Wave Forees on a Restratned Shtp tn Head-Sea Waves 
We then see that the order of magnitude of +t = = is 
w 
a =? Dies = i ia ey Sek 2 (58) 
It is obvious that 7 will be larger than 1/4. This is important, 
because the solution will be singular when 1 = 1/4 (see Ogilvie and 
Tuck (1969) ). 
There are two parts in this chapter: (1) derivation of the 
far-field source solution due toa line of pulsating, translating sources 
located on the x-axis between -L/2 and L/2 (see Figure 1) and 
derivation of a two-term inner expansion of the far-field source solu- 
tion ; (2) formulation of the near-field problem, and the matching of 
a two-term near-field solution with the far-field solution. 
IV.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. Itis 
difficult to say how differentiation changes order of magnitudes in the 
far-field. So, to be careful, we would rather keep too many terms in 
the far-field. But we have to be sure that we have a system of equa- 
tions that describes a wave motion. Using arguments similar to those 
in the section ''Far-field source solution and the inner expansion of 
the far-field source solution" in the chapter on the zero-speed pro- 
blem, we can find that ¢p must satisfy the Poisson equation, 
3 
264 pullusd of a ¢ 
Fe ee te g(a) oe = 
B(y) 6(z-z) 
where Zn 0. We write the free-surface condition as follows : 
2 (oR) 
(io + U 2 +4) oT 8 ae =O on ZS 0 
where uw is the artificial Rayleigh viscosity, which will approach 
1787 
