STRINGS 81 



the formulae (2). The value of ft in terms of F is found from 

 the consideration that the force must just balance the pull of 

 the string on this point, i.e. 



F C os(pt + ct)=P yi '-Py 2 ' ............... (4) 



for x = a. This leads to 



. px . p(l a) 

 sm sin-- - 



sn 



c c 



The formula for y z differs only in that the letters x and a are 

 interchanged ; we have here an instance of the reciprocal theorem 

 of 17, according to which the vibration at a point x due to a 

 periodic force at a must be the same as the vibration at the 

 point a due to an equal force (of the same period) at x. 



The amplitude becomes as a rule great when sm(pljc) is 

 small, i.e. when the imposed period approaches a natural period 

 of the whole string. An indeterminate case occurs when 

 sin (pale) and sin (pile) = simultaneously, the point x = a 

 being then a node. 



29. Qualifications to the Theory of Strings. 



We have in 26, 27 considered the relative amplitudes of 

 the different harmonics when a string is excited in various ways, 

 but we must not assume that the corresponding relative inten- 

 sities are accurately reproduced in the resulting sound-waves, 

 which are started indirectly through the sounding board. 

 If we neglect the reaction on the string, which may for a 

 considerable number of vibrations be insensible, we may regard 

 the string as exerting on each bridge a force proportional to 

 the value of dy/dx there*, as given by the respective formula. 

 The differentiation introduces a factor s in the coefficient of 

 the 5th harmonic, and so increases the importance of the 

 higher modes. On the other hand, the amplitude of vibra- 

 tion of the sounding board due to a simple-harmonic force 

 of given amplitude, will vary somewhat with the frequency, 



* Thus in the case of the plucked string it appears from Fig. 27 that 

 the pressure on each end alternates between two constant values of opposite 

 sign. 



L. 6 



