4 
Some imiterestine conclusions may be taken irom this 
equanon. Ris zero 1 w equals 7, and this is m faci ap- 
proximaiely the case m shallow-water; the whole group oi 
iramsyerse waves consists then ot a hump which accompanies 
the ship ai its own speed (rv), and m consequence, once the 
disturbance is tormed, no turiher supply of enerzy is needed. 
Tn deep waier, 1 cam be shown that u is $7: we limit our 
consideration here to this case. 
We musi examine now the variation ot the heighi (@) 
of the waves with the speed. The motion of the ship makes 
ditierences of fuid pressure in iis neighbourhood, so we 
may consider the problem as equivalent dynamically io a 
pressure disturbamce moving over the suriace oi the waier; 
the effecis depend both upon the speed and upon the 
characier ot the disturbance. 
As regards ihe velocity, the differences of pressure in- 
crease wiih the speed, and probably they are proportional io 
iis square. 
The distribution of pressure in the disturbance depends 
upon the form of the ship. To take an extreme ease, if the 
ship were an infinite rait moving over an ininiie sea, the 
pressure would be constant over the surface and there would 
be no waves; on the other hand, if the lines of the ship are 
abrupt the pressure changes may be sudden and concen- 
trated and the height of the waves greaier. A deiailed 
analysis confirms the impression that m general the heighi 
ot the waves is diminished by diftusing the pressure sysiem. 
Bui an merease of speed is equivalent to a diffusion of the 
pressure; hence we have a two-fold effeci, imcrease of speed 
increases the magnitude of the pressures, and is at the same 
time equivalent to diffusmg them over a greaier area. 
Thus, there appear io be two opposing tendencies, and we 
infer that the height of the waves should not merease 
indefinitely with the speed of the ship, for the two effecis 
may tend to balance ai high speeds. 
For a simple type of pressure distribution, a mathemaii- 
cal amalysis shows that the wave-making resistance varies 
62 
