288 AUDITORY BIOPHYSICS 



this increment in sensation at low than at high sensation levels, so that 

 the experienced increment in sensation (AS) is inversely proportional to 

 the stimulus (I). Combining these two into a mathematical statement, 

 we may write 



A£ = fcy 



where k is the constant of proportionality. In consequence of this 

 statement we deduce, by summing both the sensation and the stimulus 

 from their minimum threshold values to the maximum values attained by 

 each, that 



s T AI 



r as = k r — 



So lo I 



so that 



S - S = k (log / - log /„) 



When the stimulus I is Iq, the experienced sensation is (minimum) 

 threshold; i.e., S — 0. Then 



S = k log e 



where k is a constant whose magnitude depends on the kind of stimulus 

 and also upon the kind of receptor used to measure the stimulus. In the 

 case of sound the value of k varies only with the frequency of the source 

 for a normal ear excited at moderate intensities. 



The above statement is known as the psychophysical law of Weber- 

 Fechner. It states that when the experienced sensation is increased by 

 equal amounts (in arithmetic progression) the stimulus must vary in a 

 geometric progression. The experienced sensation must therefore vary 

 as the natural logarithm of the stimulus. 



Except in precision work or at great intensities it is assumed that the 

 response of the aural mechanism follows very closely such a logarithmic 

 scale. Thus, as the intensity level of the sound field rises, the amount of 

 power required to develop successive aural responses becomes increas- 

 ingly greater. Such a scale is started at the intensity level 7 , where the 

 ear first experiences a sound sensation, i.e., the " threshold of audibility " 

 identified by the contour marked in Fig. VII-19. 



As the intensity rises, an upper limit of loudness is eventually reached 

 where the experienced loudness is replaced by a tickling sensation in the 

 middle ear which sensation is rapidly followed by pain. This limit is 

 called the " threshold of feeling." Intensities producing the sensations 



