2.251 Wave Celerity, Length, and Surface Profile . It can be shovm that, 

 for second-order theories, expressions for wave celerity and wavelength 

 are identical to those obtained by linear theory. Therefore, 



and 



^ gT , /27rd' 



C = 2- tanh 



2tt \ L 



L = ^^ tanh '-^ 

 2n \ L 



(2-3) 



(2-4) 



The above equations, corrected to the third order, are given by: 



gT , /27rd^ 



C = — tanh 



271 \ L 



1 + 



and 



L = ^— tanh 



27r I L 



1 + 



T 



'nH\ 



5 + 2 cosh (47rd/L) + 2 cosh^ (47rd/L) 

 8 sinh" (27rd/L) 



5 + 2 cosh (47rd/L) + 2 cosh^ (47rd/L) 



,(2-47) 



8 sinh" (2;rd/L) 



The equation of the free surface for second-order theory is 



(2-48) 



H /27rx 



— cos I 



2 \ L 



27rt ' 



+ 



(8L 



cosh (27rd/L) 

 sinh^ (27rd/L) 



2 + cosh(47rd/L) 



cos 



u. 



t) 



For deep water, (d/L > 1/2) Equation 2-49 becomes, 



H„ J2-nx 2-ni 



■o I-..- --. ^ '^"o /4jrx 4fft\ 



T? = COS I — + cos — — - 



2 \L, TJ 4L^ \l^ t/ 



(2-49) 



(2-50) 



2.252 Water Particle Velocities and Displacements . The periodic x and z 

 components of the water particle velocities to the second order are given 

 by 



HgT cosh [27r(z + d)/L] /27rx lux^ 



U = ; ■ cos I — 



2L cosh(27rd/L) \ L T y 



3 /ttHV cosh [47r(z + d)/L] ^ttx 4jrtl 



+ — C . . ^ ,. TTTT cos 



(2-51) 



sinh* (27rd/L) 



2-37 



