Finite Constant Depth Research Articles

Integrals with a large parameter: water waves on finite depth due to an impulse

The classical Cauchy–Poisson problem, of water waves generated by an impulsive disturbance on the free surface, is treated in three dimensions for finite constant depth. The conventional solution is in the form of a Fourier-Bessel transform. We wish to find its asymptotic behaviour at large distances r and large times t . Difficulties arise at the wavefront, where r/t is equal to the maximum group velocity. In the analogous two-dimensional problem the waves near the front are associated with two nearly coincident points of stationary phase and described asymptotically by Airy functions. In the three-dimensional problem the solution is expressed as a double integral, where there are four nearly coincident points of stationary phase. A systematic procedure is given for successive terms in an asymptotic expansion, involving the square of an Airy function and its derivatives. In both two and three dimensions it is important to use an appropriate transform of the variable(s) of integration, to achieve uniform validity of the asymptotic expansion. Calculations are performed to illustrate the utility of the asymptotic results.

Transient development of axisymmetric surface waves

The initial value problem of axisymmetric wave motion generated by a cylindrical wave-maker immersed in an incompressible homogeneous fluid of constant finite depth is solved using the Laplace and Weber transforms. An asymptotic treatment of the problem for sufficiently long time and then for large distances is also carried out in detail to obtain a deeper understanding of the steady and transient motions.

Impingement of surface waves on the edge of compressed ice

The impingement of small-amplitude surface waves on the edge of a solid compressed ice sheet in a basin of finite constant depth is considered. The influence of the cylindrical rigidity and the value of the compressing force on the dependence of the amplitude coefficients of reflection and transmission on the incident wave period is analyzed.

Motion generated by a porous wave-maker in a continuously stratified fluid

The motion generated by a porous wave-maker in a continuously stratified fluid is investigated. Adopting Chwang's method the stream function is obtained for fluids of infinite and finite constant depth. The free surface depression, the hydrodynamic pressure distribution and the resultant fluid thrust on the two sides of the wave-maker are calculated for particular forms of the density stratification and wave-maker motion. Curves are drawn to show the effect of stratification on the free surface depression and force coefficients.

On oblique waves forcing by a porous cylindrical wall

The problem of oblique cylindrical linearized wave motion is considered for a fluid of infinite depth or finite constant depth in the presence of an impermeable cylindrical wall and coaxial porous wall immersed vertically in the fluid. The motion is generated once by the oscillations, which are periodic in time and in θ‐direction, of the impermeable wall and next by the porous wall. The velocity potentials have been found in closed forms in the different regions of the fluid and then calculating the hydrodynamic pressure distribution on the porous wall and the profile of the free surface. The scattering problem of oblique waves is then considered. A wave trapping phenomenon is investigated. Numerical results are given to the case of radial incident waves and the case when the angle of incident waves is 30° to the radial direction.

Computing Periodic Gravity Waves on Water by Using Moving Composite Overlapping Grids

The composite overlapping grid method is applied to compute periodic gravity waves on water of finite constant depth. One component grid is made to follow the free surface while the remaining components are independent of the location of the surface. A pseudo-arclength continuation method is used to compute the solution as function of the phase velocity of the wave. The type of equation associated with some grid points and the number of equations in the discretized problem will change when the surface moves. The author expounds a stable way of switching the composite grid during the continuation procedure that works close to limit points. An adaptive technique is also developed to efficiently resolve the solution where sharp gradients develop. Numerical examples are presented that show very good agreement with existing results.

Note on the reflexion of water waves at a nearly vertical cliff in the presence of surface tension

In this note the two-dimensional problem of incoming surface waves against an approximately vertical wall or cliff in water of infinite depth is examined, using velocity potential formulation and linearized boundary-value problem theory for time-harmonic motion. The cliff has arbitrary profile but for simplicity is taken to be vertical at the free surface. The approximate first-order solution is determined subject to a dynamical edge condition apposite to the presence of surface tension, and contains partially reflected outgoing waves. The solution is obtained by perturbation theory in a form involving known unperturbed and first-order correction potentials that is applicable also to water of finite constant depth. The motivation for the note is to point out a correction in principle to the results of a recent investigation for a specific profile, in which reflexion is ignored and another error made in obtaining a first-order solution by a method that is restricted to infinite depth.

