Ship Waves. 9 
local disturbance decreases with increasing distance when we sum for the 
system as a whole. The localisation of the disturbance into parts associated 
with special points is in general no more than a convenient help for purposes 
of calculation and description. 
6. The previous settion gives a surface elevation which is finite and con- 
tinuous throughout, and it is simple to extend the method to cover any form of 
distribution. 
We begin, for simplicity, by considering any limited distribution of which 
the graph is made up of straight lines. 
The general expression (5) gives, for infinite depth of distribution, the 
elevation along the line of motion as 
s io) aT ios) ix (w—h) cos 
cat Man |” see 900 f° ik. 
TU Jo dh 0 K — Kg sec? 8 + zu. sec 0 
—7 
Take the integration with respect to 4 along two parts of the range meeting 
at a junction h,,, and we obtain, associated with this junction 
ekg cos 0. (30) 
r 
dM 
dh 
a 
kK cos 8 
where the coefficient in straight brackets is the increase in slope of the M, h 
graph in the positive direction, or tan ¢, — tan ¢, in terms of the slopes of 
the adjacent parts of the graph. It should be noted that the positive direction 
of h, and of a, is taken here in the direction of motion, that is, from stern to 
bow. 
It is clear that for any limited distribution which is zero outside a certain 
range in h, we have from (29) and (30) the complete surface elevation in the 
7 4 cc > a ® of (7h) cos 6 
c=-+\ sect ode fe? en Sa aon r de. (31) 
o K(k — Ko sec? 6 + iu sec 8) 
form 
where the summation extends to all the junctions, including the bow and stern. 
Further, the algebraic sum of all the changes of slope is zero; hence we may 
separate out the calculation for each junction by writing (31) in the form 
s oct 0 — pix (&—h,,) cos 
ws dM | sect 0.40 | CU Cees Seo Bs 
TU Up \\> jae o K (kK — ky sec? 6 + wy sec 8) 
4 dM |s 
eteN LAS (ep a 
mir eae (x — h,s), (32) 
355 
