44 MECHANICAL VIBRATING SYSTEMS 



Following these simpler forms of vibration are combinations of nodal 

 circles and nodal diameters. The frequency of one nodal circle and one 

 nodal diameter, Fig. 3 AG, is 



/i2 = 5.98/01 



The frequency of one nodal circle and two nodal diameters, Fig. 3 AH, is 



/22 = 8.74/01 



The frequency of two nodal circles and one nodal diameter, Fig. 3.4/, is 



/i3 = 11.9/01 



The clamped plate is used in electromagnetic telephone receivers in 

 which the steel diaphragm serves as the armature. See Sec. 10. 2^^. It is 

 also used in carbon microphones. See Sec. 9.2^^. Clamped plate dia- 

 phragms have been used in miniature condenser microphones. The dis- 

 advantage of a plate is the difficulty of mounting a thin plate to give a 

 small mass per unit area for high sensitivity and still have sufficient stiff- 

 ness to yield a high fundamental frequency. 



3.6. Longitudinal Vibration of Bars ^^- ^'^' ^^' ^^. — Consider an entirely 

 free rod of homogeneous material and constant cross section. The 

 simplest mode of longitudinal vibration of a free rod is one in which 

 a loop occurs at each end and a node in the middle, that is, when the 

 length of the rod is one half wavelength. The fundamental frequency of 

 longitudinal vibration of a free rod, Fig. 3.5, is 



/■ - W! 



where / = length of the rod, in centimeters, 



p = density of the material, in grams per cubic centimeter, and 

 ^ = Young's modulus, in dynes per square centimeter. See Table 3. 1 . 

 The overtones of the free rod are harmonics of the fundamental. That is 

 /2 = 2/1, /3 = 3/1, /4 = 4/1, etc., Fig. 3.5. 



The fundamental resonance frequency occurs when the length of the rod 

 is one-half wavelength. This fact provides a means of computing the 

 velocity of sound when the density. Young's Modulus and the frequency 



" Rayleigh, " Theory of Sound," Macmillan and Co., London. 



" Morse, " Vibration and Sound," McGraw Hill Book Co., New York. 



^* Wood, " A Text Book of Sound," Bell and Sons, London. 



^^ Lamb, " Dynamical Theory of Sound," E. Arnold, London, 



