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. Several relations have been presented to determine the 

 celerity of a solitary wave; these equations differ depending on the 

 degree of approximation. Laboratory measurements by Daily and Stephan 

 (1953) indicate that the simple expression 



C = Vg (H + d)" , (2-66) 



gives a reasonably accurate approximation to the celerity. 



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

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



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



u = CN —'- (2-67) 



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



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



w = CN (2-68) 



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



where M and N are the functions of H/d shown on 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^nox' occurs when x and t are both equal 

 to zero; hence, 



CN 



"'"'« " 1 + cos(My/d) ■ ~^^' 



Total energy in a solitary wave is about evenly divided between 

 kinetic and potential energy. Total wave energy per unit crest width is, 



E = ^^pgH3/2d3/2 , (2-70) 



and the pressure beneath a solitary wave depends upon the local fluid 

 velocity as does the pressure under a cnoidal wave; however, it may be 

 approximated by 



P = Pg(y,-y). (2-71) 



Equation 2-71 is identical to that used to approximate the pressure 

 beneath a cnoidal wave. 



As a solitary wave moves into shoaling water it eventually becomes 

 unstable and breaks. McCowan (1891) assumed that a solitary wave breaks 



2-60 



