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.     

Unsteady behaviour of a heterogeneous elastic beam floating on shallow water

The unsteady behaviour of a thin elastic Euler beam with heterogeneous structural properties, floating freely on the surface of an ideal incompressible liquid is investigated using the linear theory. The unsteady behaviour of the beam is due to the incidence of a localized wave on its surface or initial deformation. Two methods of solving the problem are proposed in which the sagging of the beam is sought in the form of an expansion in eigenfunctions of the oscillations of a heterogeneous beam (the first method) or of a homogeneous beam (the second method) in the void. In both methods the problem is reduced to solving an infinite system of ordinary differential equations for the unknown amplitudes. The effect of different actions on a beam having a piecewise-constant distribution of the cylindrical stiffness and the specific mass is investigated. The eigenvalues of the systems of differential equations are determined.     

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.     

The diffraction of surface waves by an elastic platform floating on shallow water

The scattering of long gravitational waves by a floating elastic plate is investigated using linear shallow-water theory. For a plate of arbitrary shape, the solution of the problem is reduced to a system of boundary integral equations. Using the example of a rectangular plate, the solution obtained is compared with existing theoretical and experimental results. The behaviour of the buckling of a rectangular plate and of a strip of constant width is compared for oblique incidence of surface waves.     

Central-upwind scheme for shallow water equations with discontinuous bottom topography

Finite-volume central-upwind schemes for shallow water equations were proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), 133–160]. These schemes are capable of maintaining “lake-at-rest” steady states and preserving the positivity of the computed water depth. The well-balanced and positivity preserving features of the central-upwind schemes are achieved, in particular, by using continuous piecewise linear interpolation of the bottom topography function. However, when the bottom function is discontinuous or a model with a moving bottom topography is studied, the continuous piecewise linear approximationmay not be sufficiently accurate and robust.     

Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.     

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.     

Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation

In this paper the von Kármán model for thin, elastic, infinite plate strip resting on a linear elastic foundation of Winkler type is studied. The infinite plate strip is simply-supported and subjected to evenly distributed compressive loads. The critical values of bifurcation parameters and buckling modes for given frequency of longitudinal waves are found on the basis of investigation of linearized problem. The mathematical nonlinear model is reduced to operator equation with Fredholm type operator of index 0 depending on parameters defined in corresponding Hölder spaces. The Lyapunov-Schmidt reduction and the Crandall-Rabinowitz bifurcation theorem (gradient case) are used to examine the postcritical behaviour of the plate. It is proved that there exists maximal frequency of longitudinal waves depending on the compressive load and the stiffness modulus of foundation.     

Scattering of an incident acoustic wave by an elastic sphere in a shallow water

The scattering of acoustic waves by an elastic sphere in a shallow ocean wave guide is investigated taking into account the shear waves which can exist in addition to compressional waves in scatterers of solid material. Expressions for the scattered waves are given. Numerical values for a quantity called the farfield form function for various depth are presented in graphical forms.      

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.     

A robust central scheme for the shallow water flows with an abrupt topography based on modified hydrostatic reconstructions

We study a second-order central scheme for the shallow water flows with a discontinuous bottom topography based on modified hydrostatic reconstructions (HRs). The first HR scheme was proposed in Audusse et al, which may be missing the effect of the large discontinuous bottom topography. We introduce a modified HR method to cope with this numerical difficulty. The new scheme is well-balanced for still water solutions and can guarantee the positivity of the water depth. Finally, several numerical results of classical problems of the shallow water equations confirmed these properties of the new scheme. Especially, the new scheme yields superior results for the shallow water downhill flow over a step.     

On symmetries and conservation laws of the equations of shallow water with an axisymmetric profile of bottom

The local symmetries and conservation laws are calculated for the equations of shallow water with an axisymmetric profile of bottom under the assumption that the corresponding generating functions may depend only on all variables and their derivatives up to the second order. It is shown that if the bottom has the form of a paraboloid of revolution, then there are many symmetries and conservation laws generalizing those for the case of plane bottom.     