Trapped modes of internal waves in a channel spanned by a submerged cylinder

A horizontal channel of infinite length and depth and of constant width contains inviscid, incompressible, two-layer fluid under gravity. The upper layer has constant finite depth and is occupied by a fluid of constant density ρ*. The lower layer has infinite depth and is occupied by a fluid of constant density ρ > ρ*. The parameter ε = (ρ/ρ*)–1 is assumed to be small. The lower fluid is bounded internally by an immersed horizontal cylinder which extends right across the channel and has its generators normal to the sidewalls. The free, time-harmonic oscillations of fluid, which have finite kinetic and potential energy (such oscillations are called trapped modes), are investigated. Trapped modes in homogeneous fluid above submerged cylinders and other obstacles are well known. In the present paper it is shown that there are two sets of frequencies of trapped modes for the two-layer fluid. The frequencies of the first finite set are close to the frequencies of trapped modes in the homogeneous fluid (when ρ* = ρ). They correspond to the trapped modes of waves on the free surface of the upper fluid. The frequencies of the second finite set are proportional to ε, and hence, are small. These latter frequencies correspond to the trapped modes of internal waves on the interface between two fluids. To obtain these results the perturbation method for a quadratic operator family was applied. The quadratic operator family with bounded, symmetric, linear, integral operators in the space L2(−∞, +∞) arises as a result of two reductions of the original problem. The first reduction allows to consider the potential in the lower fluid only. The second reduction is the same as used by Ursell (1987).

Computing the Flow around a Submerged Body Using Composite Grids

The subject of this paper is an accurate numerical method for solving the linear two-dimensional steady potential flow around a body which moves in a liquid of finite constant depth at constant speed and distance below the free surface. The differential equation is discretized by a second-order accurate finite difference scheme on a composite grid. The composite grid consists of two overlapping component grids; one curvilinear grid close to the body and one Cartesian grid which covers the surrounding liquid. To solve the problem numerically, the infinite domain is truncated to finite length. The inflow and outflow boundary conditions are formed by making an eigenfunction expansion of the solution ahead of and behind the body. Each eigenfunction is required to be bounded and satisfy the upstream condition at infinity. This is imposed by functional relations between the solution and its normal derivative at the inflow and outflow boundaries. The method is carefully validated and the computed solutions are found to be in very good agreement with existing results.

Havelock wavemakers, Westergaard dams and the Rayleigh hypothesis

Water of constant finite depth fills a semi-infinite channel, with a wavemaker, W, at one end. The generation of small-amplitude gravity waves by harmonic oscillations of W leads to a linear boundary-value problem for a velocity potential, ϕ. For vertical, plane wavemakers, there is a theory due to Havelock in which ϕ is represented as a convergent series of eigenfunctions, with coefficients determined by the boundary condition on W. We show that the same representation (with different coefficients) can also be used for some wavemakers with other shapes; the allowable geometries and forcings are determined. This is a hydrodynamic analogue of the so-called Rayleigh hypothesis in the theory of gratings. Similar results obtain for the hydrodynamic loading of dams due to short-duration earthquakes.

A Numerical Method to Calculate the Two-Dimensional Flow Around an Underwater Obstacle

The linear two-dimensional steady potential flow around a body travelling in a liquid of finite constant depth at constant speed and distance below a free surface is considered. The analytical solutions ahead of and behind the body are used to form functional relations between the potential and its normal derivative. These relations are imposed as boundary conditions for the problem close to the body. To satisfy the radiation condition, one more relation is imposed ahead of the body than behind it. A second-order accurate finite difference discretization of the problem is also analyzed. An analogous theory is developed which results in matrix relations as boundary conditions for the problem close to the body.

