290 AUDITORY BIOPHYSICS 



small, but that at greater loudness levels the intensity must get 

 progressively larger and eventually increase enormously to produce the 

 same change in loudness. The change in intensity to produce these 

 equal increments in loudness may be written 



AL = L\ — L 2 = k log, 



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Unfortunately, logarithmic tables to the base e are not as convenient to 

 use as logarithmic tables to the base 10; therefore for convenience we 

 write 



AL = k' log 10 (jj 



where k' includes k and the conversion factor which changes base e to 

 base 10. Since the intensity is proportional to the square of the pres- 

 sure, we may write 



AL = 2k r log — 



where the root-mean-square pressure variation corresponding with the 

 threshold of audibility at 1000 cycles is taken as 0.0002 dyne/cm 2 

 (p = 2.8 X 10 -4 dyne/cm 2 ), or in power units as 10 -16 watt/cm 2 as in 

 Fig. VII-6. 



The most convenient choice of the intensity or pressure that may be 

 used as a standard for comparison depends on the problem under con- 

 sideration. In the auditory sensation area of Fig. VII-6, the intensity 

 line corresponding with 1 dyne/cm 2 was used in the literature before 

 1936 as the zero level; i.e., both p 2 and the constant were chosen as 

 unity. 



If, however, I\ and I 2 are the two different amounts of power to be 

 compared, then the logarithm of the ratio (to the base 10) is a dimension- 

 less unit designated asa" bel," a name adopted in honor of Alexander 

 Graham Bell. Its origin is traced to the telephone engineer's use of 

 a " mile of standard cable " or the actual level of sound which existed 

 after signals from a telephone transmitter had passed through a mile of 

 telephone cable. Subsequently, when a unit was needed to express the 

 rates of the smaller acoustical quantities, the " decibel," which is -^o 

 of 1 bel, was used. The decibel notation is now simply a convenient 

 means of expressing a gain or loss ratio and does not signify any definite 

 physical unit. It has no meaning unless a reference level and the units 

 in which the reference level is measured are designated. 



As an illustration of its use, let us examine what happens to a culture 



