Falttnsen 
the far-field, andso 0/d0z=0( « hye The existence of a surface 
wave implies that 9@/90@z and, say, d/ as are the same order of 
magnitude, where s is measured normal to wave fronts. In the far- 
field, we cannot in general say that the normal to the wave fronts 
should be neither along the x-axis nor along the y-axis. This im- 
plies that 0/9@x and 0@/ dy must also be of order e~l in the 
far-field. 
From a far-field point of view, one cannot see the shape of 
the hull. As ¢ —0 the disturbance from the hull to the far-field 
seems to emanate from a line of singularities located on the x-axis 
between -L/2 and L/2. The dominant far-away effect is expected 
to appear to be due to a line of sources. Since the incoming waves 
vary as etl stds wachkivg, expected that the line of sources has a 
source density of the form 
puget Me ae og vx) 
Og 
Ox 
Due to the slenderness of the ship we assume = OCs 
These physical arguments can be given a mathematical for- 
mulation. We can replace equation (1) by the Poisson equation 
i(wt - vx) 
+———- + ——— = _ a(x) e d(y) 6(z - z,) 2 <3) 
Here 6 is the Dirac-delta function, and initially we take z, < 0. 
When the solution of ¢p is found, zo will be set equal to zero. If 
we set Zz) = 0 first, we would be in difficulties solving the problem. 
We cannot expect that the far-field solution will satisfy the 
boundary condition on the hull given by (2), but it must satisfy a 
proper radiation condition. We must be sure that the diffraction po- 
tential ¢p does not contain an incoming wave. This is most easily 
taken care of by introducing the artificial Rayleigh viscosity wm (see 
for instance Ogilvie & Tuck (1969) ). The free-surface condition (1?) 
will then be modified to 
LR¥2 
