The transmission of a flexural-gravitational wave through a rigid end-stop in a floating plate

An elastic plate is located on the surface of a liquid, in continuous contact with it and rigidly clamped in a support along a certain straight line. The orthogonal incidence of a small amplitude flexural-gravitational wave on the support is considered. Exact expressions are obtained for the wave field in the fluid and the flexural field in the plate. The transmission coefficient of the incident flexural-gravitational wave through the support and its reflection coefficient from it are determined. The forces which arise in the support are found. The investigation is carried out for liquids of finite and infinite depth. The effect of the depth of the liquid on the wave processes is indicated. The liquid is assumed to be inviscid and its friction on the bottom and the lower surface of the plate in the neighbourhood of the support is therefore ignored.     

Stabilization of a transmission wave/plate equation

We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchhoff plate equation. We prove an exponential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate.     

The transmission of a plane acoustic wave through a non-uniform thermoelastic layer

The reflection and refraction of a plane acoustic wave by a thermoelastic plane layer, non-uniform in thickness, bounded by non-viscous heat-conducting liquids, generally different, is considered. The system of equations for small perturbations of the thermoelastic medium is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by two methods: the spline-collocation method and the power-series method. Analytic expressions are obtained which describe the wave fields outside the layer. The results of calculations of the intensity transmission coefficient of the acoustic wave are presented.     

Trapped modes along a periodic array of freely floating obstacles

We consider the coupled problem describing the motion of a linear array of three‐dimensional obstacles floating freely in a homogeneous fluid layer of finite depth. The interaction of time‐harmonic waves with the floating objects is analyzed under the usual assumptions of linear water‐wave theory. Quasi‐periodic boundary conditions and a simplified reduction scheme turn the system into a linear spectral problem for a self‐adjoint operator in Hilbert space. Based upon the operator formulation, we derive a sufficient condition for the nonemptiness of its discrete spectrum. Various examples of obstacles that generate trapped modes are given. Copyright © 2014 John Wiley & Sons, Ltd.     

Lamb wave resonance transmission through a system of deep notches in a plate-like waveguide

The results of theoretical and experimental investigation into resonance phenomena accompanying guided wave propagation in a metallic plate with a finite regular system of rectangular deep notches are presented. While the expected intensification of wave blocking by a system of notches comparing to the case of a single defect is confirmed, the resonance transmission at the predicted frequencies within these forming stop bands is also observed. Noticeable motion localization in the waveguide area between the notches at these frequencies is visualized with experimental surface B-scans. Their patterns coincide with the theoretical predictions obtained on the basis of a semi-analytical computer model for the notched infinite plate. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The passage of a bending wave through an inhomogeneity in an absolutely rigid rib backing an elastic plate

The wave properties of a system consisting of an elastic plate and an absolutely rigid infinite rib with a defect on a segment are examined. An elastic inclusion and a gap are two kinds of defects under study. The Green's function method is applied to the diffraction problem and transforms it to singular integro-differential equations on an interval. For the case of short defects, the nonresonance and resonance asymptotics of the scattering pattern are obtained. These results show that the coefficient of penetration for a gap is much larger than that for an elastic inclusion if the frequency is nonresonant. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210. 1994, pp. 22–29. Translated by I. V. Andronov.     

The effect of periodic surface pressure on a rectangular elastic plate floating on shallow water

The steady-state behaviour of a floating elastic plate of bounded dimensions acted upon by a localized external load is investigated using linear shallow water theory. In the case of a plate of arbitrary shape, the problem reduces to solving a system of boundary-value integral equations supplemented with differential relations for the free edge of the plate. Using a rectangular plate as the example, the effect of the frequency of the periodic actions and the positions at which they are applied on the amplitudes of the normal flexures of the plate and of the directional pattern of surface waves far from the plate is investigated. It is shown that waveguide properties occur for an elongated plate.     

Evolution of flexure-gravitational stimulated vibrations of a floating nonuniformly compressed elastic plate

We consider the evolution of vibrations of a thin nonuniformly compressed elastic plate that floats on a surface of a homogeneous ideal incompressible fluid of finite depth, by using the method of integral transformations. Waves are caused by pressures of the type of traveling waves on the surface of the plate, which is harmonic in one of the directions. An analysis of conditions of perturbations of waves that form oscillations of the plate and the wave motion of the fluid is given. A study is given of the dependence of the structure and the number of waves from stressing forces and characteristics of perturbed pressures.Translated from Dinamicheskie Sistemy, No. 7, pp. 58–62, 1988.     

