Numerical solution of the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary

A method of numerically solving the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary is described, and results are presented of calculating the wave field in the case of the first and second whispering gallery waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 207–219, 1976.     

Excitation coefficients of whispering gallery waves in a neighborhood of a flat point of a concave boundary

A method is proposed for the computer calculation of the excitation coefficients of whispering gallery waves which arise beyond a flat point of a concave boundary when a single whispering gallery wave is incident on this point. Values of the moduli of these coefficients are presented for the incidence of the first and second whispering gallery waves. From the results obtained it follows that in both cases considered the major portion of the energy of the incident wave (27 and 23%, respectively) goes over into the first wave beyond the flat point of the boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 138–145, 1980.The authors are grateful to I. Pshenchik (Geophysical Institute, Prague) for his assistance and advice.     

Diffraction of creeping waves from a point of curvature jump on the boundary of a domain (The point of transition of the convex boundary into the concave boundary)

The problem of diffraction of a creeping wave propagating in a domain near the convex part of the boundary and overrunning a point where the convex boundary transforms to the concave one is studied. The tangent to the boundary is continuous at this point, but the derivative of the tangent has the jump. The Green's function to the right of the point of jump of curvature is a superposition of whispering gallery waves. The Dirichlet, Neumann, and impedance boundary conditions are considered. The formulas for the boundary current and for the diffraction coefficients related to the problem are obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 288–299. Translated by N. Ya. Kirpichnikova     

On the dispersion of nonstationary surface waves in anisotropic elastic media

The whispering gallery modes propagating along the surface of an anisotropic elastic body are investigated with the use of space-time caustic expansions and space-time ray series. Each surface mode modulated in amplitude and frequency, is interpreted as a wave packet, with its amplitude’s maximum moving at a group velocity. On the boundary surface, asymptotic expressions for the group velocity (as a function of time and coordinates) are derived, which are in agreement with analogous formulas for Rayleigh waves of SV type in the isotropic case. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 299–312.     

Waveforms in additional components of elastic bulk waves

Additional components in displacements of elastic wave fields are those which vanish in the case of propagation of homogeneous plane waves. For P-waves in a homogeneous isotropic solid, these are transverse components. Waveforms in additional components in simple models of non-time-harmonic elastic wave propagation with plane wavefronts are analyzed. It is demonstrated that the models based on homogeneous waves with a transverse structure and on inhomogeneous waves show a qualitative difference. Bibliography: 20 titles. Dedicated to V. M. Babich on the occasion of his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 90–98.     

Diffraction of whispering gallery waves by a conjunction line

A nonhomogeneous elastic body with stress-free boundary is considered. The boundary consists of two smooth cylindrical surfaces with continuous tangent plane. The curvature of the boundary is assumed to have a discontinuity on the conjunction line. The behavior of two kinds of transversal surface whispering gallery waves passing through the conjunction line is studied. For a wave of the first kind, corresponding to the Dirichlet boundary condition, the displacement vector is normal to the boundary. For a wave of the second kind, corresponding to the Neumann boundary condition, the displacement vector is tangent to the boundary and normal to the ray, which is similar to the case of Love waves.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 95–109.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Correctness of the problem of whispering gallery waves in a neighborhood of points of zero curvature of the boundary

Problems of the propagation of whispering gallery waves in a neighborhood of boundary points where the curvature has a zero of multiplicity one to two are considered. The existence and uniqueness of a solution of these problems are established by means of wave operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 261–269, 1979.     

Concentration of the solutions near a limit ray in the neighborhood of an inflection point of the boundary

One considers the question regarding the energy concentration of a wave field in the neighborhood of a limit ray in the problem of the behavior of whispering gallery waves near an inflection point of the boundary. An estimate related to this question is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 155–158, 1988.     

Whispering gallery waves in a neighborhood of a flat point of a concave boundary

The propagation of whispering gallery waves near a concave (from the side of the wave field) boundary having a flat point is studied. As in [4], the problem that arises for an equation of Schrodinger type is solved by the method of grids. The results of the computations are presented as shadow figures.     

