OF FLUIDS ON THE MOTION OF PENDULUMS. [95] 



M. Biot's, unless we supposed the term expressing this effect to be so small that it 

 might be disregarded. I am now prepared to calculate the numerical value of the term in 

 question, and so decide whether the theory is or is not at variance with the result of M. Biot's 

 experiment. 



According to the expression given in the paper just mentioned, we have for the propor- 

 tionate diminution in the velocity of propagation 



8 ^m' 2 

 9\"V*' 



A being the length of a wave, and V the velocity of sound. To talce a case as disadvantageous 

 as possible, suppose X only equal to one inch, which would correspond to a note too shrill to 

 be audible to human ears. Taking the velocity of sound in air at 1000 feet per second, there 

 results for the common logarithm of the expression above written 11.0428, so that a wave would 

 have to travel near 100000000000 inches, or about 1578000 miles, before the retardation due 

 to friction amounted to one foot. It is plain that the introduction of internal friction leaves 

 the theory of sound just as it was, so far as the velocity of propagation is concerned, at least 

 if the sound be propagated in free air. 



The effect of friction on the intensity of sound depends on the first power of y.'. In the 

 case of an indefinite succession of plane waves, it appears that during the time t the amplitude 

 of vibration is diminished in the ratio of 1 to e _c ', and therefore the intensity in the ratio of 

 l to e~' ict , where 



8wV 



Putting A = 1 and t - l we get 1 to 0*4923, or 2 to 1 nearly, for the ratio in which the intensity 

 is altered during one second in the case of a series of waves an inch long. The rate of dimi- 

 nution decreases very rapidly as the length of wave increases, so that in the case of a series of 

 waves one foot long the intensity is altered in one second in the ratio of 1 to 0'995095, or 201 

 to 200 nearly. It appears then that in all ordinary cases the diminution of intensity due to 

 friction may be neglected in comparison with the diminution due to divergence. If we had 

 any accurate mode of measuring the intensity of sound it might perhaps be just possible, in 

 the case of shrill sounds, to detect the effect of internal friction in causing a more rapid dimi- 

 nution of intensity than would correspond to the increase of distance from the centre o*f diver- 

 gence. 



