Salvesen 



is not only the linearized wave solution but also satisfies the free-surface con- 

 ditions correctly to the second order, for the case of infinite depth. Lord Ray- 

 leigh''' pointed out that this solution can also be shown to be correct to the third 

 order. In the present work, however, we will only be concerned with first-order 

 and second-order terms. 



The equation of the wave profile (^ = o) follows from Eq. (Al) by successive 

 approximation: 



Tj - — + /? e'^^ cos vx = — + fS(l + vrj+ . . . ) cos vx 



C 



— + — B^ + B cos z^x + — /S^ cos 2vx + ... 



U 2 2 



(A2) 



For convenience, let us set 



c^ = - ^ u/3^ (A3) 



such that the wave profile becomes 



7] - fi COS VX + — vfi^ cos 2vx ... . (A4) 



We note that Eq. (A4) coincides with the equation for a trochoidal wave to the 

 given order of accuracy. 



The solution, Eq. (Al), with the wave profile, Eq. (A4), must, to be valid, 

 also satisfy the condition of constant pressure (p = 0) at the free surface. Ap- 

 plying the Bernoulli equation 



^+ i Igrad <D|' + gy = Ci (A5) 



we have that 



^U^ (1- 2v/3e^y cos vx + v"^ fi'^ e'^'^y) + gy = Cj . (A6) 



At points on the line 



T7 = - — 1//32 + ^e'"' cos vx 



we therefore have 



(g- vv\'')-n + ^u^ - Ci , 



2^^ 1 u2 . C. . (A7) 



*Lord Rayleigh, "On Waves," Philosoph. Mag., Series 5, 1:257-279 (1876). 



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