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



5 + 2 cosh(4iTd/L) + 2 cosh2(4TTd/L) 



-f"-(^)hfiJ 



8 sinh'+(2TTd/L) 



(2-47) 



and 



gT2 /27rd\ nR] 



5 + 2 cosh(4TTd/L) + 2 cosh2(4TTd/L) 



8 sinh'+(2TTd/L) 



(2-48) 



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



H /2Trx 2trt 



n = — cos I 



2 \ L T 



irH^X cosh(2TTd/L) 



8L / sinh3(2ird/L) 



2 + cosh(4TTd/L) 



4Trx 4irt 



cos 



(2-49) 



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



O / 2lTX 



2irt\ , o /4Trx 



— ■^417 ^°^ — 



/ o \ o 



4iitl 



(2-50) 



b. 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[2iT(z + d)/L] /2irx 2Trt 



u = -— cos 



2L cosh(2Trd/L) \ L T , 



3 /tthV cosh[4iT(z + d)/L] /4irx 4iTt^ 



4 \ L/ sinh'+(2ird/L) '^"^ I L T , 



ttH sinh[2Tr(z + d)/L] /2Trx 2TTt^ 



w = — C r-. sin 



L sinh(2TTd/L) \ L T , 



3 /irH Y sinh[4Tr(z + d)/L] Uwx. 4Tit' 



4 \ L / sinh'+(2Trd/L) \ L T ^ 



(2-51) 



(2-52) 



Second-order equations for water particle displacements from their mean 

 position for a finite-amplitude wave are 



2-35 



