500 Prof. T. H. Havelock. 
therefore, only straight-crested parallel waves and so emphasising the connec- 
tion between the critical velocity and that of the wave of translation, In 
the present paper I obtain an expression for the wave resistance of a surface 
pressure symmetrical about a point, and moving over water of finite depth. 
The result is in the form of a definite integral, which has been evaluated by 
numerical and graphical methods so as to give graphs of the variation of 
wave resistance with speed for different values of the ratio of the depth of 
water to the length associated with the pressure distribution. The graphs 
are of special interest in the cases intermediate between the two extremes of 
deep ‘water and shallow water. They show the double effect of limited 
depth, in lowering the normal wave-making speed of the ship and in 
increasing the magnitude of the effect as the speed approaches that of the 
wave of translation. The results are discussed in their bearing upon the 
experimental results which have just been described. 
2. In a previous paper* I worked out the case of a symmetrical surface 
pressure moving over deep water. The present analysis is on exactly similar 
lines, except for suitable changes in the expressions; it may be sufficient, 
therefore, to set forth the calculation briefly, referring to the previous paper 
for further detail in the argument. 
Take axes Ox, Oy in the undisturbed horizontal surface of water of depth h 
and Oz vertically upwards. For an initial impulse symmetrical about the 
origin, that is if the initial data are 
pho=F(m), €=0, @) 
where oa? = 2?+y", the velocity potential and surface elevation in the 
subsequent fluid motion are given by 
w= [Le («) cosh « (g-+h) sech xh Jp (wes) cos (« VE) « de, 
me = = \,f («) Jo (xe) sin («Vt) x2 V de, (2) 
where V? = (g/«) tanh ch, 
oy [Fea (oede (3) 
We obtain the effect of a travelling pressure system by integrating with 
respect to the time. We shall suppose that the system has been moving for 
a long time with uniform velocity, c,in the direction of Oz. Transferring 
to a moving origin at the centre of the system, we replace x in (2) by #+ct, 
and we find for the surface elevation 
l= — | “oo Meet dt [7 Jo [a { (wtct)?+y?}!2] sin (eVt) «2 Vde, (4) 
* Roy. Soc. Proc.,’ A, vol. 95, p. 354 (1919). 
193 
