Ship Waves. 11 
parabolic lines. We take the origin at the bow, and let the form of the half- 
section for y positive be given by 
y=bf1—(h+)P7/P}; —2<h <0. (36) 
The discontinuities of f’(h) at the bow and stern are both positive, and equal 
to 2b/l; while f’"(h) is constant throughout the range and equal to —26/I?. 
Hence from (28) and (35) we have, from the discontinuity at the bow, 
=F Qwloo, 20 
TK g 
/ 8b U 1 id 
= — —— Qo (ko 2) + = Po (ko “); a > 0. (37) 
TK ol 
There is an equal system for the discontinuity at the stern. 
Consider now the contribution due to the curved portion and take first the 
wave terms. For a point behind the stern («’ > 2I) we have 
TK gl” 0 
20 
ages | Py fico (a! — B')} ah. (38) 
We have, in a notation already used, 
| Po (u) du =1+ P, (u) 
= Pp*(u), say. (39) 
Thus from (38) and (36) the complete wave disturbance at a pomt behind the 
stern is given by 
8 {2 t = {2 = So atyh 
en Ob. [me (ee Dee OT (eect Te ce 21) | (40) 
TK ol Kol 
Taking a point between the bow and stern (0 <2’ < 21), it is easily verified 
that (40) gives the wave elevation for all points with the convention that the 
functions P, and P% ‘ are to be taken zero for negative values of their arguments. 
It may be noted that as these functions are zero for zero values of their argu- 
ments, the expression is continuous throughout. 
Similarly, if we consider the local disturbance and take first a poiwt in 
front of the bow (a > 0), we readily obtain from (28) and (35) 
- 
t= — BL atin) + Qa tka (@ + 20} + A (Qs (eye) — Q, (coe FB} | HD 
‘Ko Ko 
where 
Q, (u) = | Q, (u) du. (42) 
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