36 



d 

 he found that at a slope y = 0.4, the period increases 25 percent over 



the period in deep water. This conclusion is in conflict with wave 

 tank observations and with one principal assumption running through 

 these notes, namely, that the period of a single wave train is constant 

 over a gradually sloping bottom. The problem should be given 

 careful study experimentally. 

 Rayleigh (13) states that: 



A uniform regime being established, what we are to equate at two separated 

 places where the waves are of different character is the rate of propagation of 

 energy through these places. 



From his analysis he concludes that between very great depths 



d 

 and y = 0.04 the bottom has little effect on the wave height but that: 



* * * the wave length is diminishing, so that the waves, even though they do 

 no more than maintain their height, grow steeper. 



Although precise observations are necessary to establish the fact 

 conclusively, it will be assumed as the most reasonable approach in 

 the light of present knowledge that waves do maintain their identity 

 in running over a gradually sloping bottom. Rayleigh's assumption 

 will be made, namely, that the transmitted power is the same at all 

 points along the path of travel. 



The problem then is to trace a wave from deep water to the plunge 

 point and predict the height, length, and velocity at each point along 

 its path. The known characteristics in deep water are assumed to be 

 the period and height although any other definitive combination is 

 sufficient. From the period one obtains the velocity and length, 

 which combined with the height gives the energy per wave length 

 per unit width, and the transmitted power per unit width per second 

 in deep water is found as: 



a=5.12 T (26) 



La=5.12 T^ (2c) 



P„=^^(i-4.93A:) (34) 



Consider a wave of uniform height with crest straight and parallel 

 to the bottom contours. The problem is then two-dimensional. 

 Furthermore, the effect of the finite height on the wave velocity will 

 be neglected. In each depth, there is only one wave length and 

 velocity consistent with the period. These values are obtained from 

 figures 1 and 2. The power is known from the computed value in 

 deep water, equation 34. 



