258 AUDITORY BIOPHYSICS 



Energy of a Particle Executing Simple Harmonic Motion 



The total energy content of the moving air particle is the sum of its 

 kinetic and potential energies at any instance. Its kinetic energy, how- 

 ever, is a maximum at its midpoint, where its potential energy is zero. 

 Its kinetic energy is %mv 2 max = ^m(2TnA) 2 , so that its energy at any 

 point in its path is 



E = 2-w 2 mn 2 A 2 



The energy of an air particle in a longitudinal wave, therefore, is 

 proportional to its mass, to the square of the frequency, and to the 

 square of the amplitude. 



Intensity 



The energy is passed along from particle to particle at the velocity of 

 the propagating longitudinal wave. The rate at which this energy is 

 transferred across unit area is called the intensity of the wave. The in- 

 tensity is the product of the energy density, or energy per unit volume, 

 and the velocity of energy propagation and hence is expressed in ergs 

 per square centimeter times seconds or watts per square centimeter. 

 The energy per unit volume is 



E 2ir 2 mn 2 A 2 



Unit volume m/p 



where p is the density of the medium. Let the velocity of energy propa- 

 gation be V; then the intensity in ergs per square centimeter times 

 seconds is 



/ = 27r 2 n 2 A 2 pV 



If the longitudinal waves have a velocity V in air under standard con- 

 ditions, and the frequency is in the acoustic range, the intensity of the 

 acoustic energy in the object space is proportional to the square of the 

 frequency and to the square of the amplitude. 



It has been found experimentally that the minimum audible intensity 

 is about 10 -16 watt/cm 2 at a frequency of 1000 cycles. What is the 

 amplitude of an air particle which produces this just-audible intensity? 



Here n = 1000, V (air) - 343 meters/sec for p = 0.00121 gram/cm 3 

 at 20° C, 760 mm pressure. I = 10 -16 watt/cm 2 = 10 -9 erg/cm 2 X sec. 

 Solve for A in the intensity equation . Since 



I = 2ir 2 n 2 A 2 P V 

 10 -9 = 2tt 2 X 10 6 X A 2 X 121 X 10" 5 X 343 X 10 2 

 i = 8X 10~ 10 cm 



