The difference between this number and the 18.38 ft/sec calculated 

 under (b) above is the result of small errors in reading the 

 curves . 



2.27 SOLITARY WAVE THEORY 



Waves considered in the previous sections were oscillatory or nearly 

 oscillatory waves. The water particles move backward and forward with the 

 passage of each wave, and a distinct wave crest and wave trough are evident, 

 A solitary wave is neither oscillatory nor does it exhibit a trough. In 

 the pure sense, the solitary wave form lies entirely above the Stillwater 

 level. The solitary wave is a wave of translation relative to the water 

 mass. 



Russell (1838, 1845) first recognized the existence of a solitary 

 wave. The original theoretical developments were made by Boussinesq 

 (1872), Lord Rayleigh (1876), and McCowan (1891), and more recently by 

 Keulegan and Patterson (1940), Keulegan (1948), and Iwasa (1955). 



In nature it is difficult to form a truly solitary wave, because at 

 the trailing edge of the wave there are usually small dispersive waves. 

 However, long waves such as tsunamis and waves resulting from large dis- 

 placements of water caused by such phenomena as landslides, and earthquakes 

 sometimes behave approximately like solitary waves. When an oscillatory 

 wave moves into shallow water, it may often be approximated by a solitary 

 wave, (Munk, 1949). As an oscillatory wave moves into shoaling water, the 

 wave amplitude becomes progressively higher; the crests become shorter and 

 more pointed, and the trough becomes longer and flatter. 



The solitary wave is a limiting case of the cnoidal wave. When k^ = 

 1, K(k) = K(l) = oo, and the elliptic cosine reduces to the hyperbolic 

 secant function, y- = d, and Equation 2-59 reduces to 



d + H sech^ 



H 



(x - Ct) 



or 



1? = H sech^ 



3 H_ 



4 d^ 



(x - Ct) 



(2-64) 



where the origin of x is at the wave crest. The volume of water within 

 the wave above the still water level per unit crest width is 



V = 



16 



% 



d' H 



(2-65) 



2-59 



