Acoustic diffraction by an oscillating strip

The problem of diffraction of a plane acoustic wave from an oscillating rigid strip is studied. The problem is solved by using the temporal and spatial integral transform and the Wiener-Hopf technique. The scattered field in the far zone is determined by the method of steepest descent. The significance of the present analysis is that it recovered the results when a strip is widened to a half plane. Graphical results for the diffraction problem have also been presented.     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

Rayleigh wave attenuation due to scattering by stationary defects

We obtain expressions for the energy spectrum widths of Rayleigh waves arising because of their scattering by point and distributed defects of the surface, as well as by the edge dislocations on the surface and by the grooves of a random lattice in the surface plane. The calculations are valid when the defect density is small. Under certain conditions, our results coincide with the results of other authors who studied the scattering of Rayleigh waves by point defects and by the grooves of a random lattice. The calculations are based on the Keldysh diagram technique modified for the case of semibounded media.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 241–257, May, 2005.     

Light scattering by a homogeneous sphere near a plane boundary

The scattering of electromagnetic waves by a homogeneous sphere near a plane boundary is presented in this paper. The vector wave equations derived from Maxwell’s equations are solved by means of the two orthogonal solutions to the scalar wave equation. Hankel transformation and Erdélyi’s formula are used to satisfy the planar boundary conditions and the determination of the unknown coefficients in the scattered field and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution of the series involving these unknown coefficients are shown.     

Time‐dependent scattering of generalized plane waves by a wedge

We obtain explicit formulas for the scattering of plane waves with arbitrary profile by a wedge under Dirichlet, Neumann and Dirichlet‐Neumann boundary conditions. The diffracted wave is given by a convolution of the profile function with a suitable kernel corresponding to the boundary conditions. We prove the existence and uniqueness of solutions in appropriate classes of distributions and establish the Sommerfeld type representation for the diffracted wave. As an application, we establish (i) stability of long‐time asymptotic local perturbations of the profile functions and (ii) the limiting amplitude principle in the case of a harmonic incident wave. Copyright © 2015 John Wiley & Sons, Ltd.     

Scattering theory for the elastic wave equation in perturbed half-spaces

In this paper we consider the linear elastic wave equation with the free boundary condition (the Neumann condition), and formulate a scattering theory of the Lax and Phillips type and a representation of the scattering kernel. We are interested in surface waves (the Rayleigh wave, etc.) connected closely with situations of boundaries, and make the formulations intending to extract this connection.

The half-space is selected as the free space, and making dents on the boundary is considered as a perturbation from the flat one. Since the lacuna property for the solutions in the outgoing and incoming spaces does not hold because of the existence of the surface waves, instead of it, certain decay estimates for the free space solutions and a weak version of the Morawetz arguments are used to formulate the scattering theory.

We construct the representation of the scattering kernel with outgoing scattered plane waves. In this step, again because of the existence of the surface waves, we need to introduce new outgoing and incoming conditions for the time dependent solutions to ensure uniqueness of the solutions. This introduction is essential to show the representation by reasoning similar to the case of the reduced wave equation.

     

Water wave scattering by finite arrays of circular structures

The scattering of small amplitude water waves by a finite arrayof locally axisymmetric structures is considered. Regions ofvarying quiescent depth are included and their axisymmetricnature, together with a mild-slope approximation, permits anadaptation of well-known interaction theory which ultimatelyreduces the problem to a simple numerical calculation. Numericalresults are given and effects due to regions of varying depthon wave loading and free-surface elevation are presented.     

An improved model for noise barriers in a moving fluid

We study a problem of diffraction of a cylindrical acoustic wave from an absorbing half plane in a moving fluid introducing Myers' condition [M.K. Myers, On the acoustic boundary condition in the presence of flow, J. Sound Vibration 71 (1980) 429] and present an improved form of the analytic solution for the diffracted field. The importance of the work lies in the fact that Myers' condition (a generalization of Ingard's impedance condition) is now the accepted form of the boundary condition for impedance barriers with flow and hence yields a correct form of the field. The method of solution consists of Fourier transform, Wiener-Hopf technique and the modified method of stationary phase.     

Water wave scattering by two sharp discontinuities in the surface boundary conditions

^** Corresponding author. Email: biren{at}isical.ac.in The problem of water wave scattering by two sharp discontinuitiesin the surface boundary conditions involving infinitely deepwater is examined here by reducing it to two coupled Carleman-typesingular integral equations. The discontinuities arise due tothe presence of two types of non-interacting materials floatingon the surface, one type being in the form of an infinite stripof finite width sandwiched between another type. The non-interactingmaterials form an inertial surface which is a mass-loading modelof floating ice and is regarded as a material of uniform surfacedensity having no elastic property. The two integral equationsare solved approximately by assuming the two discontinuitiesto be widely separated, and approximate analytical expressionsfor the reflection and transmission coefficients are also obtained.This problem has applications in wave propagation through stripsof frazil or pancake ice modelled as floating inertial surfaces.Numerical results for the reflection coefficient are depictedgraphically against the wave number for different values ofthe surface densities of the two types of floating materials.The main feature of the graphs is the oscillatory nature ofthe reflection coefficient and occurrence of zero reflectionfor an increasing sequence of discrete values of the wave number.A direct analytical treatment to solve the integral equationsnumerically, when the separation length between the two discontinuitiesis arbitrary, is also indicated. For the case of more than twodiscontinuities the solution methodology of the correspondingscattering problem is described briefly.     

