394 BELL SYSTEM TECHNICAL JOURNAL 



Since, during the time of decay, Si decreases uniformly with time, and 



therefore Lt also, then for a thousand cycle note, evaluating our 



integral we have 



Lt,T, = 2K (3) 



or 



St^Ti = 2K, 

 where 



T, = t,- to. 



This last expression is practically in the form in which this condition 

 was first stated by Lifschitz.'' In (3) there are three unknowns and a 

 fourth is implied, namely, the power of the source, E. 



We now turn our attention to finding the relation between the 

 volume of a room and the reverberation time dictated by the stated 

 condition. Following P. E. Sabine let us take the rate of emission of 

 the source, E to be 10^° cubic meters (35.3 X 10^" cubic feet) of sound 

 of threshold density per second. Now ^ 



4V, 4 X 35.3 X 10'" 



Ti = — loge , 



ca c • a 



where F is the volume of the room in cubic feet. 



c is the velocity of sound, 1120 feet per second. 

 a is the number of absorption units in sq. feet and '^ 



T c ini 4 X 35.3 X 10'" 

 Li, = St, = 10 logio ~ 



If we should substitute these values in (3) we would obtain a relation 

 between V, a, and K which must be satisfied when condition (2) is 

 satisfied. In other words, this relation would specify the amount of 

 absorption, for a one thousand cycle note, a room should have if it 

 complies with (2). 



If we assume Sabine's well known formula, namely, 



^ _ .057 



where T is the reverberation time in seconds we may express this 

 relation in terms of V, T, and K with the result 



(2KYI- 

 10.40 + log To, - log V = ^ ^g3 j.^ ,., , (4) 



' See Crandall "Theory of Vibrating Systems and Sounds," page 211. 

 * See Crandall "Theory of Vibrating Systems and Sounds," page 210, and the 

 definition of sensation level. 