The action of an unsteady external load on a circular elastic plate floating on shallow water

A solution of the transient problem of the behaviour of a circular elastic plate floating on the free surface of a liquid under the action of an external load is constructed. It is assumed that the liquid is ideal and incompressible and that its depth is small compared with the radius of the plate. The compatible motion of the plate and the liquid is treated within the framework of linear theory. The flow of the liquid is assumed to be irrotational. The behaviour of the plate under different loads is investigated and it is shown that the bounded dimensions of the elastic plate have a substantial effect on its unsteady behaviour.     

Frequency-based investigation of a floating wave energy converter system with multiple flaps

In the present paper, a new Floating Wave Energy Converter System (FloWECS), consisting of a floating platform and multiple rotating flaps, is proposed and is preliminary investigated. The numerical analysis is implemented in the frequency domain under the action of regular head and oblique incident waves and it is based on a “dry” mode superposition approach. In this approach, the generalized modes concept is utilized for describing the relative to the platform rotation of the flaps, additionally to the six rigid-body modes. The required mode shapes of the generalized modes are determined through appropriate vector shape functions, which are derived in the present paper. The diffraction/radiation problem is solved using a numerical model based on the conventional boundary integral equation method. The proposed numerical formulation is, initially, validated by comparing results with the numerical ones of other investigators for the case of a single, bottom-hinged, rotating plate. Next, the exciting forces and the response of the “isolated” platform are assessed, while, finally, the hydrodynamic behaviour and the energy absorption of the FloWECS are investigated. The results illustrate, that for both examined incident wave directions, the generalized modes’ response and, thus, the system's power absorption is characterized by the existence of distinctive peaks at two different wave frequency ranges, attributed to resonance effects, as well as to the strong interaction effects between the activated flaps.     

Passage of a wave through a thin layer

The problem is considered of the passage of an acoustic wave given by its ray expansion through a thin layer. The method of boundary-layer asymptotic expansions is applied to obtain ray representations of the reflected and transmitted waves (the small parameters are the thickness of the layer and the wave length).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklov AN SSSR, Vol. 173, pp. 20–29, 1988.     

Diffractive wave transmission in dispersive media

The aim of this paper is to study the reflection-transmission of diffractive geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell-Lorentz equations with the anharmonic model of polarization.The framework is that of P. Donnat's thesis [P. Donnat, Quelques contributions mathématiques en optique non linéaire, chapters 1 and 2, thèse, 1996] and V. Lescarret [V. Lescarret, Wave transmission in dispersive media, M3AS 17 (4) (2007) 485-535]: we consider an infinite WKB expansion of the wave over long times/distances O(1/ε) and because of the boundary, we decompose each profile into a hyperbolic (purely oscillating) part and elliptic (evanescent) part as in M. William [M. William, Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. 33 (2000) 132-209].Then to get the usual sublinear growth on the hyperbolic part of the profiles, for every corrector, we consider E, the space of bounded functions decomposing into a sum of pure transports and a “quasi compactly” supported part. We make a detailed analysis on the nonlinear interactions on E which leads us to make a restriction on the set of resonant phases.We finally give a convergence result which justifies the use of “quasi compactly” supported profiles.     

The dynamics of a bar in the presence of obstacles

The dynamic evolution of visco-elastic and purely elastic bars hitting rigid and elastic obstacles are studied, from either the theoretical, numerical or computational point of view.     

Nonlinear wave propagation through a stratified atmosphere

We examine the propagation of sound waves through a stratified atmosphere. The method of multiple scales is employed to obtain an asymptotic equation which describes the evolution of sound waves in an atmosphere with spatially dependant density and entropy fields. The evolution equation is an inviscid Burger-like equation which contains quadratic and cubic nonlinearities, and a curvature term all of which are functions of the space variables. A model equation is derived when the modulations of the signal in a direction transverse to the direction of propagation become significant.     

The scattering of a plane flexural wave by a sector of a thin elastic plate with a supported edge

A solution is constructed of the problem of the diffraction of a plane flexural wave at the vertex of a thin elastic plate cut out in the form of a sector, on the edge of which the condition of a supported edge (hinged support) is specified.     

Development of three-dimensional vibrations of an elastic uniformly stretched floating plate

The influence of uniform extension on the unsteady vibrations of an elastic plate floating on the surface of a constant-depth homogeneous fluid surface is studied by the method of integral transformations. The vibrations are caused by a load moving at the constant velocity of the plate surface. The dependence of the size of the angular zones covered by waves, the wave front displacement velocities, and the critical load displacement velocity on the magnitude of the tensile force is analyzed. An estimate is given of the role of uniform compression in the formation of vibrations ahead of and behind the load.Translated from Dinamicheskie Sistemy, No. 4, pp. 72–77, 1985.     

Some comparative studies on testing parallelism of several straight lines under heteroscedastic variances

This paper examines the robustness of the Welch test , the James test as well as Tan's Anova test (to be referred as Fβ test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.