Ship Resistance. 493 
of W. Froude, with the superposition of bow and stern wave-systems, that 
is, when the whole system may be separated into two fairly distinct parts. 
I have represented this previously by a positive pressure system of type (9) 
associated with the bow, followed by a similar negative system associated 
with the middle of the run. Thus if / is the distance between the centres of 
the two systems we have in the present notation 
Ay As (13) 
PS Gy Ee 
Substituting in (8) and evaluating the integrals we find 
gd = (Ai— Aa) e~* cos 3 xl, a = (Ai + Ag) e~* sin $l, 
Hence from (7) 
gpk = K? (Ai? + A?— 2 AjAs cos kl) e72aK, (14) 
The graph of R is a mean curve similar to (12) with oscillations super- 
posed upon it, humps and hollows corresponding to minima and maxima of 
cos xl or cos (gl/v?). 
It is of interest to note that if A, and Ag are equal, we have 
R = const. x e~?™ sin? $ xl. (15) 
Thus, in a hollow, R would be actually zero if the two pressure systems 
were equal in magnitude. This, of course, follows at once from general 
principles; if we have a pressure system followed at a fixed distance by an 
equal and similar system, then there are certain wave-lengths and corre- 
sponding speeds for which the main regular waves due to the two systems 
cancel each other out exactly. A moving body which would produce such 
a state of affairs would be, in Lord Kelvin’s phrase, a waveless pontoon. 
Of course, this does not occur in ship forms, and there are several reasons 
why it could not be expected to do so. In fact we have in general to suppose 
Az much less than A; in (13). However, it is conceivable that some change 
of form might give more effective interference effects of this kind and so 
deepen the hollows in the resistance curve, though possibly as a practical 
suggestion it may be subject to the same limitation as in other cases, namely, 
even if the wave-making resistance were lessened in this way probably the 
alterations would so increase frictional and other resistances that there might 
be no gain on the whole. 
5. Baker and Kent have pointed out that in certain cases the pressure 
distribution at the entrance of a ship form is not simply a hump of excess 
pressure, but is a hump followed by a hollow of negative pressure. They 
assign to the interference of these two parts a certain subsidiary interference 
effect in the resistance which may become important when it coincides with 
98 
