For linear theory to predict accurately the wave characteristics, both the 

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

 Figure 2-7. 



5. 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 sur- 

 face can be described to the first order of approximation by equation (2-10): 



2Trt\ H 

 - = — cos 6 or a cos 9 



T y 2 



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

 bution made to the solution by higher order terms is negligible. A more gen- 

 eral expression would be 



n = a cos(9) + a2B2(L,d) cos(29) 



n 



(2-46) 



+ a^B:,(L,d) cos(39) + ...a B (L,d) cos(n9) 



n 



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

 than the second, and B^ , B^, etc. are specified functions of the wave- 

 length 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 approxima- 

 tion of the free-surface profile. The order of the approximation is deter- 

 mined 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 (1847, 1880) first developed equations for finite- 

 amplitude waves, the equations presented here are those of Miche (1944). 



a. Wave Celerity, Length, and Surface Profile. It can be shown that, for 

 second-order theories, expressions for wave celerity (eq. 2-3) and wavelength 

 (eq. 2-4) are identical to those obtained by linear theory. Therefore, 



gT ^ /2TTd\ 



C = -2- tanh 



2Tr L 



and 



L = IZ! tanh U^ 

 2ti \ L 



2-34 



