Prof. T. H. Havelock. 855 
where w? = x? +7, the velocity potential of the subsequent fluid motion, and 
the surface elevation are given by* 
ne =, T(#) &= To (wes) cos (Vt) ede, (1) 
eee = [7 In Ga) aia IO) Ze (2) 
where V? = g/x, and 
POs [,F@J (ea endles (3) 
We have assumed that it is permissible to use the integral theorem 
NG) = | “Jo (+o) ede|"F (2) Jy (Ayode. (4) 
We obtain the effect of a pressure system moving over the surface with 
uniform velocity ¢ in the direction Ox by integrating (1) and (2) after 
suitable modifications. We replace ¢ by t—7 and x by «w—cr, and integrate 
with respect to 7 over the time during which the system has been in motion. 
We shall limit the present discussion to the case when the system has been 
in motion for a long time, so that if we take an origin moving with the 
system a steady relative condition has been attained. In this case, with a 
moving origin O, we replace a by w+cw and ¢ by w in (1) and (2), and obtain 
the required resultst 
pp = \ e— 3H" du [7 & Jo [ka/ {(w+cu)?+¥"} ] cos («Vu) ede, (6) 
mee — [ee au | "Fedo [xr/ {(w-+eu)? +42} ] sin(«Vu) «2Vdee, (6) 
where f(«) is obtained from the assigned pressure distribution p = F (=) by 
means of (3). 
The introduction of the factor exp(—u/2) is familiar in these problems 
and needs only a brief explanation. It may be regarded as an artifice to 
keep the integrals determinate, it being understood that ultimately p is to be 
made infinitesimal. Or, again, it ensures that the solution is the fluid motion 
which would establish itself eventually under the action of dissipative forces, 
however small. 
In the steady motion with which we are concerned, we may imagine a 
rigid cover fitting the water surface everywhere and moving forward with 
uniform velocity c. The assigned pressure system F(a) is applied to the 
* Lamb, ‘Hydrodynamics,’ 6th edn. (1932), p. 432. 
T*Roy. Soc. Proe.,’ A, vol. 81, p. 417 (1908). 
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