Mr. R. H. M. Bosanquet on the Theory of Sound. 419 



when t= j (quarter period), there is no velocity anywhere, 

 but there is a maximum displacement, 



Y=2, 



and a maximum pressure^ 



2a cos — x, 



A. 



1 ^ 47TCI . 2-7T 



r4II.— — sin— ~ x. 

 A, A 



Consider a tube having the length of the velocity of sound, 

 in which such a stationary wave is maintained by the trans- 

 mission from the opposite ends of the equal and opposite wave- 

 systems. The energy in the tube is to be found. 



We have shown that the form of energy alternates entirely 

 between potential and kinetic. In either form it is identical 

 in distribution and magnitude with the kinetic or potential 

 energy of an ordinary wave of transmission in air of the same 

 amplitude ; so that there is only one quantity in the stationary 

 wave where there would be two in the ordinary wave of trans- 

 mission. The energy in the tube of length v is therefore 



i-in»(y) ! , (A=2«,) 



or half that in a wave of transmission of amplitude A. And 

 this may be written 



•{^^}i 



which is twice the energy of either single stream, or the sum 

 of both. 



Cor*— Putting # = 0, we have the conditions of a loop sur- 

 face ; and it follows without difficulty that, If a disk of air 

 oscillate with maximum velocity V and without change of 

 density, the energy per second through the disk consists of 



two equal and opposite streams each = *g- V 2 per second, 



making a total transfer of ^- V 2 per second through the disk. 



(Section unity.) 



Prop. II. — If any two wave-systems of the same wave- 

 length (pitch, or periodic time) meet in air, travelling in 

 opposite directions, their direct superposition carries the energy 

 of both* 



2E 2 



