the basis of the methods studied. Buryat Institute of Natural Sciences, Siberian Branch of the Russian Academy of Sciences, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 8, pp., 1043–1057, August, 1998.     

On the Born-Oppenheimer approximation of wave operators in molecular scattering theory

In this paper we study the diatomic molecular scattering by reducing the number of particles through Born-Oppenheimer approximation. Under a non-trapping assumption on the effective potential of the molecular Hamiltonian we use semiclassical resolvent estimates to show that non-adiabatic corrections to the adiabatic (or Born-Oppenheimer) wave operators are small. Furthermore we study the classical limit of the adiabatic wave operators by computing its action on quantum observables microlocalized by use of coherent states.     

Towards the bounce inflationary gravitational wave

In the bounce inflation scenario, the inflation is singularity-free, while the advantages of inflation are preserved. We analytically calculate the power spectrum of its primordial gravitational waves (GWs), and show a universal result including the physics of the bounce phase. The spectrum acquires a cutoff at large scale, while the oscillation around the cutoff scale is quite drastic, which is determined by the details of bounce. Our work highlights that the primordial GWs at large scale may encode the physics of the bounce ever happened at about \({\sim }60\) efolds before inflation.     

Accurate solution of the boundary problem for collisionless gas in the field of a plane gravitational wave

An accurate solution of the Cauchy problem is found for a general-relativity collisionless kinetic equation against the background of the metric of a nonlinear plane gravitational field with an arbitrary law of gas-particle reflection from a boundary of any specified form. It is shown that in the field of a gravitational wave, interaction of the gas with the boundary necessarily leads to the appearance of shock waves.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 41–45, December, 1985.It remains to thank the participants of the gravitation seminar at Kazan' State University for discussions of the work.     

Chaotic scattering in the gravitational three-body problem

We summarize some results of an ongoing study of the chaotic scattering interaction between a bound pair of stars (a binary) and an incoming field star. The stars are modeled as point masses and their equations of motion are numerically integrated for a large number of initial conditions. The global features of the resulting initial-value space maps are presented, and their evolution as a function of system parameters is discussed. We find that the maps contain regular regions separated by rivers of chaotic behavior. The probability of escape within the chaotic regions is discussed, and a straightforward explanation of the scaling present in these regions is reviewed. We investigate a statistical quantity of interest, namely the cross section for temporarily bound interactions, as a function of the third star's incoming velocity and mass. Finally, a new way of considering long-lived trajectories is presented, allowing long data sets to be qualitatively analyzed at a glance.     

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with -1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100x180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10x18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.     

On the motion of test particles in the field of a plane gravitational wave

The motion of test particles in the field of a plane gravitational wave is studied in order to derive some general properties of such motion, especially the possibilities of repeated meetings of two inertially moving particles.The work was done at Moscow State University, Moscow.