singular Perturbation Problems in Ship Hydrodynamics 



vary the motion amplitude for a given ship (i.e. , for fixed e) , and it 

 also guarantees that the motions are small compared with the ship 

 beam and draft, even as the latter approach zero as € "^ 0. Velocity- 

 potential, wave height, motion variables, and all other dependent 

 variables may be expected to have double asymptotic expansions, 

 valid as e ~-^ and 6 -* 0. We shall consistently carry terms 

 which are linear in 6. The steady-motion problems already treated 

 correspond to the 6 = case; at zero speed, the 6 = case is 

 trivial. The problem ahead is to solve the linear motions problem 

 -- "linear" in terms of motion amplitude. With respect to the 

 slenderness parameter, we shall consistently carry up to e terms. 



It should be noted that the slenderness assumption is not 

 needed in formulating a linear motions problem at zero forward 

 speed; it is convenient, however, in practical application of the 

 theory. 



All motions are assumed to be sinusoidal ajb radian frequency 

 o). I use a complex exponential notation, so that: |j(t) = icL4j(t). Also, 

 it is assumed that to = 0(e''^^), and so symbolically we can write: 

 a/at = 0(€-'/2). 



The potential function, ^(x,y,z,t), satisfies the Laplace 

 equation and the following boundary conditions: 



[A] = g; +<|,t +^[<^x + ^y+ ^z] . on z = C(x,y,t); (3-20a) 



[B] = ^,;, + <^y;y - ^z + ^t ' on z = ;(x,y,t); (3-20b) 



[C] = c^^Sx + <|>ySy - ((>z + S^ , on S(x,y,z,t) = 0. (3-21) 



We consider first the problem of a ship which is forced by 

 some external means to heave and pitch in calm water. In the far 

 field, the slenderness assumption leads us to expect that the potential 

 function can be represented by a line of singularities on the x axis. 

 From previous experience, we might hope that a line of sources 

 would suffice in the first approximation; this turns out to be correct. 

 Since these sources represent an oscillating ship, the strengths of 

 the sources will also vary sinusoidally. Suppose that there is a 

 source distribution on the x axis: 



Re {o-(x)e } , - oo < x < oo. 



Define (r(x) to be identically zero beyond the ends of the ship. 

 Obviously, {r(x) = o(l) as 6 ~* , since there is no fluid motion at 

 all for 6 = 0, Therefore, In the first approximation, we may 



749 



