A at a steady speed c parallel to its length. It is required to find the dynamic pressure every* 

 where on the bottom and for all time. Since the motion is steady with respect to a coordinate 

 system moving with the pressure distribution, the problem can be treated as being independent 

 of time; thus (x, y, z) can be taken to be a right-handed coordinate system moving in space with 

 speed c in the positive x direction. Alternatively, both the pressure distribution and the co- 

 ordinate system can be taken as fixed in space and with a uniform stream moving with speed 

 c in the negative x direction. In either case the origin of the coordinate system is located 



in the plane of the undisturbed free surface, the s-coordinate is taken to be positive up, and 



L B 



the pressure distribution occupies the region \x\ < — , \y\ < — (Figure 1). 



For convenience the constant atmospheric pressure will be taken to be zero so that everywhere 

 on the surface outside the rectangle p = 0.* 



With the usual assumptions of an inviscid, irrotational fluid there exists a velocity 

 potential $, whose positive gradient is equal to the fluid velocity vector; it can be written 



$ (x, y, z) = cp (x, y, z) - ex 



where <p(x, y, 2) represents the perturbation potential. It will be further assumed that the 

 disturbance on the surface is sufficiently small so that the problem can be linearized. 



The potential <f> (x, y, z) must satisfy the following conditions (see Reference 1): 



1. V 2 <p - everywhere in the fluid (Laplace's equation). 



2 

 c c 



2. 4> + — <A = p on z = (linearized free-surface condition). 



9 %X 99 X 



3. cf> = on z = - h (rigid-bottom condition). 



4. 4> \ = (x~ 1/2 ) as x -» + «= (radiation condition). 



where c is the forward speed of the pressure distribution (alternatively, c = speed of the 

 free stream); 



1L B 



P for \x\ < — and \y\ < — 

 ; 

 otherwise 



p is the fluid density; 



g is the gravitational acceleration; and the subscripts designate partial differentiation. 



*Note that there is no loss of generality provided the base pressure p Q is defined relative to atmospheric 

 pressure. 



