Resistance of a Ship among Waves 305 
We now use these measurements solely in order to take a corresponding 
velocity and corresponding distances in the expressions (19) and (20) and 
so to calculate the ratio R,,/R,. 
We have kyl = gl/c? = 2-7. For the two values of k, the corresponding 
values of kjk are 33-07, 39-74 respectively. With these we obtain from (19) 
and (20) the values — 0-24, — 0-3 respectively for the ratio R,,/R,, a relative 
reduction of resistance of between 20 and 30 %. 
7—We have considered so far only wave motion which is stationary 
relative to the ship, and we examine now a ship advancing through free 
transverse waves which are moving with the velocity appropriate to their 
wave-length. 
Suppose first that the waves are moving in the same direction as the ship. 
With a fixed origin O we now have, instead of (2), 
@ = $,("—ct, y, z) +hVex* cos k(x — V2), (21) 
where V? = g/x, and the additional surface elevation due to the free waves 
is hsink(x— Vt). 
The variable part of the pressure is p 0¢/0t, or 
0g, h K2aq = 7. 9 
— pe~ + gphe sin k(a— V2). (22) 
To calculate R from (4) and (22), transfer to an origin moving with the 
ship. Then the first term in (22) gives the same expression for R, as in (5), 
while for the additional resistance due to the second term we have 
R=- 2aph| | oH e sin{xa — k(V —c) t}dadz. (23) 
This is the same as in (6) for relatively stationary waves, except that Kp, 
¢ are replaced by x, V respectively, and that the phase # has now the 
varying value x(V —c)t. 
For transverse waves hsinx(x+Vt) moving in the opposite direction 
(21) is replaced by 
@ = $,(x —ct, y, z) —hVe cos k(x + Vt), (24) 
and it is easily seen that we get the same result as before with the phase p 
equal to —K(V +c)E. 
The result is that the additional resistance depends only upon the in- 
stantaneous position of the ship relative to the waves. This might have been 
anticipated from the various approximations which have been made. We 
435 
