Water waves over sloping beaches and the solution of a mixed boundary value problem for δ²ø−k²ø = 0 in a sector

Abstract

The linear theory of motion of surface waves in wa.ter over sloping beaches leads to the problem of finding a velocity potential @(p, 8, z, t) = (cos t or sin t ) x ( p , 8, z) such that x ( p , 8, z) satisfies 2 X P P + PXP + Xee + P 2 X a = 0Xe -PX = 0 in the wedge -y < B < 0, and the mixed boundary conditions for 0 = 0; Xe = 0 for 0 = -y. In these equations the variables are dimensionless; t is the time variable; p and e are polar coordinates in the zy-plane whose y-axis is perpendicular to the undisturbed surface of the water; and the z-axis is taken to coincide with the straight shore line. When x is known the vertical displacement n ( p , z, t) of the free surface is given by the formula n = A@., = A(-sin t or cos t) x ( p , 0, z) where A is a constant. The solutions of the potential problem which are of basic interest are those which represent progressing wavw whose curves of constant phase at great distances from the shore tend to parallel lines which make an arbitrary angle with the shore line. For the purpose of obtaining such solutions, let x(p, 6, z) = (cos lcz or sin h) $(p, 6).Then +(p, 0) has to satisfy P%PP + &PP + #ee -P W J = 0 in the sector -7 < 8 < 0; with the boundary conditions -p4]e10 = 0, = 0.

Sloshing frequencies of longitudinal modes for a liquid contained in a trough

“…where R = eiS (3.5) is the coefficient of reflections for such a beach and 8, which is a function of a and p, may be calculated from the work of Peters (1952). Thus for symmetric modes e-2iaa --R,…”

The generalized dock problem

“…In this case we have ξ = y − η(x 1 , x 2 , t) = 0 for any particle, and (1.5) leads to 9) while the Bernoulli's law gives the condition 10) where g is the gravitational constant, and P (x 1 , x 2 , y, t) prescribed over the region of disturbance. at t = 0.…”

The uniqueness question for waves against an overhanging cliff

“…in the sector -< 6 < 0, satisfying I2 -<p = 0 for 0=0, ( [10] have found solutions for this problem for k < 1 and all angles ira < x. Roseau [12] has also studied solutions for angles xa < ir/2 when k > 1.…”

“…Introduction. For the problem of three-dimensional waves over sloping beaches Roseau [11] and Peters [10] have found explicit solutions for all slope angles. These solutions are in the form of integrals in the complex plane resembling those obtained in the inversion of the Laplace transform.…”