Wave field in the caustic shadow in a neighborhood of an inflection point of the boundary

The problem of the propagation of whispering gallery waves in a neighborhood of an inflection point of the boundary is considered. It is shown that a caustic shadow zone occurs away from the boundary along the normal. The asymptotics of the wave field in the caustic shadow are obtained, and their geometric interpretation in terms of complex rays is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 246–260, 1979.     

Special Function Related to the Scattering of the Whispering Gallery Mode at a Point of Local Straightening

We consider the scattering of the whispering gallery mode at a point of local straightening of the boundary. A special function describing the main features of the wave field in the vicinity of the point of local straightening is constructed and examined. Bibliography: 12 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 180–186     

Radiation problem of normal stress loading of bore surface

The radiation of elastic waves at normal harmonic stress loading of a ring strip of the cylindrical bore-surface in the infinite elastic medium is considered. The wave field is represented by contour integrals. By using the saddle-point method stress and displacement of asymptotic expansions in the far field were obtained. It is shown that in the far field P-waves bring radial displacements and stresses, and perpendicular to these, displacements and stresses are caused by S-waves. In the paper orientation diagrams of radial and circular displacements and frequency dependencies of relative distribution of radiation power source on types of waves are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radiation field of whispering gallery waves over a concave-convex boundary

In this work the wave field arising over a concave-convex reflecting boundary is studied in the Kirchhoff approximation. The field arises as a result of the incidence of whispering gallery waves on an inflection point of the boundary from the concave side. The shortwave asymptotics of the Kirchhoff integral are obtained which is expressed in terms of special functions in a neighborhood of the inflection point of the boundary and in a neighborhood of the tangent to the boundary at the inflection point. Diagrams are constructed that illustrate the behavior of the scattered field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 49–65, 1981.     

Comparison of the field of a whispering gallery wave computed by Kirchhoff's method and by the method of the parabolic equation in a neighborhood of zero curvature of the boundary

In the paper a comparison is made of results obtained by Kirchhoff 's method and by the method of the parabolic equation in solving the problem of whispering gallery waves propagating over a concave — convex boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 139–145, 1981.     

A point source of oscillations on the boundary of a region

The problem of a point source of oscillations on a curveS with curvature which is nowhere zero is considered: The case where the whispering gallery effect arises is investigated: rays issuing from a source and reflected many times fromS create this effect. A function containing all singularities ofw is constructed explicitly. The theorem that the set of singularities of the functionw coincides with the wave fronts of geometrical optics is a consequence of these considerations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 3–20, 1976.     

Problem of whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary

The mathematical formulation of the problem on the propagation of whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary is given for the principal term of the short-wave asymptotics, and the uniqueness of the solution of this problem is demonstrated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 197–206, 1976.     

On the case of the explicit integrability of the equation for SH-waves

The equation for SH-waves is considered for the following parameters μ and ρ: μ=a(x)·b(y), ρ=a(x)b(y)(c(x)+d(y)) (a,b,c,d are known functions). For such μ and ρ the variables in this equation can be separated. An explicit solution of the problem of the interaction of a whispering gallery wave with a vertical interface of two media is obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 12–16. Translated by N. S. Zabavnikova.     

Ray dynamics in a neighborhood of flat points of the boundary

The behavior of rays is studied in a neighborhood of boundary points where the curvature has a zero of multiplicity one (an inflection point) and multiplicity two (a flat point of a boundary which is concave from the side of the wave field). The rays considered away from the flat point of the boundary are connected with a whispering gallery wave incident on this point, and they are constructed on the basis of the known asymptotics of this wave. The results are represented in figures obtained with the help of the computer.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 146–155, 1981.     

The directional diagram of whispering gallery modes for a concave reflector

We compute the short-wave asymptotic behavior of the directional diagram of the whispering gallery modes in a field scattered by a nonplanar screen. Numerical results obtained by solving a two-dimensional Dirichlet problem in the exterior of a parabolic screen are reported for various modes. The total contribution of the whispering gallery modes to the directional diagram is also given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 61–67, 1985.     

On whispering gallery wave propagation in a medium with a vertical separating boundary

The solution of the problem of transmission of a whispering gallery wave through the separating boundary is constructed in the parabolic equation approximation. Formulas for the transformation coefficients of the normal modes are constructed in the case of the Dirichlet and Neumann problems. Computational results are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 147–151, 1989.