302 Mr. stokes, ON THE FRICTION OF FLUIDS IN MOTION, 



equilibrium, and conceive a series of plane waves to be propagated in the direction of tlie 

 axis of X, so that u =/(*'> t), v = 0, w = 0. Let p^ be the pressure, and p^ the density of 

 the fluid when it is in equilibrium, and put p = p,+ p- Then we have from equations (11) 

 and (12), omitting the square of the disturbance, 



1 dp du du dp' 4 d^u 



p, dt dx dt dw 3 dx 



Let JiAp be the increment of pressure due to a very small increment Ap of density, the 

 temperature being unaltered, and let m be the ratio of the specific heat of the fluid when 

 the pressure is constant to its specific heat when the volume is constant ; then the relation 

 between p and p will be 



p'=mA{p- p) (20) 



Eliminating p and p from (19) and (20) we get 



d'u , d'u 4/u dJ'u 



^ rn A — ^ 



d^ dx^ Sp^tdx' 

 To obtain a particular solution of this equation, let u = (p (t) cos h \j/ {() sin . Sub- 



stituting in the above equation, we see that <p (t) and \j/{t) must satisfy the same equation, 

 namely, 



</>" (t) + '^-^mA<l> (i) + ^J^ <p' it) = 0, 



the integral of which is 



,/„ 2ir6r „, . ■2irbt\ 

 <p (t) = e-" IC cos — - + C sin — — j , 



where c= — — — , V = m A ; — -, C and C being arbitrary constants. Taking the same 



3X p^ S^'ft 



expression with different arbitrary constants for \|/^ (t), replacing products of sines and cosines 

 by sums and differences, and combining the resulting sines and cosines two and two, we see 

 that the resulting value of u represents two series of waves propagated in opposite directions. 

 Considering only those waves which are propagated in the positive direction of x, we have 



M= Ce-'^'cosj— (bt- X) +cA (21) 



We see then that the effect of the tangential force is to make the intensity of the sound 

 diminish as the time increases, and to render the velocity of propagation less than what it 

 would otherwise be. Both effects are greater for high, than for low notes ; but the former 

 depends on the first power of n, while the latter depends only on (ix\ It appears from the 

 experiments of M. Biot, made on empty water pipes in Paris, that the velocity of propagation 

 of sound is sensibly the same whatever be its pitch. Hence it is necessary to suppose that for air 



fX' . . • P 



— — ; is insensible compared with ^ or — . I am not aware of any similar experiments made 

 ^ P: ft 



on water, but the ratio of ( — ) to A would probably be insensible for water also. The 



diminution of intensity as the time increases is, in the case of plane waves, due entirely to 

 friction ; but as we do not possess any means of measuring the intensity of sound the theory 

 cannot be tested, nor the numerical value of y. determined, in this way. 



