130 DYNAMICAL THEORY OF SOUND 



the calculations, including the determination of the nodes &c., 

 have been greatly extended by Lissajous (1850), Seebeck* (1848) 

 and Lord Rayleigh. 



48. Summary of Results. Forced Vibrations. 



In any one of the preceding cases, and in any particular 

 mode, ra varies inversely as I, and therefore, by 46 (2), the 

 period 27r/n will for bars of the same material vary as Z 2 //c. 

 Hence for bars which are in all respects similar to one another 

 (geometrically) the period will vary as the linear scale. For 

 bars of the same section the period is as the square of the 

 length. As regards the shape and size of the cross- section, 

 everything depends on the radius of gyration K\ thus for bars of 

 rectangular section the frequency varies as the thickness in the 

 plane of vibration, and is independent of the lateral dimension. 

 This latter statement needs, however, some qualification ; it is 

 implied that the breadth is small compared with the length of 

 the bar, or (more precisely) with the distance between con- 

 secutive nodes. When this condition is violated the problem 

 comes under the more complex theory of plates ( 55). 



It is of interest to compare the frequencies of transverse and 

 longitudinal vibration of a bar in corresponding cases. For a 

 bar free at both ends we have, in the gravest transverse mode, 



?i 2 = / ^(m0 4 = ^ / ^x(l-50562) 4 , (1) 



whilst in the gravest longitudinal mode 



(2) 



Hence -, = 7'122y (3) 



71 I 



This explains the relative slowness of the transversal modes. 

 The comparison is due to Poisson. 



We pass over the question of determining the motion 

 consequent on arbitrary initial conditions, by means of the 

 normal functions. In the case of the free-free bar, for example, 

 these are given by the expressions in brackets in equations (20) 

 and (21) of 46. 



* L, F. W, A, Seebeck (180549), professor of physics at Leipzig. 



