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.

A half plane in the surface of a body of water of great depth is covered with a floating thin mat. The mat may be taken to be a first approximation to a field of broken ice, or some other field of floating material which consists of small particles which do not interact,. A progressing wave in the free surface approaches the edge of the mat. The wave proceeds parallel to the edge and has a sinusoidal form at a great distance from the mat. How is such a wave affected by the mat? This question was asked by Prof. J. J. Stoker. It initiated the analysis presented in the following sections.In section 1 it is shown that, under certain linearizing assumptions, the answer to the question can be found by solving a potential problem for a half plane. The potential problem involves mixed boundary conditions. In section 2 the problem is generalized by formulating it for a sector of arbitrary angle 0 < y 5 T.Along the positive x-axis the potential function 9(x, y) must satisfy 9, -9 = 0, and along the ray 8 = -y the condition is 9" + I . @ = 0 where 4,, is the outward normal derivative. The generalized problem includes, among others, the problem of surface waves over sloping beaches. and the "dock" problem which are the subjects of recent papers. Sections 3 to 8 are concerned with the solution of the generalized problem, and the behavior of the waves in the neighborhood of the origin and at infinity.In section 8 the development returns to the problem of the floating mat. It is shown in this section that the effect of the mat depends upon the sign of c = 6g/(6102 -Sg) where 6 is the density of water, g acceleration due to gravity, 61 the density of the floating material, and w the time frequency. If c > 0 the disturbance in the free surface is not propagated to any great distance inside the mat. The disturbance subsides not exponentially but like a power of l/d where d is the distance inside the mat. If c = 0 we have the case of waves approaching a dock. If c < 0 waves pass into the mat with an altered wave length and amplitude which are functions of I c I.In section 9 the case c = m is analyzed. It is shown that this condition also prevents the transmission of waves into the mat.

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lntrductionThis paper is concerned with the determination of the surface waves created by a concentrated pressure which moves in a straight course over the surface of a body of water whose depth is great. These waves are of interest because, as a first approximation, the moving pressure point can be considered as representing a moving ship.The general characteristics of these waves are well known. The disturbance of the surface is practically confined to a sector in the wake of the moving pressure point, 0. In this sector, whose half angle is 19"28', the displacement of the surface diminishes rapidly as the distance from 0 increases. The approximate equation of the free surface within the sector is known and it represents a wave pattern which agrees closely with experimental observations, hamely a wave system transverse to the course of 0 superposed on a wave system which diverges from 0. The lines of constant phase of these systems meet in cusps which lie on the boundary lines of the sector, and in the vicinity of these lines the displacement of the surface is most pronounced. Outside of the sector, and not near 0. the surface remains pract,ically undisturbed.The equation of the free surface is very difficult to derive if (with the assumption of irrotational motion) the exact hydrodynamical theory is used. It depends upon the determination of a potential function which satisfies certain nonlinear boundary conditions at the free surface whose shape is not given, but which must be determined as part of the solution of the problem along with such quantities as the velocity and pressure. The results mentioned above, as well as those obtained in this paper, are derived from approximate solutions of the linearized problem. The problem is linearized under the assumptions that the amplitude of the surface waves is small, and that the difference between the velocity of the water and the velocity of the pressure point is small.The linearized problem was formulated and solved long ago by Kelvin [I]. He regarded the displacement of a point in the free surface as due to a series of pressure impulses applied along the course of the pressure point and behind it. The summation of these impulses led to a representation of the surface in the form of an infinite integral which Kelvin evaluated by the use of his principle of stationary phase, which yields approximate results valid at some distance from the presure point. This procedure is open to the objections that it makes the amplitude of the surface waves infinite along the course of the pressure point and, although the procedure yields the equation along the boundary lines of the 123