243 Prof. T. H. Havelock. 
To verify the solution (8) in this case, regard « in (7) asa complex variable 
re, 
For o positive, integrate round a sector of radius R bounded by the lines 
@6=0 and @=6 (0<@<47). Under the specified conditions, it can be 
shown that the integral along the arc r=R tends to zero as R is made 
infinite. In this way the integral (7) is transformed into an integral, along 
the line 0 = £, in which we can make p zero. 
For o negative, integrate round a sector of radius R bounded by the lines 
=Oand 6=8, with —tan712u/m>8>—47. We get a similar result, 
except that the integrand has now a simple pole within the sector at the 
point x«9—2y2 approximately. The residue at this pole gives the term in (8) 
which represents the recular train of waves in the rear of thesystem. It can 
also be verified that in this case y and dy/ds are finite and continuous 
throughout. 
Returning to the general expression (6), the second integral represents the 
deviation from the steady state. It contains exp {ix (m+ct) } aga factor, and 
we see from its form that it represents the effect at time ¢ of a certain initial 
distribution of velocity and displacement. To illustrate this point, consider 
a stationary pressure system which is suddenly established at a given instant 
and maintained constant. The effect is the same as if there had been in 
existence up to the given instant two equal and opposite systems with their 
ultimate static effect upon the water surface fully established, the negative 
system being then suddenly annulled. Thus the subsequent effect is the 
steady state of the positive system combined with the effect of an initial 
displacement equal to the steady state of an equal negative system. In the 
same way, for a pressure system which is suddenly established and started in 
uniform motion, the effect is the superposition of the steady state of this 
system and the disturbance due to initial conditions given by the steady state 
of an equal negative system in uniform motion. We shall find this principle 
of use in a later section. 
4. The wave resistance R; in the steady state is usually obtained from 
energy principles applied to the regular waves. The front of the train 
advances with velocity c, while the rate of flow of energy across any fixed 
vertical plane in the rear is the corresponding group velocity $c; from the 
amplitude of the regular waves in (8), by equating the net rate of gain of 
fluid energy to Rye, it follows that 
Ry = Ko? {$ (0) }?/ 9p. (10) 
Some consideration is necessary before we can apply this method to the 
motion before the steady state has been attained. 
108 
