wave phenomena. For given values of H, d and T, Figure 2-7 may be used 

 as a guide in selecting an appropriate theory. The magnitude of the 

 Ursell parameter Ud shown in the figure may be used to establish the 

 boiondaries of regions where a particular wave theory should be used. 

 The Ursell parameter is defined by 



T 2 14 



"« = y^ . (2-45) 



For linear theory to predict accurately the wave characteristics, both 

 wave steepness, H/gT^, and the Ursell parameter must be small as shown in 

 Figure 2-7. 



2.25 STOKES' PROGRESSIVE, SECOND-ORDER WAVE THEORY 



Wave formulas presented in the preceding sections on linear wave 

 theory are based on the assumption that the motions are so small that the 

 free surface can be described to the first order of approximation by 

 Equation 2-10: 



H /277X 27rt\ H 



v = — cos 1 — I = — cos or a cos d . 



2 \L T I 2 



More specifically, it is assumed that wave amplitude is small, and the 

 contribution made to the solution by higher order terms is negligible. A 

 more general expression would be: 



T? = acos(0) + a^Bj (L, d) cos (20) 



(2-46) 

 + a'Bj (L, d) cos(3fl) + • • • a"B„ (L, d) cos(n0) , 



where a = H/2, for first- and second-orders, but a < H/2 for orders 

 higher than the second, and Bg, B3 etc. are specified functions of the 

 wavelength L, and depth d. 



Linear theory considers only the first term on the right side of 

 Equation 2-46. To consider additional terms represents a higher order of 

 approximation of the free surface profile. The order of the approxima- 

 tion is determined by the highest order term of the series considered. 

 Thus, the ordinate of the free surface to the third order is defined by 

 the first three terms in Equation 2-46. 



When the use of a higher order theory is warranted, wave tables, such 

 as those prepared by Skjelbreia (1959), and Skjelbreia and Hendrickson 

 (1962), should be used to reduce the possibility of numerical errors made 

 in using the equations. Although Stokes (1880) first developed equations 

 for finite amplitude waves, the equations presented here are those of 

 Miche (1944). 



2-36 



