On the excitation of a closely spaced array by a line source

An infinite row of periodically spaced, identical rigid circularcylinders is excited by an acoustic line source which is parallelto the generators of the cylinders. A method for calculatingthe scattered field accurately and efficiently is presented.When the cylinders are sufficiently close together, Rayleigh–Blochsurface waves that propagate energy to infinity along the arrayare excited. An expression is derived which enables the amplitudesof these surface waves to be computed without requiring thesolution to the full scattering problem.     

Scattering relations for point‐source excitation in chiral media

A spherical electromagnetic wave propagating in a chiral medium is scattered by a bounded chiral obstacle which can have any of the usual properties. Reciprocity and general scattering theorems, relating the scattered fields due to scattering of waves from a point source put in any two different locations are established. Applying the general scattering theorem for appropriate locations and polarizations of the point source we prove an associated forward scattering theorem. Mixed scattering relations, relating the scattered fields due to a plane wave and the far‐field patterns due to a spherical wave, are also established. Copyright © 2005 John Wiley & Sons, Ltd.     

Rayleigh wave scattering by a plane strip in a deep ocean

A theoretical formulation to study the problem of scattering of Rayleigh waves due to the presence of a rigid plane strip in a deep ocean is presented. A rigid plane strip (0 ≤z ≤ H, 0 ≤x ≤ l) is fixed in the surface of the ocean occupyingz ≥ 0. Fourier transformation and Wiener-Hopf technique are used to arrive at the solution. The scattered Rayleigh waves behave as cylindrical waves emerging out of the corner of the strip and its image in the free surface of the ocean. The scattered waves are obtained in terms of Bessel functions whose behaviour near and far from the strip is well-known. The numerical calculations for the scattered waves show that their amplitude increases rapidly for a small increase in the value of the wave number. Scattering of Rayleigh waves due to a thin plane vertical barrier and a thin barrier in the free surface of the ocean has been considered as the special cases.     

Fourier Expansion of Disk Automorphisms via Scattering in Layered Media

A family of orthogonal polynomials on the disk (which we call scattering polynomials) serves to formulate a remarkable Fourier expansion of the composition of a sequence of Poincaré disk automorphisms. Scattering polynomials are tied to an exotic Riemannian structure on the disk that is hybrid between hyperbolic and Euclidean geometries, and the expansion therefore links this exotic structure to the usual hyperbolic one. The resulting identity is intimately connected with the scattering of plane waves in piecewise constant layered media. Indeed, a recently established combinatorial analysis of scattering sequences provides a key ingredient of the proof. At the same time, the polynomial obtained by truncation of the Fourier expansion elegantly encodes the structure of the nonlinear measurement operator associated with the finite time duration scattering experiment.     

Inverse scattering problem for Sturm–Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition

The inverse problem of the scattering theory for Sturm–Liouville operator on the half line with boundary condition depending quadratic on the spectral parameter is considered. Scattering data are defined, some properties of the scattering data are examined, the main equation is obtained, solvability of the integral equation is proved and uniqueness of algorithm to the potential with given scattering data is studied. Copyright © 2010 John Wiley & Sons, Ltd.     

Scattering of plane waves by the junction of transmissive and soft-hard half-planes

A uniform asymptotic high frequency solution is developed for the diffraction of plane waves by the junction of two half-planes. One of the half-planes is assumed to be characterized by Seniors resistive-type partially transmissive boundary conditions and the other is soft at the top and hard at the bottom. The related boundary-value problem is formulated as a matrix Wiener-Hopf equation which is solved explicitly through the Daniele-Khrapkov method. Some graphical results are also presented.     

Free convection from a horizontal line source of heat

Summary In this paper a study is made of finite Grashof number effects upon the flow, in an unbounded domain, in the buoyant plume which forms over a horizontal line source of heat. The consequences of introducing a plane boundary into the domain are also assessed for two particular configurations.

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Zusammenfassung In dieser Arbeit wird der Effekt der endlichen Grashof-Zahl auf die Strömung untersucht, in einem unbegrenzten Medium mit natürlicher Konvektion, die durch eine horizontale linienförmige Quelle erzeugt wird. Die Folgen einer ebenen Begrenzungsfläche werden für zwei besondere Konfigurationen untersucht.

     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. II. Green’s function for two-phase infinite plane

Green’s function for isotropic thermoelastic two-phase infinite plane under a line heat source is established in this paper. By virtue of the fourth compact general solutions in Part I which is expressed in three harmonic functions, six new suitable harmonic functions with undetermined constants are constructed for the two semi-infinite planes of the two-phase infinite plane, respectively. The corresponding thermoelastic field can be obtained by substituting these harmonic functions into the general solution, and the undetermined constants can be determined by compatibility conditions and the equilibrium conditions. Numerical results are given graphically by contours.     

Toda lattice with a special self-consistent source

We describe a method for integrating the Toda lattice with a self-consistent source using the inverse scattering method for a discrete Sturm-Liouville operator with moving eigenvalues. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 305–315, February, 2008.     

