This simplification is justifiable only under the assumption that 

 the slope of the surface waves is very small; this is equivalent to 

 k — « 1 where k is the wave number and H the amplitude of the surface 

 wave. When the water depth is not very large (d< TT/k) the solution 

 of the simplified equation indicates that the water particles describe 

 elliptical orbits around their mean position with component displace- 

 ments in the x and y directions given by the expressions: (Lamb, 1932 

 sec. 229). 



H cosh k (y + d) ,, ,_. ,„ . 



X = . - - - y cos (kx-ujt) (3-6) 



2 smh kd 



H sinh k (y + d) . ,. ,,. ._ _.,. 



Y = 2 sinh kd Sin (kx - «*> (3 " 7) 



It is evident that the vertical component of the displacement be- 

 comes smaller as the distance from the surface increases and that right 

 at the bottom (y = -d) the motion degenerates into a simple harmonic os- 

 cillation along the x direction. The corresponding velocity components 

 are obtained by differentiation of equations (3-5) and (3-6); so that 



^X H 

 u ± (x,y,t) = |-j: = - a) TTZZ IV "' sin < kx - U)t) (3-8) 



v, (x,y,t) = |J= -% u) ^ "" "^ ' — cos (kx - u)t) (3-9) 

 1 ' J ' ■ St 2 sinh kd 



cosh 



k (y + d) 



sinh 



kd 



sinh 



k (y + d) 



As y — => -d v (-d) — > 0, and 



(3-10) 



u (x,-d,t) = — u) cosech kd sin (kx - ujt) 

 or u (x,-d, t) = auj sin (kx - ujt) = u Q sin (kx - u)t) 



where obviously a = — cosech kd and u = aoo 



On the basis of dimensional considerations it is reasonable to postu- 

 late that the thickness of the boundary layer is very small -j 6 = / — \ 



compared to 1/k so that for all practical purposes u-^ (x,t) may be 

 assumed constant within the boundary layer and approximately equal to 

 u-i(x,-d,t); as it is customary we will use the notation U^ for the free 

 stream velocity at the outer edge, so that 



U ro = u^ (x,-d,t) = ayj sin (kx - (j)t) (3-11) 



with surface waves of large wave length (ka«l) equation (3-11) becomes 

 U ot = auj sin oft = u Q sin ujt (3-12) 



