the shore and thus the linearized theory breaks down. 
Another class of solutions can be obtained under the 
assumption that the water is shallow. The shallow water theory 
of the solitary wave, for example, as obtained by Airy is treat- 
ed in Lamb [1932]. Recent refinements in the theory have been 
obtained by Keller [1949]. Lowell [1949a] has studied the 
propagation of waves in shallow water. Munk [1949] has studied 
the breaking of solitary waves in shallow water. Stoker [1949] 
has applied the non-linear shallow water theory to the formation 
of breakers and bores and to the breaking of waves in shallow 
water. 
The results which will be obtained in this paper will 
hold only up to the shallow water zone. It will be possible 
to generalize the theory of ocean wave refraction to the dis- 
turbances studied herein. The breaker zone, however, will not 
be treated, although an extension of the results obtained by 
Biesel [1951] may make this possible. Biesel's graphs of waves 
just before breaking appear to be the most realistic mathemati- 
cal breakers ever presented. 
Non-periodic solutions 
One final important class of solutions which has been ob- 
tained in classical wave theory remains to be discussed. They 
are the solutions which have been obtained by the use of Fourier's 
Integral Theorem for waves in infinitely deep water. The gen- 
eral procedure is’ to integrate the potential function and the 
representation of the free surface given in equations (2,18) 
and (2.19) over a continuous spectrum of angular wave frequencies 
“a6. 
