Numerical study of the stability of breakwater built on a sloped porous seabed under tsunami loading

Strong earthquake induced huge tsunami has occurred for three times in Pacific ocean in recent ten years; for example, the tsunami triggered by the Sumatra earthquake in 2004, Chile earthquake in 2010 and Tohoku earthquake (Japan) in 2011. Tsunami carrying huge energy always would bring high risks to the population living near to coastline. Breakwater is widely used to dissipate the wave energy, and protect coastline and ports. However, they are vulnerable when being attacked by tsunami wave. At present, the interaction mechanism between tsunami, breakwater and its seabed foundation is not fully understood. In this study, the dynamics and stability of a breakwater under the attacking of tsunami wave is investigated by adopting an integrated model PORO-WSSI 2D, in which the VARANS equation for wave motion, and the Biot’s dynamic equation for soil are used. Based on the numerical results, it is found that offshore breakwater interacts intensively with tsunami wave when it overtopping and overflowing over a breakwater. The impact force on the lateral side of breakwater applied by tsunami wave is huge. The shear failure is likely to occur in the seabed foundation of breakwater. The liquefaction is unlikely to occur due to the fact that there is basically no upward seepage force in seabed foundation in the process of tsunami wave passing through the breakwater.     

A ray‐integral solution for the wave‐field in the Sommerfeld model

A ray‐integral solution is presented for the wave‐field in the Sommerfeld model (liquid half‐space over solid half‐space), where a point source is placed in the fluid and the two media (fluid and solid) of contrasting densities and wave speeds are homogeneous. This exact closed form solution is then used to evaluate complete time records of the acoustic pressure (at a point receiver located in the fluid in the vicinity of the penetrable fluid‐solid interface) for two Sommerfeld models: one where the shear wave speed in the solid bottom is lower than the sound speed in the fluid and the other where the shear wave speed is higher. These pressure response curves indicate the relative importance of the various wave‐forms (the critically refracted longitudinal and shear waves, the pseudo‐Rayleigh and Stoneley interface waves, the direct wave, and the totally reflected wave) contributing to the solution and the possibility of utilizing the arrival times of the refracted and interface waves to determine the bottom rigidity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generating Boundary Conditions for the Calculation of Tsunami Propagation on Nested Grids

Some boundary conditions used to numerically simulate tsunami generation and propagation are studied. Special attention is given to generating boundary conditions thatmake it possible to simulate tsunami waves with desired characteristics (amplitude, time period and, in general, waveform). Since the water flow velocity in a propagating tsunami wave is uniquely defined by its height and ocean depth, one can simulate a wave propagating from the boundary into the simulation area. This can be done by specifying the wave height and water flow velocity on the boundary. This method is used to numerically simulate the propagation of a tsunami from the source to the coast on a sequence of refined grids. In this numerical experiment the wave parameters are transferred from the larger area to the subarea via boundary conditions. This method can also generate a wave that has certain characteristics on a specified line.     

Interfacial long traveling waves in a two‐layer fluid with variable depth

Long wave propagation in a two‐layer fluid with variable depth is studied for specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow‐water theory and determined by a family of solutions of the second‐order ordinary differential equation (ODE) with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the point where the two‐layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow‐water theory is used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation point corresponds to an infinite increase in water depth, which contradicts the shallow‐water assumption. This point is eliminated by matching the “nonreflecting” bottom profile with a flat bottom. The wave transformation at the matching point is described by the second‐order Fredholm equation and its approximated solution is then obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by Prof. Roger Grimshaw, to the new solutions types.     

Homogenization Method in the Problem of Long Wave Propagation from a Localized Source in a Basin over an Uneven Bottom

In the framework of the linearized shallow water equations, the homogenization method for wave type equations with rapidly oscillating coefficients that generally cannot be represented as periodic functions of the fast variables is applied to the Cauchy problem for the wave equation describing the evolution of the free surface elevation for long waves propagating in a basin over an uneven bottom. Under certain conditions on the function describing the basin depth, we prove that the solution of the homogenized equation asymptotically approximates the solution of the original equation. Model homogenized wave equations are constructed for several examples of one-dimensional sections of the real ocean bottom profile, and their numerical and asymptotic solutions are compared with numerical solutions of the original equations.     

Properties and exact solutions of the equations of motion of shallow water in a spinning paraboloid

A transformation is found and, using this, the non-linear system of equations describing the spatial oscillations of a thin layer of liquid in a spinning circular parabolic basin is reduced to the conventional equations of the model of shallow water over a level fixed bottom. This transformation is obtained by analyzing the properties of the symmetry of the equations of motion of spinning shallow water. The existence of non-trivial symmetries in the case of the model considered enabled group multiplication of the solutions to be carried out. Using the known steady-state rotationally symmetric solution, a class of time-periodic solutions is obtained that describes the non-linear oscillations of the liquid in a circular paraboloid with closed or quasiclosed (ergodic) trajectories of the motion of the liquid particles.     

