Vibrations by Forces of Double Frequency. 157 



The general equation of vibration for a stretched string of 

 periodic density is 



(2ttx , . 2irx Attx 



p + p } cos —j- + pi sin —j- + p2 cos — y~ 



+p/sm __ + ...)_ =T _, . . . (68) 



/ being the distance in which the density is periodic. We 

 shall suppose that p/, p 2 ', . . . vanish, so that the sines dis- 

 appear, a supposition which involves no loss of generality 

 when we restrict ourselves to a simple harmonic variation of 

 density. If we now assume that w a e ipt , or oc cos pt, we 

 obtain 



^ + (© + 2e i cos2f + 2© 2 cos4| + ...)^=0, . . (69) 

 at; 



where %=7rx/l, and 



© =fe 20 1 =fe&c.;. . . (70) 



and this is of the form of Mr. Hill's equation (2). 



When c is real, we may employ the approximate solutions 

 (41), (44). The latter (with £ written for t) gives, when 

 multiplied by cos pt or smpt, the stationary vibrations of the 

 system. From (41) we get 



cos[> + (c-2)g] cosfrl + cg] 



in which, if c = 1 nearly, the two terms represent waves pro- 

 gressing with nearly equal velocities in the two directions. 

 Neither term gains permanently in relative importance as x is 

 increased or diminished indefinitely. 



It is otherwise when the relation of <s) to % x is such that c 

 is imaginary. By (44) the solution for w, assumed to be 

 proportional to e ipt , now takes the form 



w = Re-* \ ®^ pt - + ( 1 - O - 2i S y< pt +& \ ' 



(72) 



+ $e s t {0^-fl + (1 -0 O + 2isy( pt +^\ 



: 



Whatever may be the relative values of R and S, the first 

 solution preponderates when x is large and negative, and the 

 second preponderates when x is large and positive. In either 

 extreme case the motion is composed of two progressive waves 

 moving in opposite directions, whose amplitudes are equal in 

 virtue o/(43). 



The meaning of this is that a wave travelling in either 



