Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.

Complementary approximations to wave scattering by vertical barriers

Roundoff error m the solution of hnear algebraic systems is stud,ed using a more reahstsc notion of what st means to perturb a problem, namely, that each datum :s subject to a relatwely small change Th:s ,s particularly appropriate for sparse linear systems The condition number :s determined for th:s approach The effect of scahng on the stabdlty of Gaussmn ellmmat,on is stud:ed, and st is d:scovered that the proper way to scale a system depends on the right-hand s:de However, ff only the norm of the error is of concern, then there ~s a good way to scale that does not depend on the right-hand stde

Scaling for Numerical Stability in Gaussian Elimination

This paper presents a numerical method for solving linearized water-wave problems with oscillatory time dependence Specifically it considers the diffraction problem for oblique plane waves incident upon an infinitely long fixed cylinder on the free surface The numerical method is based on a variational principle ezuivalent to the linearized boundary-value problem Finite-element techniques are used to represent the velocity potential; and the variational principle is used to determine the unknown coefficients in the solution throughout the fluid domain To illustrate this method, reflection and transmission coefficients and the diffraction forces and moment are computed for oblique waves incident upon a vertical flat plate, a horizontal flat plate and rectangular cylinders, where the comparison is made with the existing results by others

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075001802

Diffraction of oblique waves by an infinite cylinder

Two-dimensional waves are incident upon a pair of vertical flat plates intersecting the free surface in a fluid of infinite depth. An asymptotic theory is developed for the resulting wave reflexion and transmission, assuming that the separation between the plates is small. The fluid motion between the plates is a uniform vertical oscillation, matched to the outer wave field by a local flow at the opening beneath the plates. It is shown that the reflexion and transmission coefficients undergo rapid changes, ranging from complete reflexion to complete transmission, in the vicinity of a critical wavenumber where the fluid column between the obstacles is resonant.

Interaction of water waves with two closely spaced vertical obstacles

In a recent paper (Evans 1976) a theory was presented for the behaviour of an oscillating two-dimensional cylinder of any shape which was capable of absorbing energy from a given regular sinusoidal wave. In particular an expression was derived for the efficiency of power absorption of the cylinder when oscillating in a single mode in terms of properties of the solution of the so-called radiation problem in which the cylinder is forced to oscillate in the appropriate mode in the absence of the incident wave train. In the present paper this theory is extended to two independent cylinders of arbitrary shape each oscillating in a single mode and capable of absorbing energy in that mode. A general expression for the efficiency is derived which depends on properties of the solution to a new radiation problem, in which one cylinder is forced to oscillate in the presence of the other cylinder, which is held fixed in its equilibrium position. In this case, the efficiency also depends on cross-coupling coefficients related to the force on the fixed cylinder due to the motion of the oscillating cylinder. It is shown that the cylinders can be tuned to absorb all the incident wave energy at a given frequency even for symmetric cylinders, in contrast to the single symmetric cylinder, for which the maximum efficiency has been shown to be 50%. The general solution to the new radiation problem is derived in terms of the solution to the radiation problem for a single cylinder, by assuming that the cylinders are far enough apart for local wave effects to be negligible. The special case of two widely spaced rolling vertical plates is considered in detail and curves showing the variation of efficiency with wavelength are given for a variety of plate spacings and points of rotation.

C. A. N. Morris

Papers

Complementary Approximations to the Solution of a Problem in Water Waves

The Effect of a Fixed Vertical Barrier on Obliquely Incident Surface Waves in Deep Water