The Interaction of a Strong Bore with Small Disturbances in Shallow Water

This paper gives an extension of previous work [2] on weakly nonlinear shallow water waves over a variable bottom to include the effects of strong bores and small surface disturbances. We first consider the interaction of a strong bore with quiescent water over an isolated bottom disturbance to highlight some of the modifications that are introduced in our results for both noncritical and transcritical Froude numbers. We also exhibit the secular effect on the bore trajectory of a bottom disturbance that has a nonzero average. In a second example, we consider the interaction of a strong bore with a small amplitude periodic surface disturbance upstream. We show that downstream of the bore, the wave length of this disturbance increases, whereas its amplitude increases (decreases) depending on whether the bore speed is larger (smaller) than a critical value. We also use this example to illustrate the derivation of the solution and bore trajectory to second order accuracy. All our asymptotic results, obtained in the form of multiple scale expansions, are compared with numerical solutions for a number of illustrative cases.     

Reflection of harmonic elastic waves from a convex cylindrical cavity of arbitrary section

We describe an approach to the solution of problems of reflection of harmonic elastic waves from a convex indrical cavity of arbitrary section, on the basis of Debye ray, series. We obtain a system of linear algebraic equations to determine the constants of integration and expressions for determining the stresses in the elastic medium surrounding the convex cylindrical cavity. We give the results of computation of the stresses that arise as a result of the action of a planar harmonic rarefaction wave on a cylindrical cavity in the shape of elliptic and parabolic cylinders and on a cylindrical cavity whose section is a Munger oval. Four figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 10–16, 1991.     

Bremmer series that correct parabolic approximations

A Bremmer type series solution of the three dimensional reduced wave equation is obtained. The series is obtained by iterating generalizations of the Bellman-Kalaba integral equations. The lowest order term is the solution of the parabolic approximation to the reduced wave equation. The series thus provides systematic corrections to the parabolic approximation. New derivations of the parabolic approximation are also provided. These are based on the idea of splitting a solution to the reduced wave equation into “upward” and “downward” components.     

Two-dimensional travelling waves over a moving bottom

Two-dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied. The bottom and the wave travel with a same speed. The exact Euler equations are formulated as a spatial dynamic system by using the stream function. A manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions. A homoclinic solution to the normal form of this reduced system persists when higher-order terms are added, which gives a generalized solitary wave—the homoclinic solution connecting a periodic solution.     

A Long Wave Approximation for Capillary-Gravity Waves and an Effect of the Bottom

The Korteweg–de Vries (KdV) equation is known as a model of long waves in an infinitely long canal over a flat bottom and approximates the 2-dimensional water wave problem, which is a free boundary problem for the incompressible Euler equation with the irrotational condition. In this article, we consider the validity of this approximation in the case of the presence of the surface tension. Moreover, we consider the case where the bottom is not flat and study an effect of the bottom to the long wave approximation. We derive a system of coupled KdV like equations and prove that the dynamics of the full problem can be described approximately by the solution of the coupled equations for a long time interval. We also prove that if the initial data and the bottom decay at infinity in a suitable sense, then the KdV equation takes the place of the coupled equations.     

Determination of a wave field inside a hollow cone with a notch along the generator

The aim of this investigation is to determine the wave field inside a part of a conic domain filled with an acoustic medium subjected to the action of a nonstationary pressure. The method of solution is based on the discretization of the problem with respect to time by replacing the second derivative by a difference scheme and using new integral transformations with respect to other variables. A recurrent solution of the problem is obtained, and the calculation of a wave field for different geometric parameters of the domain is performed.     

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.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

On transforms reducing one-dimensional systems of shallow-water to the wave equation with sound speed c 2 = x

We obtain point transformations for three one-dimensional systems: shallow-water equations on a flat and a sloping bottom and the system of linear equations obtained by formal linearization of shallow-water equations on a sloping bottom. The passage of these systems to the Carrier-Greenspan parametrization is also obtained. For linear shallow-water equations on a sloping bottom, we obtain the solution in the form of a traveling wave with variable velocity. We establish the relationship between the resulting solution and the solution of the two-dimensional wave equation.     

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh–Nagumo, Aliev–Panfilov and MacCulloch models.     

Martingale solutions of a stochastic wave equation with reflection

We discuss an initial boundary value problem for a one-dimensional stochastic wave equation with reflection. For stochastic parabolic equations with reflection, there are some well-known results. However, there seems to be no existence result for a stochastic wave equation with reflection. Even for a deterministic wave equation, the problem has not been completely resolved. Our goal is to establish the existence of a martingale solution for this problem.     

Fock asymptotics in the problem of diffraction by a moving smooth contour

In the paper the asymptotics are obtained of the solution of the diffraction problem in the Fock region for the incidence of a nonstationary modulated wave on a smooth moving contour. The formulas obtained generalize the solution of V. A. Fock to the stationary problem in a neighborhood of a point of tangency of an ray to the case of a nonstationary wave and a moving scatterer. A new characteristic of the problem is introduced the effective radius of curvature of the contour, depending on the geometric and kinematic properties of the scatterer. The boundaries are established of the applicability of the asymptotic formulas.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 195, pp. 69–81, 1991.In conclusion I want to thank the participants of the seminars of LGU and LOMI on the diffraction and propagation of waves for useful discussions.     

Long-time behaviour of solutions of a class of nonlinear parabolic equations

In this paper, the long-time behaviour of solutions of a class of nonlinear parabolic equations is studied. It is shown that the solutions of initial-boundary value problem to the equations converge to a travelling wave solution of the equation or a self-similar solution of a Hamilton–Jacobi equation under certain conditions on initial and boundary values of the solutions.