refraction which are based upon the linear theory fail in one or 
more important aspects in the breaker zone. Therefore the methods 
developed in this paper cannot be applied to the breaker zone. 
Between deep water and the coast there will first be found a 
zone which will be referred to in this paper as the transition zone. 
Between the transition zone and the coast there is a possibility 
of a solitary wave zone, a shallow water wave zone, and a breaker 
zone. If boundaries between the transition zone and the above three 
zones can be defined, then in this paper the theory will apply to 
the transition zone as marked by deep water on one side and the 
boundary of that zone (of the above three zones) which is farthest 
from the coast. Non-linear effects of great importance must be pre= 
sent in these near-shore zones, and they will not be treated in this 
paper. 
It might also be noted that conditions can occur in which the 
solitary wave zone, the shallow water zone, and the breaker zone 
would not occur. Also any two of the above zones or any one of 
the above zones might be missing. For example, waves approaching 
a vertical cliff rising sheerly out of a depth of forty feet at the 
edge of a bottom of variable depth could be reflected back out to 
deep water without ever undergoing any of the above suggested modi- 
fications. 
The invariance of discrete spectral periods 
Consider the following experiment in a very long deep wave tank. 
Waves with a period of exactly two seconds are generated in a forty 
foot depth at one end of the tank. The water for all practical 
purposes is infinitely deep, and the waves can be expressed as a 
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