Doctors 



u = 2 rrm/B , (39) 



m 



and 



2 2 2 (40) 



k = w + u 



2 

 7 = gk tanh(k d) , (41) 



e =1/2, e =1 for n> 

 o n 



I V . 2 . The Wave Resistance 



The method of obtaining the wave drag is the same as in the 

 previous section and utilizes Eqs (24) and (38). After some algebra, 

 one obtains : 



2 J 



R = 



c( t ) dr 2 « n 2 « w ■ cos] \gk ■ tanh(k d) • (t - t ) [• 

 nto n m,0 m n < s mn v mn ' v ' ' 



2 

 P jcos(w (s(t) - s(t ))) - cos(w (s(t) + s( t) + 2 <r))\ + . . . 



mn « n n ' 



2 

 + Q |cos(w (s(t) - s(t))) + cos(w (s(t) + s(t ) + 2 a )) + . . . 



mn ( n n 



+ 2P Q sin(w (s(t) + s(t) + 2er )) , (42) 



mn mn n 



•]■ 



where 



and 



P = P (w , u ) 



mn n m 



Q = Q (w , u ) 



mn n m 



It is clear that the fluid motion in the tank consists only of wavelets 

 whose wavenumbers are given by Eqs (37) and (39), and that in the 

 limit of L — * oo and B — ► <*> , the result for a longitudinally and 

 laterally unbounded region is recovered. The terms containing fl- 

 are due to reflections off the starting end of the tank, and as a — > oo 

 they contribute nothing to the wave resistance. 



50 



