Symmetric and anti-symmetric vibrations in micropolar thermoelastic materials plate with voids

The problem of symmetric and anti-symmetric vibrations in micropolar thermoelastic plate with voids has been investigated. The dispersive frequency equations are obtained for different surface waves propagating in the plate. The velocity curves are depicted for the symmetric and anti-symmetric vibrations, plate, Rayleigh and flexural waves. It is found that there exist two modes in the solution of frequency equation for the surface waves in micropolar thermoelastic plate with voids. We have observed that the first modes of velocity ratios of corresponding surface waves are lesser than those of second mode of vibration.     

An exact solution of a linearized problem of the radiation of monochromatic internal waves in a viscous fluid

The eigenvalue method is used to construct an exact solution of the linearized boundary-value problem of the generation of internal waves in an exponentially stratified fluid, when the source is part of a plan which vibrates along its surface. The spatial structure of the solution obtained describes two well-known types of wave beams-unimodal and bimodal. In the limiting cases the phase pattern of the waves is identical with well-known asymptotic forms and laboratory experiments. The exact solution is compared with the solution of the model problem of the generation of waves by force sources, constructed using homogeneous fluid theory. The phase patterns of the waves in both cases agree everywhere with the exception of critical angles, when the wave propagates along the radiating surface. The amplitudes of the radiated waves are the same only for certain ratios of the angles of inclination of the plane and the direction of propagation of the beams.     

Anomalous waves as an object of statistical topography: Problem statement

Based on ideas of statistical topography, we analyze the boundary-value problem of the appearance of anomalous large waves (rogue waves) on the sea surface. The boundary condition for the sea surface is regarded as a closed stochastic quasilinear equation in the kinematic approximation. We obtain the stochastic Liouville equation, which underlies the derivation of an equation describing the joint probability density of fields of sea surface displacement and its gradient. We formulate the statistical problem with the stochastic topographic inhomogeneities of the sea bottom taken into account. It describes diffusion in the phase space, and its solution must answer the question whether information about the existence of anomalous large waves is contained in the quasilinear equation under consideration.     

The unknown object problem in a sea with sloping bottom

This article considers the acoustic unknown object problem for a shallow ocean with a sloping seabed. The incident waves are sent from point sources along a s raight line parallel to the sea surface, and the corresponding scattered fields are measured from a line above the unknown object. We prove a uniqueness theorem for the inverse problem,and describe a generalizeddual space indicator method for numerical solution.Numerical results are given in Section 4.     

Singular boundary integral equations of boundary value problems of the elasticity theory under supersonic transport loads

We consider a transport boundary value problem for an isotropic elastic medium bounded by a cylindrical surface of arbitrary cross-section and subjected to supersonic transport loads. We pose the corresponding hyperbolic boundary value problem and prove the uniqueness of the solution with regard to shock waves. To solve the problem, we use the method of generalized functions. In the space of generalized functions, we obtain the solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem.     

On stable composite capillary-gravitational waves of finite amplitude on the surface of a fluid of finite depth : PMM vol. 39, n 6, 1975, pp. 1023–1031

The problem of stable plane capillary-gravitational waves of finite amplitude on the surface of a perfect incompressible fluid stream of finite depth is considered. It is assumed that the waves are induced by pressure periodically distributed along the free surface, and that these, unlike induced waves, do not vanish when the pressure becomes constant, are transformed into free waves. Such waves are called composite; they exist similarly to free waves, for particular values of velocity of the stream.The problem, which is rigorously stated, reduces to solving a system of four nonlinear equations for two functions and two constants. One of the equations is integral and the remaining are transcendental. Pressure on the surface is defined by an infinite trigonometric series whose coefficients are proportional to integral powers of some dimensionless small parameter; these powers are by two units greater than the numbers of coefficients.The theorem of existence and uniqueness of solution is established, and the method of its proof is indicated. The derivation of solution in any approximation is presented in the form of series in powers of the indicated small parameter. Computation of the first three approximations is carried out to the end, and an approximate equation of the wave profile is presented.Composite capillary-gravitational waves in the case of fluid of infinite depth were considered by the author in [1].     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The scattering of plane elastic waves by a one‐dimensional periodic surface

The two‐dimensional scattering problem for time‐harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi‐periodic solutions similar to the condition utilized in the scattering of acoustic waves by one‐dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary‐value problems is established. We then proceed by introducing a quasi‐periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary‐value problem is sought as a superposition of single‐ and double‐layer potentials defined utilizing this quasi‐periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Qualitative theory of nonlinear steady water waves

In this paper, the nonlinear boundary problem describing two-dimensional steady waves on the surface of water with finite depth is discussed. The problem is formulated in the conventional statement (the gravity is taken into account, but the surface tension is neglected). The latter one allows discussing the whole class of bounded waves that includes periodic waves, solitary waves, and other types of waves (for instance, almost-periodic waves, although their existence has not been established yet). This fact suggests that the results obtained fall within the domain of the qualitative theory of differential equations (investigation of the properties of solutions without finding them). In this paper, two approaches to the qualitative theory are discussed. The first approach consists in averaging the solution along the vertical sections of the region, and the second approach is based on the authors’ modification of Byatt-Smith’s integro-differential equation. Thus, the paper contains an overview of the results obtained for the problem of nonlinear stationary waves on water with finite depth. Two approaches to this problem form a basis of the qualitative theory of such waves, because there are no constraints imposed on the waves except for the boundedness of their profiles and steepness restrictions.     

