dt 



U'Vp = 



(10) 



V.u = 



(11) 



The symbols have the following meaning: u = (u, v, w) is the fluid velocity, 

 p the pressure,^ the gravitational potential, and p the density. The effect of 

 the earth's rotation is neglected. The boundai'y conditions ai-e 



w = -I -^ + u-MC,\ at the sea surface, z = ~C, (x,t) (12) 



p = const at the sea surface, z = —^(T,t} (13^ 



ii) = at the bottom, z = H = const (14) 



In order to apply the boundary conditions (12) and (13) at the undisturbed sea 

 surface ^ = 0, we expand them about 2 = 0. The Taylor series of (12) is 



dw (^ d^w 



W - C + K- 



dz 2 dz- 



equation (13) yields 



dc /- , aT (^ d^ 



dt \ - dz 2 dz^ 



vc 



at 2 = (15) 





const at 2 = 



(16) 



THE PERTURBATION EQUATIONS 



Suppose the variables in (9) to (11) can be written as perturbation series 



p^-p^^^.p^^Up^'^\ 



^^^l),^2. 



(17) 



with p*°' = p (z) and p*°' = pfz), then, entering the equations (9) to (11) and (15) 

 to (16), we arrive at the following equations, omitting terms of higher than second 

 order. 



11 



