388 BELL SYSTEM TECHNICAL JOURNAL 



For sounds other than the 1000-cycle reference tone a relation 

 similar to Eq. (5) can be derived, namely, 



/3 = ^1 + 10 log {WIW,), (11) 



where /3i and Wi are corresponding values found from loudness balances 

 for each frequency or complex wave form of interest. If, as is usually 

 assumed, a linear relation exists between /3 and 10 log W, then de- 

 terminations of j3i and Wi at one level are sufficient and it follows that 

 a change in the power level of A decibels will produce a corresponding 

 change of A decibels in the intensity of the sound generated. Ob- 

 viously the receivers must not be overloaded or this assumption will 

 not be valid. Rather than depend upon the existence of a linear 

 relation between /3 and 10 log W with no confirming data, the receivers 

 used in this investigation were calibrated at two widely separated 

 levels. 



Referring again to Table I, the data are expressed in terms of voltage 

 levels instead of power levels. If, as was the case with our receivers, 

 the electrical impedance is essentially a constant, Eq. (11) can be put 

 in the form: 



/3 = /3i-f 20 1og(I7Fi) (12) 



or 



/3 = 20 log V + C, (13) 



where V is the voltage across the receivers and C is a constant of the 

 receivers to be determined from a calibration giving corresponding 

 values of /3i and 20 log Vi. The calibration will now be described. 



By using the sound stage and the technique of measuring field 

 pressures described by Sivian and White ^ and by using the technique 

 for making loudness measurements described in Appendix A, the 

 following measurements were made. An electrical voltage V\ was 

 placed across the two head receivers such that the loudness level pro- 

 duced was the same at each frequency. The observer listened to the 

 tone in these head receivers and then after \\ seconds silence listened 

 to the tone from the loud speaker producing a free wave of the same 

 frequency. The voltage level across the loud speaker necessary to 

 produce a tone equally loud to the tone from the head receivers was 

 obtained using the procedure described in Appendix A. The free wave 

 intensity level /3i corresponding to this voltage level was measured in 

 the manner described in Sivian and \\'hite's paper. Threshold values 

 both for the head receivers and the loud speaker were also observed. 

 In these tests eleven observers were used. The results obtained are 

 given in Table II. In the second row values of 20 log Fi, the voltage 



^ L. J. Sivian and S. D. White, "Minimum Audible Sound Fields," Jour. Acous. 

 Soc Am. 4, 288 (1933). 



