Bow and Stern Waves and Snnall-Wave-Ship Singularities 

 If we consider, for convenience, a polynomial source distribution 



N 

 n = 



in < X < L and < Zj < H, then Eq. (13) becomes 



^1 = -r^ A(0,^) sin (kgX sec ^) + B(0, ^) cos (koxsec^) - B(x,^) d^ , (16) 



4ko '-"' 



'-77/2 



where 



and 



... - 2n 



n=0 kn sec 



B(L,5) = 1-e j L 



Equation (16) is an analytic function of x. Therefore, Eq. (16) or (13) can be 

 anal3rtically continued to x > L, and we may call these the physical bow waves 

 on the x axis for the source distribution given by Eq. (15); then the physical 

 stern waves will be, from Eq. (14), 



4ko r^/^ r'' 



^2 "^ V ^^ \ U(koH,x,^) cos [ko(x-Xi) sec 9] dxj 



4ko .-/2 



- —rr- A(L,6) sin (kgX-L sec 6) 



''-77/2 



+ BCL.ei) cos (kgT^T sec 6) - B(x,0) 66 (17) 



for x > L. 



This is essentially a fictitious consideration of the analytically continued 

 source distribution in x > L for the bow waves and a sink distribution with the 

 same magnitude of strength from x = L to x > L for the stern waves, so that the 

 total source strength in x > L is zero. In this sense we may extend our defini- 

 tion of the physical bow and stern waves to any point on the free surface (x,y,o). 



In Eqs. (16) and (17) the integrals of B(x,6) are canceled out in a total wave 

 (bow waves plus stern waves) for x > l . Therefore 



685 



