Mr. stokes, on THE THEORY OF OSCILLATORY WAVES. 447 



8. When the depth of the fluid is very great compared with the length of a wave, we may 

 without sensible error suppose h to be infinite. This supposition greatly simplifies the expressions 

 already obtained. We have in this case 



<l> = - ace''"" sin ma; (21), 



J/ = a cos TO*' — 1 jna" cos 2m.r (22), . 



„ TO TT 2 ^X 



A; = 0, A = - = - , c^ = 5— , 



2 \ 2iv 



the y in (22) being the ordinate of the surface. 



It is hardly necessary to remark that the state of the fluid at any time will be expressed by 

 merely writing x — ct in place of x in all the preceding expressions. 



9. To find the nature of the motion of the individual particles, let x + ^ be written for x, y + ]] 

 for y, and suppose x and y to be independent of t, so that they alter only in passing from one 

 particle to another, while ^ and r) are small quantities depending on the motion. Then taking the 

 case in which the depth is infinite, we have 



— = ?/ = - mace"'"'*'*''' cos to (x + ^ - cf) = - mac e'"'" cos m (x - ct) + nv'ac e'""" sin to (x - ct) .P 



+ m'ace''"'' cos m (x - ct) . >;, nearly, 



— = u = TOace"*"'"'''''' sin m (x + ^ - ct) = mace^*"^ sin to {x - ct) + m^ace'"'" cos to (x - ct) . ^ 

 (if 



- ni'ac e'"'" sin to (x - ct) . r), nearly. 



To a first approximation 



^ = ae'"" sin to {x ~ ct), r) = ae'"" cos to (x - ct), 



the arbitrary constants being omitted. Substituting these values in the small terms of the preceding 

 equations, and integrating again, we have 



^ = oe""^ sin TO (ai - cO + M'a'c^e-'"'^ 



ri = ae"*"* cos m {x - ct). 



Hence the motion of the particles is the same as to a first approximation, with one important 

 difference, which is that in addition to the motion of oscillation the particles are transferred forwards, 

 that is, in the direction of propagation, with a constant velocity depending on the depth, and 

 decreasing rapidly as the depth increases. If U be this velocity for a particle whose depth below 

 the surface in equilibrium is y, we have 



/ 2 7r\ i -1 "» 

 f/=TO=a=ce--'"-^ = aM— g' e' ~ (23). 



The motion of the individual particles may be determined in a similar manner when the deptli 

 is finite from (18). In this case the values of ^ and >/ contain terms of the second order, involving 

 respectively sin 2to {x - ct) and cos 2to {x — ct), besides the term in ^ which is multiplied by t. 



The most important thing to consider is the value of U, which is 

 (e"'* _ e-'«'')V 



U = m'a'c , "^ -, — (24). 



Since U is a small quantity of the order a', and in proceeding to a second approximation the 

 velocity of propagation is given to the order a only, it is immaterial which of the definitions of 

 velocity of propagation mentioned in Art. .3, we please to adopt. 

 Vol. VIII. Part IV. 3 M 



