500 VISCOSITY. [CHAP, xi 



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



we find X 2 + /*X + & 2 c 2 = 0, 



whence 



(3). 

 If we neglect the square of jju/kc, this gives, in real form, 



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



The modulus of decay is 2/A" 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, 



" 2 - 



t JL ____ _ ________ g2t(n' 



*' 'c'-wV + Jvm'a* 

 Hence, for the surface-elevation, we have 



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



To put these expressions in real form, we write 



where < ^ < 90. We thus find that to the tidal disturbing force 



+\ ..................... (9) 



corresponds the horizontal displacement 



and the surface-elevation 



x 



"= 



