275-277] EFFECT OF FRICTION ON PHASE. 499 



On the other hand, when 6 2 &amp;gt; 4ac the amplitude continually 

 increases as the speed cr diminishes, tending ultimately to the 

 equilibrium-value C/c. 



It also appears from (15) and (12) that the maximum dis 

 placement follows the maximum of the force at an interval of 

 phase equal to e 1? where 



(19). 



If the period be longer than the free-period in the absence of 

 friction this difference of phase lies between and 90 ; in the 

 opposite case it lies between 90 and 180. If the frictional 

 coefficient b be relatively small, the interval differs very little from 

 or 180, as the case may be. 



The rate of dissipation is bq 2 , the mean value of which is easily 



found to be 



bC 2 



This is greatest when &amp;lt;r = (c/a)*. 



As in Art. 165, when the oscillations are very rapid the formula 

 (11) gives 



q = - Q/o*a (21), 



approximately ; the inertia only of the system being operative. 



On the other hand when a is small, the displacement has very 

 nearly the equilibrium -value 



n _ n/ r f99\ 



(/ V/v \L&). 



277. An interesting example is furnished by the tides in an 

 equatorial canal*. 



The equation of motion, as modified by the introduction of a 

 frictional term, is 



where the notation is as in Art. 178-f. 



* Airy, &quot; Tides and Waves,&quot; Arts. 315... 



f In particular, c 2 now stands for gh, where li is the depth. 



322 



