Scattering of acoustic waves on a sloping beach

A classical problem of acoustics is considered by the Sommerfeld-Maliuzhinets method. All geometric waves are computed explicitly. For diffracted waves, a numerical procedure based on a uniquely solvable Fredholm integral equation is elaborated. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 57–64. Translated by B. V. Budaev.     

On computing uniformly valid approximations for viscous waves on a plane beach

Existing expansions for the near- and far-field flow of linear waves in a viscous liquid on a plane beach are supplemented by an approximation in the intermediate region which enables the flow to be matched from the shore line to infinity. Methods of computation are discussed and incorporated are recent improvements on the performance of numerical computation of oscillatory integrals. Sidi's user-friendly W-transform (1988) and Köhler's parameter optimisation (this journal, 1993) for generalised Newton—Cotes quadrature are found to be significant elements particularly with regard to the inversion of the Kontorovich—Lebedev transform. Comprehensive computations are undertaken and diagrams shown to display the behaviour of the stream function for all depths. An attempt is also made to construct a composite expansion of the vorticity and this is computed on a ray bisecting the wedge.     

Hydrodynamic processes on beach: Wave breaking,up-rush,and backwash

This paper presents two-dimensional numerical predictions of wave breaking, up-rush, and backwash in inner surf and swash zones and analyzes the hydrodynamic processes involved. In the numerical simulations, the Reynolds Averaged Navier–Stokes (RANS) equations, a non-linear k–ε turbulence closure, and a piecewise linear interface construction volume of fluid (PLIC-VOF) method are employed. On the basis of a series of model calibration using experimental data, plunging and spilling breakers are simulated at different wave parameters and slope angles. The numerical results indicate that there are non-linear interactions between hydrodynamic characteristics in surf zones such as wave breaking heights and those in swash zones such as up-rush heights, and the breaker type plays an important role in hydrodynamic processes in the two zones.     

The similarity solution for a bore on a beach

A similarity solution is found for the asymptotic behavior of a bore as it approaches the shoreline on a sloping beach. This gives direct confirmation of earlier results on the motion of the bore and adds details of the associated flow field. It also makes explicit the analogy with Guderley's implosion problem in gas dynamics; the solution is constructed closely following Guderley's arguments.     

Two-phase flow modelling of spilling and plunging breaking waves

A two-phase flow model, which solves the flow in the air and water simultaneously, has been employed to investigate both spilling and plunging breakers in the surf zone with a focus during wave breaking. The model is based on the Reynolds-averaged Navier–Stokes equations with the k–?$k''–''?$ turbulence model. The governing equations are solved using the finite volume method, with the partial cell treatment being implemented in a staggered Cartesian grid to deal with complex geometries. The PISO algorithm is utilised for the pressure–velocity coupling and the air–water interface is modelled by the interface capturing method via a high-resolution volume of fluid scheme. Numerical results are compared with experimental measurements and other numerical studies in terms of water surface elevations, mean flow and turbulence intensity, in which satisfactory agreement is obtained. In addition, water surface profiles, velocity and vorticity fields during wave breaking are also presented and discussed. It is shown that the present model is capable of simulating the wave overturning, air entrainment and splash-up processes.     

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ?Ω of the domain as the square root of the distance to ?Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.     

On the breaking of water waves in a channel of arbitrarily varying depth and width

Summary By generalising an approach due to Gurtin, the Bernoulli equation is derived which governs the amplitude of an acceleration wave propagating on the surface of water at rest in a vertical walled channel of arbitrarily varying width and depth. Integration of the equation yields an explicit expression for the amplitude of the acceleration wave as a function of position, together with a simple criterion for determining when a wave will break. A special case of a non-breaking elevation wave propagating into deepening water is also found.

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Résumé En généralisant une méthode trouvée par Gurtin, on dérive l'équation de Bernoulli qui gouverne l'amplitude d'une onde d'accélération qui se propage sur la surface d'une quantité d'eau immobile qui se trouve dans un canal aux berges verticales, dont la profondeur et la largeur varient d'une façon arbitraire. L'intégration de cette équation fournit une formule explicite pour déterminer l'amplitude de l'onde d'accélération en fonction de sa position, aussi bien qu'un critère simple avec lequel on peut déterminer le moment où l'onde va se déferler. On découvre en même temps le cas spécial d'une onde d'élévation qui ne se déferle pas, et qui se propage dans l'eau qui devient de plus en plus profonde.

     

On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (_t0,_x0)∈R⁺×S¹$(t_0,x_0)\in {\mathbb{R}}^{+}\times {\mathbb{S}}^1$ is the identically zero solution. Such conclusion holds provided the physical parameter γ   of the model (related to the Finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa–Holm equation, corresponding to γ=1$\gamma =1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.     

Nonlinear breaking of waves in an electrically conducting and radiating gas

The effects of radiative transfer are treated by the use of a differential approximation which is valid over the entire optical depth range from the transparent limit to the optically thick limit. The singular surface theory is used to determine the modes of wave propagation and to evaluate the behaviour at the wave head. It is shown that there are two modes of wave propagation namely (i) the radiation induced waves which are always damped, and (ii) the modified magnetogasdynamic waves which break at the wave front if the initial discontinuity is sufficiently strong. The effects of thermal radiation, the magnetic field intensity, the finite electrical conductivity and the initial wave front curvature on the non-linear breaking of modified magnetogasdynamic waves are discussed.

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Zusammenfassung Der Einfluß der Strahlung wird mit Hilfe von Differential-Näherungen behandelt, die im ganzen Bereich von optischen Tiefen Gültigkeit haben, von der Transparenz bis zur optisch dicken Grenze. Die Theorie singulärer Flächen wird benützt um die Wellenausbreitung und das Verhalten an der Wellenfront zu behandeln. Es wird gezeigt, daß es zwei Formen von Wellenausbreitung gibt, nämlich (i) Wellen erzeugt durch Strahlung, die immer gedämpft sind, und (ii) die modifizierte magnetogasdynamische Welle, die an der Front immer bricht, wenn die ursprüngliche Diskontinuität stark genug ist. Es werden die Einflüsse der Wärmestrahlung, der magnetischen Feldstärke, der endlichen elektrischen Leitfähigkeit und der ursprünglichen Wellenkrümmung auf die nicht-lineare Frontbildung untersucht.

     

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.     

Soft supersymmetry breaking as a spontaneous breaking in the auxiliary sector

Soft supersymmetry breaking is accomplished as a spontaneous symmetry breaking by extending the auxiliary field content of the model. The renormalization procedure in such models is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp. 325–329, November, 1998.     

The influence of a crack in a medium on the interference waves

The interference Kr-waves in an elastic half-space with a crack are considered. The processes of dispersion and attenuation of such waves are described and theoretical seismograms are constructed. A number of frequency and velocity characteristics are discussed, which can help to identify the waves in mining investigations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 110–115, 1989.     

Modelling and control of the motion of a disk rolling on a sloping plane

This work deals with the stabilization and control of the motion of a disk rolling on a sloping plane. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. By using a kind of an inverse control transformation a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any smooth trajectory which is located on the sloping plane.