The exact solution of a non-linear boundary-value problem of the theory of waves on the surface of a liquid of finite depth

A non-linear boundary-value problem of the theory of waves on the surface of a heavy ideal incompressible liquid, which arises as a result of the expansion of the required functions in amplitude, taking quadratic terms into account, is investigated. A solution is constructed, on the one hand, suitable for describing long waves, and on the other, matched to the Stokes expansion (i.e., with the expansion in amplitude of the first order of infinitesimals). A function is sought which conformally maps a strip into the plane of the complex potential in the flow region. An exact solution is obtained for this problem, defined by fairly simply formulae. This solution, in the limit of long and short waves, gives linear sinusoidal waves and cnoidal waves respectively.     

Short-wave scattering near the boundry inflection point

We investigate the problem of tangential incidence of short waves onto a surface with an inflection point. Formal solutions of the corresponding equation are constructed near the inflection point in the form of a quasihomogeneous function series. The formal solution is joined with the geometrical optics solution far from the inflection point of the boundary. The problem is restated as a scattering problem for the Schrodinger equation; existence, uniqueness, and smoothness theorems are proved. The formal asymptotic expansions are proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 152–166, 1985.In conclusion, I would like to thank M. M. Popov for suggesting the problem, and also V. M. Babich and M. M. Popov for useful comments.     

Nonlinear waves in two-dimensions generated by variable pressure acting on the free surface of a heavy liquid

The development of nonlinear waves on the free surface of a heavy liquid initially at rest is treated analytically in cases where the external pressure force of limited power is distributed over a large area in the free surface but is otherwise arbitrary. In [1] approximate (up to small terms of higher order) solution of the problem is obtained in the form of functional series. In the present article the convergence theorems for the series are proved. When the pressure varies with time sinusoidally, the sums of the series are found in closed form. By passing to the limit in the solution as time goes to infinity, the form of the nonlinear steady-state wave is found. According to the solution, when the steady-state wave gets away from the variable pressure zone, a long chain of structures develops similar to so called Kelvin-Helmholtz billows. The existence of nonlinear standing waves is discovered, which have a finite number of nodes in the free surface infinite in extent, and the frequency spectrum and the form of these waves are found explicitly.     

The long‐wave limit for the water wave problem I. The case of zero surface tension

The Korteweg‐de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two‐dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long‐wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg‐de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.     

The singing wineglass: an exercise in mathematical modelling

Lecturers in mathematical modelling courses are always on the lookout for new examples to illustrate the modelling process. A physical phenomenon, documented as early as the nineteenth century, was recalled: when a wineglass ‘sings’, waves are visible on the surface of the wine. These surface waves are used as an exercise in mathematical modelling. Based on assumptions about the wine in the glass and observations illustrated with photographs, a mathematical problem is set up. This problem includes a non-homogeneous Neumann boundary condition on the lateral side of the glass. The solution to the mathematical problem is animated using Mathematica?. The predictions of the model are tested by comparing them with the known facts. The predictions of the model agree with the actual observations.     

Formation of singularities for viscosity solutions of hamilton—jacobi equations in one space variable

In this work we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in one space variable, under no assumption on the convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which is the correct class of weak solutions. We obtain the exact global structure of the shock waves by studying the way the characteristics cross. We construct the viscosity solution by either selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics or constructing explicitly the proper rarefaction waves.     

Exact geophysical waves in stratified fluids

When studying water waves travelling over an inviscid fluid at the Earth's surface there are additional Coriolis and centrifugal forces which influence the motion of the fluid particles. In particular, for waves propagating near the Equator the geophysical wave problem can be modelled by the so-called f-plane approximation. In this paper, we provide an explicit exact solution to the edge wave problem for stratified geophysical flows in the equatorial f-plane approximation.     

Reflection of plane waves from a rigid wall and a free surface in a transverse isotropic medium

For the general solution of two-dimensional equations of dynamics of a transverse isotropic medium with the Carrier–Gassmann condition, we give a representation in terms of two resolvent functions satisfying two separate wave equations. The problem of reflection of plane waves from a rigid wall and a free surface is solved. The coefficients of reflection and transformation of the plane waves are found. These formulas yield a solution for isotropic media too. Some special cases are consideredwhere the shapes (amplitudes) of the reflectedwaves are not uniquely determined, but linearly related with the shape of the incident wave.     

Diffraction of sound waves by a rigid cylindrical cavity of finite length with an internal impedance surface

A rigorous solution is presented for the problem of diffraction of plane harmonic sound waves by a cavity formed by a terminated rigid cylindrical waveguide of finite length whose interior surface is lined by an acoustically absorbent material. The solution is obtained by a modification of the Wiener-Hopf technique and involve an infinite series of unknowns, which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.     

Scattering of Love waves due to the presence of a rigid barrier of finite depth in the crustal layer of the earth

The problem of scattering of Love waves due to the presence of a rigid barrier of finite depth in the crusfal layer of the earth is studied in the present paper. The barrier is in the slightly dissipative surface layer and the surface of the layer is a free surface. The Wiener-Hopf technique is the method of solution. Evaluation of the integrals along appropriate contours in the complex plane yields the reflected, transmitted and the scattered waves. The scattered waves behave as a decaying cylindrical wave at distant points. Numrical computations for the amplitude of the scattered waves have been made versus the wave number. The amplitude falls off rapidly as the wave number increases very slowly.