64 BELL SYSTEM TECH. MCA L JOURNAL 



it is best to depend upon experimental measurements if accurate results are 

 a factor. 



6.32 Extensional Frequencies 



The resonant frequencies of a bar vibrating along its length, commonly 

 called an extensional mode of motion is derived quite easily from the wave 

 equation in one dimension and is given by 



U y Sin 



'--kVt, " 



where ( = length 



Sii = elastic constant in the direction of propagation 



p = density 



n — 1, 2, 3, 4, etc. 

 This is the case when the length is the greatest dimension. When we con- 

 sider extensional modes along the thickness of a plate, it can be shown that 

 the c constants be employed to account for the lateral inertia in the two 

 directions at right angles to the direction of propagation, (provided that the 

 resulting motion is nearly along the thickness direction). Hence, for thin 

 plates 



As an example of the use of the above equation an X-cut bar vibrating along 

 its length would result in a series of resonant frequencies defined by equation 

 6.4. An X-cut plate vibrating along its thickness would result in a series of 

 frequencies defined by equation 6.5. Applying the appropriate constants 





2Y y 127.9 X 2.65 



V77 — r n kilocycles 6.6 



7(cm) 



and 



, _ ^ / 86.05 X 10^ 

 ^' IX y 2.65 



285 



X(cm) 



n kilocycles 6.7 



This shows that although Young's Modulus is the same in the two directions 

 the resulting frequency constants are different because of the conditions at 

 the boundaries. 



