Radiation Of Water Waves Research Articles

Wave-power absorption by a line of submerged horizontal cylinders

This paper considers the radiation of water waves by a submerged, horizontal circular cylinder in a channel. The cylinder axis is perpendicular to the channel walls and the length of the cylinder is less than the width of the channel. The method of solution combines both multipole and eigenfunction expansions. The effects of finite cylinder length and of the channel walls are discussed and application is made to a line of Bristol cylinder wave power devices.

On the application of Modified Green function for wave field.

For the problem of diffraction and radiation of water waves from an arbitrary shapcd body, John's Green function (1950) has been used by several authors. However this John's Green function seems to have a ill behavior when the problem solved numerically through discretization.In this paper we point out that this ill behavior is caused from the definition of delta function which is used by John, and propose a new Green function as a Modified Green function, and check the application of this Green function for wave field.

On a bound for the frequency of surface waves trapped near a cylinder spanning a channel

There has been substantial effort recently put into proving, for a variety of different geometries, the existence of trapped waves, that is unforced time-harmonic motions which do not radiate energy to large distances. Thus it is known that such motions can exist in a deep channel which includes a cylinder spanning the channel, for various shapes of cylinder. The converse problem of proving the absence of such trapped waves has received much less consideration, and the only relevant uniqueness proof for a channel spanned by a cylinder is that of McIver (1991). In an appendix to that paper, McIver demonstrates that no trapped-wave motions can exist for the case in which the cylinder is surface piercing and is entirely contained between vertical planes through the free-surface intersections. This is exactly the same geometrical condition which John (1950) found would ensure uniqueness in water-wave radiation and scattering problems, in finite or infinite depth. Both John and McIver achieved their uniqueness results by consideration of integrals of the potential along vertical lines down from the free surface. John's work was extended by Simon and Ursell (1984) who established uniqueness for a wider class of two-dimensional radiation and scattering problems by consideration of integrals along nonvertical lines. The work presented in this paper is the corresponding extension of McIver's work; although this extension does not rule out trapped waves at all frequencies for any geometry except that already considered by McIver, it does yield an easy lower bound for the ratio of the trapped-mode frequency to the cut-off frequency, in finite or infinite depth.

The quest for a three-dimensional theory of ship-wave interactions

The radiation and diffraction of water waves by ships can be analysed in classical terms from potential theory. The linearized formulation is well studied, but robust numerical implementations have been achieved only in cases where the vessel is stationary or oscillating about a fixed mean position. Slender-body approximations have been used to rationalize and extend the strip theory of ship motions, providing analytic solutions and guidance in the development of more general numerical methods. The governing equations are reviewed, with emphasis on the interactions between the steady-state velocity field due to the ship’s forward translation and the perturbations due to its unsteady motions in waves. Recent computations based on the boundary-integral-equation method are described, and encouraging results are noted. There is growing evidence that the influence of the steady-state velocity field is important, and the degree of completeness required to account for the steady field depends on the fullness of the ship. Benchmark computations are needed to test theories and computer programs without the uncertainty inherent in experimental comparisons.

Radiation and diffraction of water waves by a submerged sphere in finite depth

The problems of radiation (both heave and sway) and diffraction of water waves by a submerged sphere in finite depth are formulated using the multiple method. In each case this leads to an infinite system of linear equations which are easily solved numerically. Simple expressions are derived for the hydrodynamic characteristics of such a body. A far-field approximation for the scattering problem is also presented. Results showing the effect of varying the depth on both the hydrodynamic characteristics and the diffraction forces are given and it is shown how the method can be used to predict free-surface elevations in the vicinity of the sphere.

Infinite elements for water wave radiation and scattering

AbstractThe infinite element method is employed to approximate the solutions of Webster's horn equation and Berkhoff's equation for water wave radiation and scattering in an unbounded domain. Functionals based on the first variational principle are presented. Two new infinite elements, which exactly satisfy the one‐ and two‐dimensional Sommerfeld radiation condition, are presented; the simple shape functions are constructed on the basis of the asymptotic behaviour of the scattered wave at infinity. All the integrals in the functionals involving each infinite element are integrated analytically and, as a result, no numerical integration is required. The programming requirements and computational efficiency are essentially no different than those of the conventional finite element method. For the test cases presented, the numerical results are acceptably accurate when compared with the existing solutions and laboratory data.

