15C 159.] WAVES IN DEEP WATER. 189 



If we compare (32) with (30), we see that a surface-particle 

 moves in the direction of the waves when it coincides with a 

 crest, and in the opposite direction when it coincides with a 

 trough. 



If the ratio - be infinite, the expressions (32) become 

 A&amp;lt; 



f = ae ky cos k(x -ct), y = ae* 1 * sin k (x ct), 



so that each particle describes a vertical circle with uniform angu 

 lar velocity. The radii of these circles, given by the formula oe**, 

 diminish rapidly with increasing depth*. At a depth below the 

 surface equal to the wave-length the motion is to that of the sur 

 face in the ratio e~- n : 1, or 1 : 535 nearly. 



158. The energy of a system of waves of the kind now under 

 examination is found as follows. Suppose two vertical planes 

 drawn perpendicular to the ridges of the waves, at unit distance 

 apart. The potential energy of the fluid between these planes is 

 ^gpfifdx, where y is the elevation above the mean level at the 

 point x. Substituting from (30), and integrating, we find \gp&amp;lt;r\ 

 for the potential energy per wave-length. 



The kinetic energy is 



6-2*(y + *) + 2 cos %k(x - ct)}dxdy. 



This gives when integrated with respect to x over a wave-length, 

 and with respect to y between the limits h and 0, 



or, by (28) and (31), J 



The kinetic and potential energies are therefore equal. 



159. So long as we confine ourselves to a first approximation 

 all our equations are linear; so that if ^, &amp;lt;/&amp;gt; 2 , c. be the velocity- 



* These results were given by Green, Catnb. Trans. Vol. vn. 1839. 