Turbulent mass transport and attenuation in Stokes waves

The motion of turbulent Stokes waves on a finite constant depth fluid with a rough bed is considered. First and second order turbulent boundary layer equations are solved numerically for a range of roughness parameters, and from the solutions are calculated the mass transport velocity profiles and attenuation coefficients. A new mechanism of turbulent mass transport is found which predicts a reduction and reversal of drift velocity in shallow water in agreement with experimental observations under turbulent conditions. This transpires because the second order Stokes wave motion, in a turbulent boundary layer, can directly influence the mass transport velocity by mode coupling interactions between different second order Fourier modes of oscillation. It is also found that the Euler contribution due to the radiation stress of the first order motion is reduced to half of it's corresponding laminar value as a consequence of the velocity squared stress law. The attenuation is found to be of inverse algebraic type with the reciprocal wave height varying linearly with either distance or time. The severe wave height restriction applicable to the Longuet-Higgins [4] solution is shown not to apply to progressive waves on a finite constant depth of fluid. The existence of sand bars on sloping beaches exposed to turbulent waves is predicted.

Transient development of surface waves in the presence of porous wave makers

The initial-value problem of surface waves generated by a harmonically oscillating porous wave maker immersed in an incompressible fluid of infinite horizontal extent but finite constant depth is prescribed. By the use of generalized functions and asymptotic analyses, it was possible to get both the steady-state solution and the transient solution without the need of imposing a radiation condition. Two limiting cases are also investigated: in the first the fluid is of infinite depth, while in the other the fluid is shallow.

Wave‐source expansions for water of infinite depth in the presence of surface tension

In this paper two expansions are obtained by contour integration methods for the velocity potential describing two-dimensional time-harmonic surface waves due to a free-surface wave source on water of infinite depth in the presence of surface tension. First the series expansion at r = 0 is found and then the asymptotic expansion as Kr®¥, where K is the wave number for progressive waves and r the radial distance from the source. The corresponding expansions for the more important submerged wave source in terms of the radial distance from the image source in the free surface may then easily be deduced. The latter are required in a number of surface wave problems, particularly those of a short-wave asymptotic nature, and are also relevant in obtaining expansions for finite constant depth.

A note on the plane vertical wavemaker in the presence of surface tension

This paper presents another method for finding the known wave motion due to a vertical wavemaker in the presence of surface tension, either for finite constant or infinite depth, by using the Fourier integral transform technique and suitably exploiting the regularity condition of the transform.

Forced surface waves in the presence of porous walls

A linearised surface wave motion is considered for a fluid of infinite extent and of infinite or finite constant depth in the presence of an impermeable plate and a porous wall immersed in the fluid parallel to each other. The motion is generated once by the plate and next by the wall. Analytical solutions are obtained for the velocity potentials of the motions, the hydrodynamic pressure distribution on the porous wall and the profile of the surface and the effects of porosity on these quantities are discussed. A scattering problem is investigated in conclusion.

A three-dimensional time-dependent model on coastal upwelling

Hu (1979) proposed a two-dimensional, unsteady, homogeneous coastal upwelling model with finite constant depth. Lu and Zhao (1985) extended Hu’s model by specifying a linear bottom slope. A three-dimensional, unsteady, homogeneous, coastal upwelling model on f-plane with arbitrarily linear bottom slope is suggested in this paper as an extension of Lu and Zhao’s model. The solution for such a problem has been obtained, and the properties of upwelling and horizontal current are examined.

Asymptotic analysis of surface waves due to oscillatory wave maker

The initial value problem of surface waves generated by a harmonically oscillating vertical wave-maker immersed in an infinite incompressible fluid of finite constant depth is presented. The resulting motion is investigated using the generalized function method, and an asymptotic analysis for large times and distances is given for the free surface elevation.

Evolution of Wave Groups in Shallow Water and Wave Group Properties of Random Waves

This paper firstly examines the evolution of wave groups of finite length having a single carrier frequency in finite constant water depth, by both hydraulic experiments and numerical simulations. ...

Ring source potentials in a fluid with an inertial surface in the presence of surface tension

The velocity potential due to the presence of a horizontal circular ring of wave sources of time-dependent strength in a fluid of finite constant depth with an inertial surface composed of uniformly distributed floating particles is obtained when the effect of surface tension at the inertial surface is included.