Falttnsen 
which is a condition to be satisfied by ¢, . Equation (47) gives us 
Abel's integral equation, and it can be solved in principle. It is to be 
noted that the term in the brackets is a function of x, whichis de- 
termined in practice by numerical computation. 
It is the near-field solution that has the primary interest. So 
let us summarize our result : A two-term near-field solution of the 
diffraction potential is given by 
_iV/2 2 im /4 
(¥, +¥,) ailwt - x) _ ilt - vx) ree Vric(x+L/2) ° 
se Hiew Bako 
C(+) 
kZ 
le ~ ieee [G(kY,kZ;k&(s),kn(s)) + G(kY,kZ;-ké(s), 
c(+) n(s)) as]| (48) 
where k and C are given by (26) and (27), and G is given by 
(42). The first term in (48) is just the negative of the incident wave 
and so (48) tells us that the total (incident-plus diffracted-wave) 
potential near the body (except near the bow and stern) will have a 
decay factor 
OE wile (49) 
in the x-direction. But note that mu in (48) is alsoa function of x, 
and so (49) does not give the total x-dependence. However, uw will 
be the same for similar cross-sections. So, if the cross-sections 
are not varying much inthe x-direction, the potential will, roughly 
speaking, drop off with the factor (x + L/2) -1/2 in the lengthwise 
direction. Note that we have assumed that the wave length is of the 
order of magnitude of the transverse dimensions of the ship. 
1784 
