253 The Initial Wave Resistance of a Moving Surface Pressure. 
In Case ii, r = 2 and Ay = 257. We find that when 2 = 9, the deviation 
is already less than 0:06 per cent. 
10. Consider now a simpler type of localised pressure distribution, namely, 
mE (x) = r/ (7? +2"). (33) 
This type leads to a steady wave resistance whose variation with the 
velocity is more like that of a ship model. We have ¢(«) = e-”, and (30) 
gives 
1/2 
soap 027?" 008 (gt [4+ A), (34) 
pR= ae pa 
The relative deviation is now 32 times as large as in the previous case, since 
3mr/Ag 
= = ee cos 4(2n+1)rr. 
With 7 = 1, »=2, the value of Ro/ Ry is about 0°5 for »=100. We 
should have to take n of the order 10,000 before bringing the deviation from 
the steady value below 5 per cent. 
On the other hand, with 7 = 2,» = 25m, the deviation is under 2 per cent. 
when n = 9, or at about 35 seconds after the beginning of the motion; it is 
less than 2 per cent. when n = 4, or after a travel of rather more than 
300 feet. 
11. The waves produced by the horizontal motion of a circular cylinder of 
small radius travelling at a considerable depth 4 below the surface may be 
cormpared with those produced by the surface pressure 
mE (x) = Ac? (h?—2*)/(h? +2). (35) 
We assume that the intensity of the system is proportional to the square 
of the velocity. It appears that the steady wave resistance is then the 
same function of the velocity as in the motion of the cylinder ;* for we have 
h(x) = Actxe*, 
and hence 
pR = A?g? en 2ghlc = A2g'/? 
ct 
64 q7}/2 ¢7/2 2 
As a numerical example, take the case when the velocity is such that 
the steady resistance R,; has its maximum value; that is, when c? = gh. 
e~9/2e* cos (gt/4¢e+ 47). (36) 
Then we have 
Ro 63? 
Ri ~ 913/272 
The value of the ratio means a deviation from the steady value of about 
0:8 per cent. when n = 33, that is, when the system has travelled through a 
distance 77h. 
cos }(2n+1) 7. (37) 
* Lamb, ‘Hydrodynamics,’ 1932 edn. p. 410. 
118 