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.     

Simulation of shallow water flows with shoaling areas and bottom discontinuities

A numerical method based on a second-order accurate Godunov-type scheme is described for solving the shallow water equations on unstructured triangular-quadrilateral meshes. The bottom surface is represented by a piecewise linear approximation with discontinuities, and a new approximate Riemann solver is used to treat the bottom jump. Flows with a dry sloping bottom are computed using a simplified method that admits negative depths and preserves the liquid mass and the equilibrium state. The accuracy and performance of the approach proposed for shallow water flow simulation are illustrated by computing one- and two-dimensional problems.     

A parallel well-balanced finite volume method for shallow water equations with topography on the cubed-sphere

A finite volume scheme for the global shallow water model on the cubed-sphere mesh is proposed and studied in this paper. The new cell-centered scheme is based on Osher’s Riemann solver together with a high-order spatial reconstruction. On each patch interface of the cubed-sphere only one layer of ghost cells is needed in the scheme and the numerical flux is calculated symmetrically across the interface to ensure the numerical conservation of total mass. The discretization of the topographic term in the equation is properly modified in a well-balanced manner to suppress spurious oscillations when the bottom topography is non-smooth. Numerical results for several test cases including a steady-state nonlinear geostrophic flow and a zonal flow over an isolated mountain are provided to show the flexibility of the scheme. Some parallel implementation details as well as some performance results on a parallel supercomputer with more than one thousand processor cores are also provided.     

Balance-characteristic scheme as applied to the shallow water equations over a rough bottom

The CABARET scheme is used for the numerical solution of the one-dimensional shallow water equations over a rough bottom. The scheme involves conservative and flux variables, whose values at a new time level are calculated by applying the characteristic properties of the shallow water equations. The scheme is verified using a series of test and model problems.     

An operator model for the oscillation problem of liquids on an elastic bottom

This paper deals with the problem of small oscillations in a liquid layer of finite depth under the assumption that the bottom is an elastic medium. The system of equations corresponding to the problem is written out and explained. The main aim of the paper is to recast these equations in the form

The torsion of a viscoelastic plate in an unsteady flow

The stability of the equilibrium position of a viscoelastic plate, subjected to torsional strain and effect of the free airflow is investigated. The unsteadiness of the flow is taken into account by introducing integral terms into the moments of the aerodynamic forces acting on the plate. In a neighbourhood of the equilibrium position, a general solution of a Volterra-type integro-differential equation with partial derivatives is constructed in the form of a Fourier series, as a function of the longitudinal coordinate of the plate with coefficients that are the power series in the small parameters introduced. The stability of the plate equilibrium in the unstrained state is analysed in the case when there are small perturbations (possibly, discontinuous) of the flow velocity. The stability under persistent perturbations of the equilibrium of the strained plate with respect to non-linear perturbing forces and perturbations of its shape at the instant of time preceding the specified initial instant is also investigated.     

Dynamic snap-through of an elastic imperfect simple shallow truss

Zusammenfassung Das dynamische Durchschlagen eines masselosen symmetrischen elastischen Dreigelenkbogens, der eine Punktmasse am mittleren Gelenk trägt, wird untersucht. Eine längs der Symmetrieachse wirkende Last wird plötzlich aufgebracht und weiterhin konstant gehalten. Es zeigt sich, dass die kritische Last durch geometrische Unvollkommenheiten stark vermindert wird.

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Research sponsored by the Air Force Office of Scientific Research Office of Aerospace Research, United States Air Force under AFOSR Grant AF-AFOSR 1226-67.     

Diffraction of a plane acoustic wave on an elastic sphere located near the bottom

On the basis of numerical computations we analyze the echo-signals from a steel spherical shell located near the bottom of water under the action of a quasimonochromatic pulse. We study the effect of the distance between the sphere and the bottom on the echo-pulse. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, pp. 79–84.