376 SURFACE WAVES. [CHAP. IX 



and also enables us to write down at once the proper modifications 

 of the remaining formulae of the preceding Art. 



Thus, we find, for the component displacements of a particle, 



cosh k (y + h) n ^ 

 x = x x - x.j = a - . ,,, - cos UUf - crt), 



Smhkh . ..(8). 



sinh k (y + Ti) . /7 J 

 y = yi -y a =a . - am (to - 



This shews that the motion of each particle is elliptic-harmonic, 

 the period (2?r/cr, = X/c,) being of course that in which the dis 

 turbance travels over a wave-length. The semi-axes, horizontal 

 and vertical, of the elliptic orbits are 



cosh k (y + h) siuhlc(y + h) 



(L - : \ T~l -- dilU. (Jb - : ; f^ -- , 



sinh kh smh kh 



respectively. These both diminish from the surface to the bottom 

 (y = h), where the latter vanishes. The distance between the 

 foci is the same for all the ellipses, being equal to a cosech kh. It 

 easily appears, on comparison of (8) with (3), that a surface-particle 

 is moving in the direction of wave-propagation when it is at a crest, 

 and in the opposite direction when it is in a trough*. 



When the depth exceeds half a wave-length, e~ kh is very small, 

 and the formulae (8) reduce to 



x = ae ky cos (kx - a-t), \ ,~. 



&quot; 



so that each particle describes a circle, with constant angular 

 velocity cr, = (^X/27r)^t. The radii of these circles are given by 

 the formula a^ y , and therefore diminish rapidly downwards. 



In the following table, the second column gives the values of sech kh 

 corresponding to various values of the ratio A/A. This quantity measures the 

 ratio of the horizontal motion at the bottom to that at the surface. The third 

 column gives the ratio of the vertical to the horizontal diameter of the elliptic 

 orbit of a surface particle. The fourth and fifth columns give the ratios 

 of the wave-velocity to that of waves of the same length on water of infinite 

 depth, and to that of long waves on water of the actual depth, respectively. 



* The results of Arts. 217, 218, for the case of finite depth, were given, substan 

 tially, by Airy, &quot;Tides and Waves,&quot; Arts. 160... (1845). 

 t Green, /. c. ante p. 875. 



