choice of a theory for a particular problem. It should be kept in mind 

 that linear (or first-order) theory applies to a wave which is symmetrical 

 about Stillwater level and has water particles that move in closed orbits. 

 On the other hand, Stokes' second-order theory predicts a wave form that 

 is unsymmetrical about the Stillwater level but still symmetrical about a 

 vertical line through the crest, and has water particle orbits which are 

 open. 



************** EXAMPLE PROBLEM 



************** 



GIVEN : A wave traveling in water depth of d = 20 feet, with a wavelength 

 of L = 200 feet and a height of H = 4 feet. 



FIND : 



(a) Compare the wave profiles given by the first- and second-order 

 theories . 



(b) What is the difference between the first- and second- order 

 horizontal velocities at the surface under both the crest and 

 trough? 



(c) How far in the direction of wave propagation will a water particle 

 move from its initial position during one wave period when z = 0? 



(d) What is pressure at the bottom under the wave crest as predicted 

 by both the first- and second-order theories? 



(e) What is the wave energy per unit width of crest predicted by the 

 first-order theory? 



SOLUTION : 



(a) The first-order profile Equation 2-10 is: 



H 



7? = — cos 6 , 

 2 



where 



(2iTX 27rt 

 L T 



and the second-order profile Equation 2-49 is: 



H ttH^ cosh(27rd/L) 



V = — cos 6 + — ; 



2 8L sinh^(27rd/L) 



for 



d 20 



- = = 0.1 , 



L 200 



2-40 



2 + cosh I 



cos 26 



