Wave Forces on a Restratned Shtp tn Head-Sea Waves 
determine the velocity potential. From (1), (2), (3), (4) and the 
assumption about linearity it follows that @¢ satisfies 
2 2 2 
SS Bit 1 bald tee EA Go dnidhe Clad depesueie 
2 2 2 
Ox dy dz 
8 = ORE Gi? Baz rain oe y Y (2) 
Sitar marrage ge = 0 on z=0 outside the body (11) 
In addition, the diffraction part of the potential must satisfy a radia- 
tion condition. 
We will write @ as 
a i a i — a + © (12) 
Here ?D denotes the diffraction potential. To find ?D weare 
going to use the method of matched asymptotic expansions (Van Dyke 
(1964) , Ogilvie (1970) ) and set up an asymptotic theory valid as 
¢ — 0. Asis usual, we introduce a far-field description and a near- 
field description. The far-field description is expected to be valid at 
distances which are 0(1) and larger from the ship. The near-field 
description is valid near the ship at distances which are 0( € ), but 
not near the ends of the ship. 
The steps we are going through is first to obtain a far-field 
description and then obtain an inner expansion of the far-field des- 
cription. So finally we are studying the near-field solution and the 
matching between the near-field and the far-field solution. 
III, 1. Far-field source solution and the inner expansion of the far- 
field source solution. 
In the far-field description, we expect to have waves. In 
order to have waves, we must satisfy the condition (11). This means 
that the two terms in (11) must be of the same order of magnitude in 
yf 