Scattering of guided SH-wave by a partly debonded circular cylinder in a traction free plate

A solution of the scattering problem of guided SH-wave by a partly debonded circular cylinder centered in a traction free plate has been set up. The plate is divided up into three regions with two imaginary planes perpendicular to the plate walls. In the central region where the partly debonded cylindrical obstacle is posted, the wave field is expanded into the cylindrical wave modes and Chebyshev polynomials. In the other two exterior regions the fields are expanded into the plate wave modes. A system of fundamental equations to solve the problem is obtained according to the traction free boundary condition on the plate walls and the continuity condition of the traction and the displacement across the imaginary planes. The approximate numerical method termed mode-matching technique is used to construct a matrix equation to obtain curves showing the coefficient of reflection and transmission versus the ratio of the cylinder’s radius to the plate’s half-thickness and the angular width of the debonded region. A comparison of the numerical results between the welded interface condition and the debonded interface condition is made, and the results are discussed.     

DN‐Scattering of a plane wave by wedges

We continue our study of a nonstationary scattering by wedges. In this paper we consider nonstationary scattering of plane waves by a ‘hard–soft’ wedge. We prove the uniqueness and existence of a solution to the corresponding DN‐Cauchy problem in appropriate functional spaces. We also give the explicit form of the solution and prove the Limiting Amplitude Principle. Copyright © 2011 John Wiley & Sons, Ltd.     

The scattering of a plane wave by a trap in the critical case

The scattering of a plane wave by a resonator with a narrow coupling channel is considered. The velocity potential of the scattered wave in this resonator has two series of poles with small imaginary parts, corresponding to the main trap and the coupling channel, the effect of which inside the trap differs by an order of magnitude. The critical case, when the limiting value for the poles from both series is the same, is investigated. It is shown that in this case two poles exist, which converge to this limiting value, and they both inherit resonance properties, characteristic for poles generated by the main trap. The principal terms of the asymptotic forms of the poles and the scattered wave are constructed.     

Asymptotics of optimal control in the problem of harmonic wave scattering by an obstacle

Optimal impedance control for the Helmholtz equation in an unbounded domain is studied. Asymptotics of the optimal control with respect to a regularization parameter are constructed.     

Locating an obstacle in a 3D finite depth ocean using the convex scattering support

We consider an inverse scattering problem in a 3D homogeneous shallow ocean. Specifically, we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle. This reconstruction is obtained from the far-field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency. The technique is based on an implementation of the theory of the convex scattering support [S. Kusiak, J. Sylvester, The scattering support, Commun. Pure Appl. Math. (2003) 1525–1548]. A few examples are presented to show the feasibility of the method.     

Electromagnetic scattering by a smooth convex impedance cone

The problem of the diffraction of an electromagnetic planewave by a convex cone of arbitrary smooth cross-section withimpedance (Leontovich) boundary conditions is studied. The vectorproblem is reduced to that for the Debye potentials. By meansof Kontorovich–Lebedev integrals, two spectral functionsare introduced and the corresponding boundary value problemis formulated. The spectral functions for the potentials arefound to satisfy the Helmholtz equations on the unit sphereand to be coupled through non-traditional boundary conditionsof the impedance type with shifts on the spectral variable.The use of the Green theorem permits us to establish an integralformulation of the boundary value problem for the spectral functions.The formal asymptotic solution of the problem is then givenfor the case of a narrow cone. For this, two different methodsare given: a method of perturbation applied to the spectralintegral equations and an adaptation of the method of matchingthe asymptotic series in spectral domain. Both methods leadto the same closed-form result for the leading term of the scatteringdiagram asymptotics.     

Diffraction of sound waves by a finite barrier in a moving fluid

The diffraction of a line source by an absorbing finite barrier, satisfying Myers' impedance condition [M.K. Myers, On the acoustic boundary condition in the presence of flow, J. Sound Vibration 71 (1980) 429-434] in the presence of a subsonic flow is studied. The problem is solved analytically by using Integral transforms, Wiener-Hopf technique and the asymptotic methods. The expression for the diffracted field is shown to be the sum of the fields produced by the two edges of the strip and a field due to the interaction of the two edges. The diffracted field in the far zone is determined by the method of steepest decent.     

Reflection of plane waves by rough surfaces in the sense of Born approximation

The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, that is, by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns’ formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions where the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far‐field terms such as those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces. Copyright © 2013 John Wiley & Sons, Ltd.     

Features of the linear stage of development of 3D wave packets in a plane Poiseuille flow

In the framework of the asymptotic theory of free interaction, the linear stage of the development of 3D wave packets in a plane Poiseuille flow is studied. Numerical results show the presence of “ripples” in the lateral direction in the first phase of the linear stage. The disturbances propagate within a certain angle.     

Nonlinear scattering theory for a class of wave equations in H

For the semi-linear (higher order) wave equation and the nonlinear (higher order) Schrödinger equation, we show that the scattering operators map a band in H^s into H^s if the nonlinearities have (sub-)critical powers in H^s. The smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing NLS equation are also shown provided that the nonlinearities have subcritical growth in H¹. Moreover, the spatial decaying behavior of solutions in energy space for the defocusing NLS equation are obtained.