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) = <», and the elliptic cosine reduces to the hyperbolic secant 

 function, y^ = d, and equation (2-59) reduces to 



or 



n = H sech^ 



(2-64) 



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

 wave above the Stillwater level per unit crest width is 



V = 



iid3 



3 



1/2 



(2-65) 



An equal amount of water per unit crest length is transported forward past 

 a vertical plane that is perpendicular to the direction of wave advance. Sev- 

 eral relations have been presented to determine the celerity of a solitary 

 wave; these equations differ depending on the degree of approximation. Labo- 

 ratory measurements by Daily and Stephan (1953) indicate that the simple 

 expression 



C = Vg(H + d) 

 gives a reasonably accurate approximation to the celerity. 



(2-66) 



The water particle velocities for a solitary wave, as found by McCowan 

 (1891) and given by Munk (1949), are 



u = CN 



1 + cos(My/d) cosh(Mx/d) 

 [cos(My/d) + cosh(Mx/D)]2 



(2-67) 



w = CN 



sin(My/d) sinh(Mx/d) 

 [cos(My/d) + cosh(Mx/d)]2 



(2-68) 



where M and N are the functions of H/d shown in Figure 2-16, and y is 

 measured from the bottom. The expression for horizontal velocity u is often 

 used to predict wave forces on marine structures sited in shallow water. The 

 maximum velocity u occurs when x and t are both equal to zero; hence. 



CN 



max 1 + cos(My/d) 



(2-69) 



2-57 



