Surface waves due to blasts on and above inviscid liquids of finite depth

For the problem of waves due to an explosion above the surface of a homogeneous ocean of finite depth, asymptotic expressions of the velocity potential and the surface displacement are determined for large times and distances from the pressure area produced by the incident shock. It is shown that the first item in Sakurai's approximation scheme for the pressure field inside the, blast wave as well as the results of Taylor's point blast theory can be used to yield realistic expressions of surface displacement. Some interesting features of the wave motion in general are described. Finally some numerical calculations for the surface elevation were performed and included as a particular case.     

Rayleigh wave scattering due to a rigid barrier in a liquid halfspace

The paper presents a theoretical formulation for studying scattering of Rayleigh waves due to the presence of rigid barriers in oceanic waters. The Wiener-Hopf technique has been employed to solve the problem. Exact solution has been obtained in terms of Fourier integrals whose evaluation gives the reflected, transmitted and scattered waves. The scattered waves have the behaviour of cylindrical waves originating at the edge of the barrier. Numerical results for the amplitude of the scattered waves have been obtained for small depth of the barrier.     

Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity

In this paper, mathematical modeling of the propagation of Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity has been considered. The equations of motion have been formulated separately for different media under suitable boundary conditions at the interface of porous layer, elastic half-space under gravity and rigid layer. Following Biot, the frequency equation has been derived which contain Whittaker’s function and its derivative that have been expanded asymptotically up to second term (for approximate result) for large argument due to small values of Biot’s gravity parameter (varying from 0 to 1). The effect of porosity and gravity of the layers in the propagation of Love waves has been studied. The effect of hydrostatic initial stress generated due to gravity in the half-space has also been shown in the phase velocity of Love waves. The phase velocity of Love waves for first two modes has been presented graphically. Frequency equations have also been derived for some particular cases, which are in perfect agreement with standard results. Subsequently the lower and upper bounds of Love wave speed have also been discussed.     

Diffraction of Kelvin waves by a finite barrier

Summary The diffraction of Kelvin waves by a finite barrier and depth discontinuities is considered using Wiener-Hopf technique. Diffracted wave and generation of double Kelvin wave are studied and comparison has been made with the case of semi-infinite barrier and depth discontinuity.

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Zusammenfassung Die Beugung Kelvinscher Wellen an einer endlichen Schranke und mit Tiefen-Unstetigkeiten wird mittels der Wiener-Hopf-Methode betrachtet. Es werden die gebrochene Welle und die Erzeugung doppelter Kelvin-Wellen untersucht, und die Resultate werden mit dem Fall einer halbunendlichen Schranke verglichen.

     

Diffraction of water waves due to a presence of two headlands

This paper is concerned with a boundary value problem for the Helmholtz equation on a horizontal infinite strip with obstacles. The derivation of Helmholtz equation from shallow water equations is given and the boundary value problem with an arbitrary shap of headland is stated. The boundary conditions are of the general Neumann type, and thus we use the finite difference method in numerical solution. Helholtz equation is replaced by the five-points formula and for the points close to the boundary, Taylors expansions are made useful with non-uniform spacing. For solving the resulting system of linear equations, the “Mathematica” package is used. The graphs show the velocity potential contours in the cases, of semielliptic, semicircular and narrow headland. Also, we discuss the problem in the presence of two headlands.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

Diffraction of sound waves by a finite barrier in a moving fluid

The diffraction of a line source by an absorbing finite barrier, satisfying Myers' impedance condition [M.K. Myers, On the acoustic boundary condition in the presence of flow, J. Sound Vibration 71 (1980) 429-434] in the presence of a subsonic flow is studied. The problem is solved analytically by using Integral transforms, Wiener-Hopf technique and the asymptotic methods. The expression for the diffracted field is shown to be the sum of the fields produced by the two edges of the strip and a field due to the interaction of the two edges. The diffracted field in the far zone is determined by the method of steepest decent.     

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.     

Dense sets and the projection theorem for acoustic harmonic waves in a homogeneous finite depth ocean

The scattering of a ‘plane wave’ off a submerged body situated in an ocean of finite depth is investigated. The index of refraction is considered to be depth-independent. It is shown that the far field is not unique; hence, the problem of determining the shape of an object from its far field is not well-posed. If solutions are sought among a restricted class of problems the ‘dense set’ property implies that the problem can be made well-posed.     

The propagating solutions and far-field patterns for acoustic harmonic waves in a finite depth ocean

The existence of extremal solutions of the Nonlinear Integro-differential system of the initia-Value problem with singular coefficients are proved by monotone iterative technique. This is achieved by developing a comparison result. This does not require the usual classical assumption of the increasing nature of the integral term. This is turn includes the earlier known scalar results asa special case.     

Rectilinear distance to a facility in the presence of a square barrier

This paper derives analytical expressions for the rectilinear distance to a facility in the presence of a square barrier. The distribution of the barrier distance is derived for two regular patterns of facilities: square and diamond lattices. This distribution, which provides all the information about the barrier distance, will be useful for facility location problems with barriers and reliability analysis of facility location. The distribution of the barrier distance demonstrates how the location and the size of the barrier affect the barrier distance. A?numerical example shows that the total barrier distance increases as the barrier gets closer to a facility, whereas the maximum barrier distance increases as the barrier becomes greater in size.     

Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier

The problem of scattering of two dimensional surface water waves by a partially immersed rigid plane vertical barrier in deep water is re-examined. The associated mixed boundary value problem is shown to give rise to an integral equation of the first kind. Two direct approximate methods of solution are developed and utilized to determine approximate solutions of the integral equation involved. The all important physical quantity, called the Reflection Coefficient, is evaluated numerically, by the use of the approximate solution of the integral equation. The numerical results, obtained in the present work, are found to be in an excellent agreement with the known results, obtained earlier by Ursell (1947), by the use of the closed form analytical solution of the integral equation, giving rise to rather complicated expressions involving Bessel functions.     

Invasion waves in the presence of a mutualist

This paper studies invasion waves in the diffusive competitor–competitor–mutualist model generalizing the two‐species Lotka–Volterra model studied by Weinberger et al. The mutualist may benefit the invading or the resident species producing two different types of invasions. Sufficient conditions for linear determinacy are derived in both cases, and when they hold, explicit formulas for linear spreading speeds of the invasions are obtained by linearizing the model. While in the first case the linear speed is increased by the mutualist, it is unaffected in the second case. Mathematical methods are based on converting the model into a cooperative reaction–diffusion system. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.Received: December 12, 2001     

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.