A variational method for water wave radiation and diffraction problems

It is shown that the linear and nonlinear exciting force coefficients, the added mass and radiation damping matrices, and the far-field wave amplitude (reflection and transmission coefficients in the two-dimensional case) can all be written as stationary values of well-defined functionals. As a consequence these quantities can be accurately determined with relatively crude approximations for the diffraction and radiation potentials. Numerical experiments confirm this feature: by inverting a 4 × 4 real symmetric matrix the results obtained by Vugts (1968), who computed the added mass and radiation damping matrices for several different geometries, were recovered over the whole range of frequencies.

Radiation and diffraction of water waves by a submerged sphere at forward speed

A submerged sphere advancing in regular deep-water waves at constant forward speed is analysed by linearized potential theory. A distribution of sources over the surface of the sphere is expanded into a series of Legendre functions, by extension of the method used by Farell ( J . Ship Res . 17, 1 (1973)) in analysing the wave resistance on a submerged spheroid. The equations governing the velocity potential are satisfied by use of the appropriate Green function and by choosing the coefficients in the series of Legendre functions such that the body surface condition is satisfied. Numerical results are obtained for the wave resistance, hydrodynamic coefficients and exciting forces on the sphere. Some theoretical aspects of a body advancing in waves are also discussed. The far-field equation of Newman ( J . Ship Res . 5, 44 (1961)) for calculation of the damping coefficients is extended, and a similar equation for the exciting forces is derived.

A hybrid integral equation method for diffraction and radiation of water waves by three-dimensional bodies

A hybrid integral equation method is formulated for the analysis of water wave diffraction and radiation by arbitrarily-shaped three-dimensional bodies. It is based upon the direct application of Green's second identity and uses the simple fundamental solution (i. e. a simple source) rather than the special Green's function. The boundary element idealisation is used only in an inner fluid region close to the body and local depth irregularities, while an analytical solution is employed in the outer region of constant depth extending to infinity. The two representations are matched on a fictitious vertical cylindrical surface. Special treatment is discussed to reduce computational efforts when the body has one or two planes of symmetry. Numerical results are presented for a variety of geometries to illustrate the applicability and potentialities of the method.

On the solvability of a two-dimensional water-wave radiation problem

The existence of a unique weak solution for the two-dimensional water-wave radiation problem arising when a floating rigid body oscillates on the free surface is established for all but a discrete set of oscillation frequencies. The body boundary is taken to be of C ∗ 1 , α C_ * ^{1,\alpha } class (see Sec. 2) and the body boundary condition is satisfied in the L 2 {L^2} -sense. The proof relies on an expansion theorem (Athanassoulis [1]) and on the property of the associated water-wave multipoles being a Riesz basis of L 2 ( − π , 0 ) {L^2}\left ( { - \pi ,0} \right ) , a fact which is established in the present paper. Under stronger geometrical restrictions on the body boundary it is proved, using a method due to Ursell [10], that the weak solution is actually a classical one; that is, the velocity field is continuous up to ana including the body boundary.

Numerical Evaluation of the Green Function of Water-Wave Radiation and Diffraction

This study presents a simple, accurate, and efficient method for numerically evaluating the Green function, and its gradient, of the theory of water-wave radiation and diffraction. The method is based on five expressions for the Green function that are useful in complementary regions of the quadrant in which the Green function is defined. These expressions consist of asymptotic expansions, ascending series, two complementary Taylor series, and a numerical approximation based on a modified form of the Haskind integral representation. The four series representations are refinements of the series obtained previously in Noblesse [1].2 These series express the Green function and its gradient as sums of power series and terms involving functions of only one variable. The power series can be evaluated quickly by using recurrence relations; and the functions of one variable, by using rational approximations. The method permits the Green function and its gradient to be evaluated with an absolute error smaller than 10–6 very efficiently (with computing time less than 6 × 10–5 sec on a CDC CYBER 176 computer). A listing of the FORTRAN subroutine is included in the paper.

The high-frequency radiation of water waves by oscillating bodies

The aim of this work is to extend the known asymptotic forms, in heave and sway, of the wave amplitude radiated by bodies at high frequency. Both two- and three-dimensional geometries will be considered, the prototype problems being the circular cylinder and sphere respectively, each with its centre in the mean free surface. The method is, in principle, applicable to other geometries by way of the solution of certain potential problems much simpler than the finite-frequency surface-wave problem.

