490 Dr. T. H. Havelock. 
theory by working out the wave-making properties of other distributions of 
pressure. Although no attempt has been made to connect the distribution 
directly with ship form, the following examples have been chosen with a view 
to general inferences which can be drawn in this respect. In particular, the 
distributions graphed by Baker and Kent can be represented, in type at least, 
by a mathematical expression for which the corresponding Fourier integral 
can be evaluated, so that one can compare the result with that obtained from 
simpler forms. Although the expression for the wave-making resistance 
becomes more complicated, it is not essentially different from that obtained 
previously ; it appears in general to be built up of terms involving the same 
type of exponential e~*”, together with oscillating factors representing inter- 
ference effects between prominent features of the pressure distribution. 
2. We confine our attention to two-dimensional fluid motion. We may 
imagine it to be produced in a deep canal of unit breadth, with vertical sides, 
by the horizontal motion of a floating pontoon with plane sides fitting closely 
to the walls of the canal but without friction. We assume that, as regards 
transverse wave-making, this is effectively equivalent to some travelling 
distribution of pressure impressed upon the surface of the water. 
Let Ox be in the direction of motion of the disturbance, and let y be the 
surface elevation of the water. Suppose the distribution of pressure to be 
given by 
p=f(). (1) 
For a line distribution we may suppose the disturbance to be inappreciable 
except near the origin and to be concentrated there in such a manner that the 
integral pressure P is finite, where 
Des | VOU (2) 
When this disturbance moves along the surface of water, of density p, with 
velocity v, the main part of the surface disturbance consists of a regular train 
of waves in the rear given by 
gpy = —2xP sin xx, (3) 
where the length ) of the waves is 
Ne 20 ie Qa" 
K g 
We can generalise this result for any form of pressure distribution / (2), 
which is likely to occur, by the Fourier method. We have in general 
apy = —2e | f(Osinn wf) dé (4) 
—oO 
= —2«(psin cu—W cos Kz), (5) 
95 
