The Inatial Wave Resistance of a Moving Surface Pressure. 242 
Consider, then, a pressure system 
p= FQ), (3) 
which is suddenly established, and is at the same instant set in motion with 
uniform velocity c along the axis of z. 
Putting «= a+ct, the surface elevation at any time ¢ after the start is 
given by 
t e) e) 
—Tg py = | eK dy | KVeEK(@+eX) sin (Vw) al F (a) e-**da, (4) 
0 0 —2» 
For simplicity, we shall confine ourselves to pressure systems which are 
symmetrical with respect to the origin ; so that 
ADE [ F@e*ae =) (j2 (3) ea rales (5) 
Also we shall use only localised distributions for which the integrals are 
finite and determinate; the systems will be finite and continuous and such 
that the integral pressure is finite, that is, the integral | F(a) da 
convergent. Carrying out the integration with respect to w, we obtain 
0 Re 1 1 
— 27g py = , KV G(x) ¢ Lavagnin an aaa 
; [" Vv : etx (V+e)t en (V—e)t Te 6 
Eel hog 
Cale EO 1: (Wim “avo aal = © 
The first integral represents the steady state, while the second gives the 
deviation from it when we take into account the beginning of the motion. 
3. From the first integral in (6) we have, with «y = g/c?, 
Sunt fee ee Clie 
Sh Sn | aaa 
PSG Ee | 2 (o—«) + 2px © 
The integral is to be evaluated first, before we make mw zero, otherwise it is 
indeterminate. The interpretation for certain types of localised pressure 
system is well known; in such cases the solution takes the form 
y=f(s), o>, 
y= — 72 $ («sin aoe +f (—2), a< 0. (8) 
This solution represents an infinite train of regular waves in the rear 
of the moving system, together with a disturbance symmetrical fore and aft 
which becomes negligible at a distance depending upon the concentration and 
the velocity. For our present purpose, all the examples we use are included 
under the case 
d(x) = n%e-*, n> 0, a> 0. (9) 
107 
