640 BELL SYSTEM TECHSICAL JOURNAL 



(2) Estimates or approximations concerning the total power from 

 the voice. 



Regarding (1) we note that the di\'ergence of waves causes some 

 frequency distortion which is greater, the nearer the source, and be- 

 comes neghgible as the distance increases (see the appendix). We 

 should accordingly expect the peak factors to he different at the 

 speaker's lips. The estimates of total power, however, are as close 

 as their importance necessitates. 



When the data are applied to a case in which the speaker's distance 

 is other than 9 cm., the required power intensity is found by the law 

 of inverse squares and the pressure by the law of inverse distance. 



APPENDIX 



Frequency Distortion in Spherical Waves 



A spherically diverging sound wave (see H. Lamb: ''Dynamical 

 Theory of Sound," page 206) is represented by 



rcS,=J{vot-r) 



where r = radius of the wave front 



(^= velocity potential 



/ = time 



z'o = velocity of sound 



Po = mean density of air 



The pressure 



p=- PoVod<t>/dr 



= PoVo[yf'{vot-r)+j,J{vJ-r)^ 



Let /(!'„/ -r)= sin w ('-^ ), 

 so that 



/, = ^f°(^^ COS. (/-;_-) + ! sin . (/-^)). 



When a wave composed of any number of such components (each 

 having a different pair of values for w and a) diverges from one radius 



to a larger one, it mn nnW changes in size, due to the factor -^ 



r 



but also in sha|)e, liui' lo the factor in the second term. Wiieii r 



