280 Dr. T. H. Havelock. [Apr. 1, 
The effect of thus diffusing the pressure system is expressed by the 
introduction of a factor ¢ (x) into the amplitude of the regular waves, where 
2m/« is the wave-length and 
o(«) = | f) cos Kw da. (6) 
Using (5) in (6), we find 
(Ce) = LGR es LCE 
Hence the amplitude of the waves is given by 
AG eee res 
a= eal (7) 
Further, since « = 2?/g, the group velocity u=d(xv)/de =4v. Hence 
the wave-making resistance R is given by 
2p2 
R = FE ¢-2ag/v*, 8 
wet (8) 
We have to examine the variation of these quantities with the velocity v 
under the supposition that the pressure system is due to the motion of a 
body either floating on the surface or wholly immersed in the water. The 
pressures concerned being the vertical components of the excess or defect 
due to the motion, it seems possible to assume as a first approximation that 
P varies as v?; this is the case in the ordinary hydrodynamical theory of 
a solid in an infinite perfect fluid, and a similar assumption is also made 
in the theory of Froude’s law of comparison. This being assumed, we find 
a = Ae~esi™, 18, == 1g (9) 
We see that both the amplitude and the resistance increase steadily from 
zero up to limiting values. 
If we draw the curve representing this relation between R and 1, there is 
a point of inflection when 
a?R 
7 =), oF v? = Aga. (10) 
Writing v’ for this velocity, we see that dR/dv imcreases as the velocity 
rises to v’ and then falls off in value as the velocity is further increased. 
We can write the relation now in the form 
R = Ben? (11) 
The character of this relation is shown by the curve in fig. 2, which 
represents the case 
1 = Bilbo Ve, (12) 
R being in tons, and V in knots. 
38 