Behavior of surface waves at transition through a junction line on the boundary of an elastic homogenous isotropic body

A homogeneous elastic body with stress-free boundary is considered. The boundary of the body, which consists of a smooth cylindrical surface and a half-plane, has a continuous tangent plane, but the curvature of the normal section of the boundary has a discontinuity of the first kind at each point of the junction line. The behavior of two kinds of “whispering gallery” transversal surface waves at transition through the junction line is studied. For waves of the first kind (corresponding to Dirichlet boundary conditions), the displacement vector is normal to the boundary, whereas for waves of the second kind (corresponding to Neumann boundary conditions) the displacement vector is tangent to the boundary and normal to the ray, similarly to the case of Love waves. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 86–102. Translated by N. Ya. Kirpichnikova.     

A new approach to Cagniard's problem

Cagniard problem refers to the class of linear reflection and transmission problem for pulsed line and point sources, which have solution methods leading to exact algebraic representations of the wave fields. All previous methods have relied heavily on integral or differential transforms. We present in this paper a new and direct approach to the problem which involves only the wave equation and its associated characteristic equation. We illustrate the new method by applying it to the problem of the reflection and transmission of acoustic waves radiating from a line source in the vicinity of a plane boundary separating two uniform acoustic media.     

Sommerfeld's solution as the limiting amplitude and asymptotics for narrow wedges

We analyze the Sommerfeld solution to the stationary diffraction by a half‐plane. We prove that this solution is the limiting amplitude for time‐dependent scattering of incident plane waves with a broad class of the profile functions. We also show that this solution is the asymptotics of the limiting amplitudes of solutions to time‐dependent scattering problem with narrow wedges when the angle of the wedge tends to zero.     

A point source ofSH wave in a medium with vertical boundary in the case where the velocity of wave propagation is a smoothly increasing function of depth

By the method of separation of variables, the solution of the equation that describes propagation of the SH waves in the case of a point source is constructed in the half-plane y≤0, The coefficients of this equation are piecewise constant functions. The asymptotics of the solution as w→+∞ is found. Bibliography: 6 titles. Translated fromapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 44–55. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by T. N. Surkova.     

Elliptic grid generation methods for scattering from multiple obstacles

A novel finite difference time domain method for acoustic scattering on generalized curvilinear coordinates is briefly described. Scattering over two-dimensional complex regions consisting of multiple scatterers are analyzed. The grid generation algorithm decomposes regions with finite number of holes to contiguous single-hole subregions. Individual grids are obtained for each subregions and they are matched with smoothness across interfaces. The new algorithm is an extension to multiple obstacles of the technique introduced in [V. Villamizar, M. Weber, Boundary-Conforming Cordinates with Grid Line Control for Acoustic Scattering from Complexly Shaped Obstacles, Numer. Meth. Part Differ. Equ. 23 (2007) 1445–1467]. The method is successfully applied to approximate the pressure field resulting from the acoustic scattering of a plane wave from two complexly-shaped obstacles. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flag-Transitive Affine Planes of Order 64

We determine all the flag-transitive affine planes of order 64. There are three: the Desarguesian plane, the Lüneburg plane and the plane given by a construction due to Kantor.     

Scattering of Acoustoelectric Waves on a Cylindrical Inhomogeneity in a Transversely Isotropic Piezoelectric Medium

Scattering of acoustoelectric waves on a inhomogeneity is studied. The scatterer is a circular, continuous piezoelectric cylinder (fiber) embedded in a transversely isotropic piezoelectric medium. Expressions are found for the scattering amplitudes and total cross sections of three acoustoelectric waves propagating in the direction normal to the fiber axis. In the long-wave approximation, these expressions are obtained in an explicit form.     

Systems of hyperbolic conservation laws with a resonant moving source

In this paper, we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source. Compared with the previous results [W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075-1098; T.P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (2) (1982) 243-260] on this stability problem, in this paper, the transonic ith shock is assumed to be relatively strong and stable in the sense of Majda. Then the framework of [M. Lewicka, L¹ stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (4) (2000) 1515-1537; M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (5) (2001) 1094-1116 (electronic)] can be applied. A new criterion is obtained to test whether such a shock is time asymptotically stable or not. And by constructing the Liu-Yang functional, one can prove the L¹ stability of the shock under the stability condition. This is an extension of the result [S.-Y. Ha, T. Yang, L¹ stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (5) (2003) 1226-1251 (electronic); W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075-1098] to a more general case.     

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

An Affine Characterization of Moufang Projective Klingenberg Planes

It is shown that with minor exceptions all projective Klingenberg planes are Moufang exactly if they have a trilateral each of whose sides is the line at infinity of a translation plane; but there are non-Moufang projective Klingenberg planes which have a single line which produces an equiaffine translation plane or a point P so that all lines through P, except perhaps for some neighbour lines, produce translation planes.     

The combined method for calculating the field generated by a point source in a surface waveguide

The two-dimensional problem of propagation of waves generated by a point source in a plane boundary-layer waveguide is investigated. The Dirichlet condition is given on the surface of the waveguide. With the help of cumbersome transformations, the solution is represented as a sum of geometric-optical waves, normal waves, and a remainder. Sufficient conditions on the general amount of detached normal and geometric-optical waves are obtained. The remainder is expressed by a simple formula. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 110–120.