Some applications of Maz'ja's uniqueness theorem to a class of linear water wave problems

AbstractThis paper describes a criterion which, if satisfied, guarantees uniqueness of solution for the problem involving the radiation or diffraction of small amplitude water waves by a totally submerged body. This criterion was originally stated by a Russian mathematician, V. G. Maz'ja, although his work does not appear to be widely known. For this reason a brief outline of Maz'ja's proof is given here, together with a discussion of the condition to be satisfied by the submerged body.Maz'ja's criterion is not satisfied by all submerged bodies and so his result does not provide a general proof of uniqueness. However, the criterion is satisfied in some important cases and examples are presented here. The partial nature of this result is due entirely to the method of proof and not to any intrinsic difficulty with the underlying hydrodynamics.

Integral identities of potential theory of radiation and diffraction of regular water waves by a body

The study presents a new general integral identity for the velocity potential of flow about a body in regular water waves. This integral identity is valid outside, inside, and exactly on the surface of the body, and is equivalent to the set of three classical identities valid strictly outside, inside, and on the body, respectively. For the usual problem of wave radiation and diffraction by a body, the integral identity yields an integral equation for determining the potential on the body surface. An interesting feature of the integral identity and related integral equation obtained in this study is that they involve an integral of the Green function over the waterplane inside the body in the case of an open sea-surface piercing body. Alternatively, these equations can be expressed in terms of a modified Green function involving the previously-noted waterplane integral of the Green function.

The Green function in the theory of radiation and diffraction of regular water waves by a body

This study is concerned with the Green function of the theory of potential flow about a body in regular (time-harmonic) water waves in deep water, that is with the linearized velocity potential of the flow due to a source of pulsating strength at a fixed position below the free surface (or a pulsating flux across the free surface) of a quiescent infinitely deep sea. An asymptotic expansion and a convergent ascending-series expansion for the Green function are obtained from two alternative complementary ‘near-field’ and ‘far-field’ single-integral representations in terms of the exponential integral. The asymptotic expansion and the ascending series allow efficient numerical evaluation of the Green function for large and small distances, respectively, from the mirror image of the singularity (submerged source or free-surface flux) with respect to the mean sea surface.

On the null-field equations for water-wave radiation problems

Consider a rigid body which is performing simple harmonic oscillations of small amplitude in the free surface of deep water under gravity. Under certain geometrical conditions on ∂D, the wetted surface of the body, it is known that the linear boundary-value problem [Pscr ] for a corresponding velocity potential ϕ is uniquely solvable at all frequencies. The usual method for solving [Pscr ] is to derive a Fredholm integral equation of the second kind over ∂D. There are two familiar ways of doing this: (i) represent ϕ as a distribution of simple wave sources over ∂D, leading to an integral equation for the unknown source strength; (ii) apply Green's theorem to ϕ and a simple wave source; when the field point lies on ∂D, this gives an integral equation for the boundary values of ϕ. It is well known that both of these integral equations have unique solutions, except at the same infinite discrete set of frequencies (the irregular frequencies).In this paper, we shall describe an alternative method for solving [Pscr ]: when the field point, in (ii), lies inside the body, we obtain an integral relation. If the simple wave source has a suitable bilinear expansion, this integral relation may be reduced to an infinite set of equations for the boundary values of ϕ. These equations, called the ‘null-field equations for water waves’, appear to be new; equations of this type were first obtained by Waterman for electromagnetic and acoustic scattering problems. The required bilinear expansion has been given by Ursell (1981) for two dimensions, and is given here for three dimensions. Using these, we show that the null-field equations always have a unique solution – irregular frequencies do not occur. This result is proved here for water waves in two and three dimensions. Similar results may be obtained for water of constant finite depth.

Radiation and Diffraction of Shallow Water Waves by an Arbitrary Number of Bodies

A shallow water theory is newly formulated retaining higher order terms, and a numerical procedure is developed placing emphasis on the application to the diffraction and radiation of waves by a group of floating bodies. In the course of the numerical analysis an irregular frequency phenomenon is observed ; however it is both analytically and numerically proved that the irregularity can be removed by a minor modification to the analysis.The results have been checked against analytical ones and also against those obtained by three dimensional numerical approaches. The present method appears to be a practical means to solve a multi-body problem, from the viewpoint of computation time as well; some of the results on the diffraction of waves by two and three circular discs are presented.

Some properties of a hybrid element method for water waves

AbstractTwo properties of hybrid element method for diffraction and radiation of water waves are examined. For long waves in shallow water the method is shown to give a unique solution for all frequencies. Thus, unlike several other known methods, there are no irregular frequencies for which the approximating matrix equation is singluar. For a sea of arbitrary depth, it is shown that all known global identities such as reciprocity and energy theorems are preserved by the discrete solution. Thus, satisfaction of these identities by the numerical solution is only a necessary but by no means sufficient condition for accuracy.