500 VISCOSITY. [CHAP, xi 



In the case of free waves, putting X = and assuming that 



we find X 2 + yuX + & 2 c 2 = 0, 



whence 



X = -i/ii(^c 2 -^ 2 ) .................. (3). 



If we neglect the square of p/hc, this gives, in real form, 



% = Ae-^cos{k(ctx) + 6} .................. (4). 



The modulus of decay is 2/ur 1 , and the wave-velocity is (to the 

 first order) unaffected by the friction. 



To find the forced waves due to the attraction of the moon we 

 write, in conformity with Art. 178, 



where n is the angular velocity of the moon relative to a fixed 

 point on the canal, and a is the earth s radius. We find, assuming 

 the same time-factor, 



Hence, for the surface-elevation, we have 



* 



( *7\ 



where H/a =f/g, as in Art. 177. 



To put these expressions in real form, we write 



where &amp;lt; ^ &amp;lt; 90. We thus find that to the tidal disturbing force 

 X = -fsm2(rit+^ + e^ ..................... (9) 



corresponds the horizontal displacement 



and the surface-elevation 



x 



= * cos 



