(May 5, 1916] 



SCIENCE 



647 



turn of all the molecules o£ this class during 

 the time t^ is 



The average pressure during the interval t^ is 

 %f.dt/t^, therefore, 



Njn ( f/j + Ml) - =■ Pi- 

 Similarly for all the strips as they successively 

 strike the surface up to the last, where 



Squaring and adding for all values of u 

 from Mj to Ms, 



N,m{sU^^ + m,s-%u, + 2m/) = %Vn- 



But 2ms throughout the wave is zero, 5p/s is 

 the average pressure during the impact of the 

 whole wave, and 2m-/s is u^, the mean square 

 velocity due to vibration, hence after divid- 

 ing by s, 



^nd the same is true of all the other classes of 

 molecules with velocities from U^ to Z7„. If 

 the total number of molecules of all classes is 

 N^ = N^--\-N^-\-N^, etc., the final resultant 

 ■effect after adding all the expressions for P„ 

 Trill be 



Nm{m + M=) = %Pn = P, 



where U^ is the mean square translational 

 velocity =.2F„f7„2/iV. The kinetic theory 

 shows that the pressure is NmV^ when no 

 •sound waves are passing. Hence the increased 

 pressure due to the waves is 

 Nmu'^ = PM-, 



where P is the density of the gas. 



If the equation of the wave motion is 



y=.a cos (Stt/A) (k — Vt), 



u = dy/dt = a(2VA) V sin (2VA) {x — Vt), 



and, since the mean value of sin^ is 1/2, 



M = ^a2(2Vr)' = ia=«)=, 



and the pressure due to the waves is 

 Pu''' = jPa^ti)^, which also represents the maxi- 

 mum kinetic energy or mean total energy of 

 the waves per unit volume, in agreement with 

 Eayleigh's conclusion. 



The same result might have been reached 

 directly by assuming that the pressure of a 

 gas is proportional to the mean square velocity 

 of the molecules, however that velocity may be 

 produced. The symmetrical positive and 

 negative values of u would cause the products 

 C„Ms to vanish in forming the squares of the 

 resultant velocities, so that u^ would be the 

 increase in the mean square velocity, leading 

 to the same result as that given above. 



When we consider the propagation of sound 

 waves in air in molecular rather than in mass 

 terms the expression potential energy loses its 

 meaning. The entire energy of the waves may 

 be expressed in terms of molecular kinetic 

 energy. The conclusion that p = Pu^ is equiv- 

 alent to saying that the pressure due to sound 

 waves is equal to twice the mean density of 

 kinetic energy in the medium. When stated 

 in this form, the results agree with those ob- 

 tained by Planck for the corpuscular theory. 

 The mean kinetic energy is twice as great in 

 one case as in the other. 



In the case of stationary waves, the energy 

 density is evidently twice as great as in the 

 incident waves alone; and the mean square 

 velocity from node to node deduced from the 

 mathematical expression for the wave dis- 

 turbance, and hence the pressure, is likewise 

 twice as great. 



The absolute temperature of a gas is pro- 

 portional to the mean square velocity of the 

 molecules. Ordinarily we should limit this 

 relation to the case where the motion is en- 

 tirely chaotic, not en masse. In either pro- 

 gressive or stationary waves there is an in- 

 creased mean square velocity in the direction 

 of propagation which would record itself as 

 an increase of temperature on any measuring 

 instrument. In particular, at the loops of 

 stationary waves where there are no density 

 changes no lateral change of pressure would 

 occur, while in the direction in which the 

 waves travel there would be an increase of 

 mean square velocity. In a sense there would 

 be a state of polarized temperature. A thin 

 bolometer strip would undoubtedly indicate 

 a higher temperature when the waves are inci- 



