134 Prof. C. V. Raman and Mr. Ashutosh Dey on the 



expression for the forced vibration is then of the form 



2,1 . nirx . mrosn 2nirrt 

 + S-aSia -sin cos — ^— , 



and is thus similar to that of a string plucked at x in the 

 same direction as the periodic impulses or in the opposite 

 direction, according as the natural frequency is greater or 

 less than the frequency of the impulses *. If the periodic 

 force, instead of being impulsive in character, has a finite 

 constant value during a part 2/3 of the period and zero at 

 other times, the maintained vibration in the two extreme 

 cases assumes the form 



^ 1 . nirx . W7ro? . 2nirrQ 2wrrt 



+ 2,-ssm ,sin- ,sin — rr. — •> cos — ~j— . 



— i n 6 a u T 1 



If ft be small, this is practically of the same type as the 

 expression for a plucked string in respect of the first few 

 harmonics, but would differ appreciably from it in respect of 

 the harmonics of higher order. 



The next step is to introduce a correction in the expression 

 for the impressed force on account of the neglected terms 

 by , cy 2 , &c. On substituting the value of y first found in 

 these terms and simplifying the product F(y )f(t), it is 

 seen that the correction results only in alterations of the 

 amplitudes and phases of the harmonic components of the 

 impressed force, but no new terms are introduced of which 

 the frequency is not the same as that of the field or a 

 multiple thereof. This shows that the corrections cannot, 

 by themselves, result in an alteration of the frequency of 

 the forced vibration, so long as we assume, in the first 

 instance, that y has the same frequency as the field of force. 

 They may, however, result in the impressed force (and 

 therefore also the maintained motion) including such partial 

 components as are absent in the field itself. 



A consequence of the preceding formulae, which is of 

 particular importance, is that, when the impressed force is 

 of an impulsive character, the corrections by , cy 2 , &c, when 

 introduced cannot result in any alterations in the relative 

 amplitudes and phases of the components of the maintained 

 vibration. For the product ~F(y )f(t) is zero at all times, 

 except at the particular instant in each period at which the 

 impulse acts, and, as these epochs are fixed, any change in 



* It assumed, of course, that the free periods of the wire form an 

 harmonic series. This may be subject to modification if the wire is- 

 imperfectly flexible or yields at the ends. 



