534 XL HYDRODYNAMICS. 



152) gp = Ar n sin no, ty = Ar n cos ncc>, 



o being the angle measured from the vertical. We have for the 

 radial velocity 



G OP . 



q r = -- = Anr n ~- 1 smn(o, 



and if a is the inclination of the surface to the vertical at the crest 

 q = Anr n ~ 1 smn a. But we have g 2 = 2gy = 2gr cos a and accord- 



o 



ingly 2 (n 1) = 1, n = - Also as in 129), cos na = 0. Thus 



The problem of waves in water of finite depth may be treated 

 in a similar manner, by putting instead of 134), 



cp -f ty = az + Ae~ ikz -f jBe*'**, 



qp = # -f (^le*y -f Se~ ky ) coshx, 



153) il> = ay (Ae k v - Be~*y) sin A; a;, 

 w = a - 



v= 

 If the depth is h, we must have v = for y = h 9 giving 



Ae~ kh = Be kh . 

 Calling this value C, we have 



154) ^ = -ay- <7(e *(*+*) - e -*(*+y)) sinA;^ = 0, 



as the equation for the wave -profile. For the first approximation, 

 for waves whose height is small compared to their length, replacing 

 e ky ,e~ k y by unity, we have 



155) ay = C(e kh e- kh } sin&#, 

 and neglecting (G Y &) 2 , 



156) u 2 + v 2 = a 2 + 2CW (e kh -f e~ k/< ) sin kx. 

 Thus the surface equation 139) becomes 



157) const = ^--^ (e kh - e~ kh ) sin Jcx 



^ 



a 



-f a 2 



which is satisfied by 

 158) a~k(e ] 



giving the velocity 

 159) 



