discussed in detail in a later chapter. 
The remaining known solutions to these reduced equations 
have been obtained by Stoker, [1947], and his co-workers for 
the problem of a linearly sloping beach with waves parallel 
to the beach and recently by Peters for waves at an angle to 
the beach (unpublished). For additional information, see the 
paper referred to above. 
No exact solutions for the reduced equations have been 
obtained under the condition that the depth is an arbitrary 
function of x and ye Graphical methods of solution based upon 
the principles of geometrical optics have been given by Sver- 
drup and Munk [1944], and by Johnson, O'Brien and Isaacs [1948]. 
Pierson [195la] has discussed these results and formulated the 
problem which would have to be solved in order to proceed from 
equations (2.14) through (2.17) to a result which would prove 
that the principles of geometrical and physical optics are ap- 
plicable to problems of ocean wave refraction and diffraction. 
Eckart [1951] has obtained an approximate solution to the com- 
pletely general problem, accurate everywhere to within a few 
percent. 
The solutions which have been discussed so far apply to 
depths which range from infinite to one or two tenths of the 
deep water wave length. The solutions do reduce to the shallow 
water theory, if h is picked smaller and smaller, but a wave 
of the finite height progressing from deep to shallow water in 
Stoker's work [1947] becomes infinitely high as it approaches 
*See References. 
Sache 
