hersey: vibrations of elastic systems 443 



For two bodies of the same material and, also, of the same 

 generalized shape, differentiating (11) gives for K in (12) the 

 more simple expression. 



K = c + 2(1 +*) - (13) 



t 



which does not require observations of stiffness. 



In any event the term C n vanishes when complex stresses 

 are absent, for S will then be independent of Poisson's ratio, 

 and/ therefore a constant. But C i2 is the only term in (12) which 

 involves the masses of the bodies in any way. Hence the curious 

 conclusion that, when complex stresses are absent, n will be 

 independent of both the absolute mass and the mass distribution. 



SOME EXAMPLES 



1. Waves in an infinite elastic plate. The most recent work 

 on this subject 2 affords an instance of equation (1); for while the 

 result as published does not resemble equation (1), it reduces to 



it identically on setting $ (°\) = ^ (2& + 1) \-. r— r^ r 



1 (1 + <?) (1 — 2<r) 



in which k is any integer, the length L of equation (1) being 



taken to be the thickness of the plate. 



2. Resonance periods of similar structures. Let quantities 

 pertaining to the original, full-sized structure be distinguished 

 by the subscript 0; while those referring to the model are without 

 subscripts. Then by (1) and (6) 



7 h = k J#o Jp_ = JSo J> 



n L lElp Q IS 1i 



the first form being preferable when stiffness measurements can 

 not be made; the second when the properties of the material 

 are not known. 



3. Temperature compensation of ordinary tuning fork. For 

 any case of pure bending or stretching, /3 can not enter (4) ; 

 hence c = 0, and n = \ {a + 7), which agrees with (2). As- 

 suming a negative and 7 positive, as is true for steel, compensa- 



2 Lamb, H. Proc. Roy. Soc. 93 (A): 114-128, eq. (80). 1917. 



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