TRANSVERSE VIBRATION OF BARS 



39 



where a = thickness of the bar, in centimeters, in the direction of vibration. 

 For a circular cross section 



where a — radius of the bar in centimeters. 



The modes of vibration of a bar clamped at one end are shown in Fig. 3.2. 

 The table below gives the position of the nodes and the frequencies of the 

 overtones. 



It will be seen that the overtones are not harmonics. The first overtone 

 of a bar or reed has a higher frequency than the sixth harmonic of a string. 

 The tuning fork is the most common example of a bar clamped at one 

 end, because it can be considered to be two vibrating bars clamped at the 

 lower ends. The overtone or the high frequency sound of a tuning fork 

 is quickly damped out leaving almost a pure sound. 



B. Bar Free at Both Ends. — Consider a perfectly free bar (Fig. 3.2). 

 The fundamental frequency is given by 



h = 



1.1337r 



^K^ 



3.3 



where / = length of the bar, in centimeters. 



All the other quantities are the same as in equation 3.2. 

 The modes of vibration of a perfectly free bar are shown in Fig. 3.2. 

 The table which follows gives the position of the nodes and the frequencies 

 of the overtones. 



