198 Dr. T. H. Havelock. [June 7, 
conventional use of the term, are discussed in connection with the residuary 
resistance curves of finer-ended models. 
2. A transverse pressure disturbance travelling over the surface of water at 
right angles to its axis leaves in its rear a procession of regular waves; on 
account of the supply of energy needed to maintain this system, there is an 
effective resistance which may be called the wave-imaking resistance of the 
given disturbance. An illustration of a simple type of disturbance, 
symmetrical fore and aft with respect to its axis, is afforded by the function 
P/(p?+*), where Ox is in the direction of motion. The length p may be 
used to define the distribution of pressure to this extent: when p is decreased, 
the changes are more concentrated and abrupt, and conversely ; we may, as a 
convention, call 2p the effective width 0 of the disturbance. If the dis- 
turbance is made to move with uniform velocity v at right angles to its axis, 
the height of the waves can be calculated, and thence, from considerations of 
energy, the corresponding wave-resistance R. If the quantity P which defines 
the magnitude of the disturbance is supposed an absolute constant, the 
calculation of R as a function of v gives an expression which rises to a 
maximum and then diminishes ultimately to zero with increasing values of 
the velocity.* But if the pressure disturbance is associated with a moving 
ship, it seems reasonable to suppose that P depends upon the velocity, and in 
fact the assumption is that P varies as the square of the speed. 
In this way we obtain the result 
R = Ber, (1) 
where B is independent of v. According to this expression, It increases from 
zero up to a limiting value B; at any given speed R is a certain fraction of 
the value B, and if the quantity ’ were increased the same value of R would 
only be reached at some higher speed. Further if we have a second 
expression R, with constants Bi, d, greater than B, 6, respectively, the curve 
for R, will intersect the curve for R at a certain velocity; at lower speeds 
Ri <R, while at higher values Ij > R. 
Suppose now that a similar negative pressure system, with a different 
coefficient P, but with the same width 6, is situated behind the first system, 
with a fixed distance 7 between the two axes. The wave-making resistance 
of the combined system is given by an expression B(1 — y cos gl/v*)e~'9™, 
where 8 and y are independent of v. In applying this result to the case of a 
ship, we can of course only expect agreement if the type of model is such that we 
may imagine distinct, but mutually interfering, wave-systems originating at 
the bow and stern; it is, in fact, an attempt to describe the wave-making 
* Of. Lord Kelvin, ‘ Math. and Phys. Papers,’ vol. 4, p. 396. 
70 
