251 Prof. T. H. Havelock. 
driven at any other steady velocity ; it will have a steady resistance, which 
we can calculate from the formula (10); in this case it is 
gpR = x? (x—0:08)?e*, Kk = g/V?. (29) 
This steady wave resistance has a maximum at a velocity of about 
5:25 ft./sec., and the value of gpR is then 164x10-% Hence the maximum 
resistance due to the sudden starting of the system at its waveless speed is 
about one-half the maximum steady resistance at any uniform speed. 
8. We have been able to obtain an exact solution for a special type of 
waveless system ; we leave this now to consider more generally asymmetrical 
localised pressure system, which is suddenly established and set in motion. 
We have seen that the surface elevation at any time is found by super- 
posing the steady state of the system and the effect due to initial conditions 
given by the steady state of an equal negative system in uniform motion. 
Apply this to a case in which the steady state consists of an infinite train of 
regular waves in rear of the system, together with a localised dispiacement 
symmetrical with respect to the moving origin. Let O be the fixed origin 
and starting point, and C the position at time ¢. The deviation from the 
ultimate steady state consists of the effect due to a certain initial distribution 
of displacement and velocity localised round O, together with the subsequent 
state of a semi-infinite train of regular waves, which at the initial instant 
had a definite front at the point O. We may describe the latter part in 
general terms as a regular train with a front, more or less detinite according 
to the time, at a point G corresponding to the group velocity, and in 
advance of G a disturbance which may be called the forerunner. If OC is 
sufficiently large, and if we require the surface elevation only at points 
sufficiently far in advance of G, the forerunner is given with considerable 
accuracy by Kelvin’s group method of approximation. The argument is 
represented diagrammatically in fig. 3, the continuous line showing the 
elevation and the dotted line the travelling pressure system. 
FIG 3 
G 
The wave resistance being defined as the total horizontal component of the 
pressure system, we divide it into two parts. The first part is the final 
steady value o?{@(«o)}2/gp as given in (10), and the second is the deviation 
given by the integral in (13). The latter represents the resolved pressure 
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