646 



SCIENCE 



[N. S. Vol. XLIII. No. 1114 



Loria; Eeview of Pierpont's Functions of a 

 Complex Variable, by H. P. Manning; 

 " Sborter if^otiees " : Snyder and Sisam's 

 Analytic Geometry of Space, by E. M. 

 Winger; Slichter's Elementary Mathematical 

 Analysis, by L. C. Karpinski; Ford's Auto- 

 morpbic Functions, by A. Emcb ; Gibb's Inter- 

 polation and ISTiimerieal Integration and Carse 

 and Shearer's Fourier's Analysis and Periodo- 

 gram Analysis, by M. Bocher; Herglotz's 

 Analytische Fortsetzung des Potentials ins 

 Innere der anziehenden Massen, by W. R-. 

 Longley; Lange's Das Schachspiel, by L. 0. 

 Karpinski; Ince's Descriptive Geometry and 

 Pbotogrammetry, by V. Snyder; "!N"otes"; 

 and " l^ew Publications." 



SPECIAL ARTICLES 

 THE PRESSURE OF SOUND WAVES 



In his " Warmestrahlung "i Planck, aft'^r 

 proving from electromagnetic theory that the 

 pressure of radiation equals the volunae den- 

 sity of radiant energy, shows that the corpus- 

 cular theory of light would give a pressure 

 twice as great. From this he infers that the 

 Maxwell radiation pressiire can not be deduced 

 from energy considerations, but is peculiar to 

 the electromagnetic theory and is a confirma- 

 tion of that theory. The implied conclusion 

 is that mechanical waves would not exert a 

 pressure of this magnitude. It may be well to 

 recall, therefore, that Lord Rayleigh has 

 shown, from energy consideration,^ that trans- 

 verse waves in a cord exert a pressure equal 

 to the linear energy density, and that sound 

 waves in air must cause a pressure equal to the 

 volume density of energy in the vibrating 

 medium. Altberg' has made the conclusions 

 of Rayleigh the basis of a method of deter- 

 mining the intensity of sounds. 



As the pressure due to sound waves in a gas 

 must be ultimately the result of molecular im- 

 pacts, it would seem probable that the magni- 

 tude of this pressure may be determined from 

 the elementary kinetic theory, and this proves 



1 ' ' WaimestraHung, ' ' 2d ed., p. 58. 



2 Phil. Mag., 3, 338, 1902. 



3 Ann. der Fhys., 11, 405, 1903. 



to be the case. Consider an extended wave 

 incident normally on a unit surface. Accord- 

 ing to the kinetic theory, the molecules which 

 strike this surface are reflected with the same 

 velocity that they had just before impact. As 

 the surface is small in comparison with the 

 extent of the wave front, we need not follow 

 the history of these reflected molecules, which 

 will immediately become dispersed in the pass- 

 ing wave in all directions. In other words, 

 under these conditions no stationary waves 

 will be formed by reflection, and we may con- 

 fine our attention to the efl^ect of the incident 

 wave. Of course there will also be increased 

 pressure on the rear surface due to the dif- 

 fracted waves, but this will not aifeet the pres- 

 sure on the front surface. At the instant that 

 the wave front strikes the surface imagine the 

 whole wave length divided into thin strips 

 parallel to the surface, s in number and each 

 of thickness x, so that sx is equal to one wave- 

 length. The velocities of displacement due to 

 the wave are mass effects, but it seems proper 

 to add them to the different individual veloc- 

 ities of the gas molecules which move en 

 masse. Let the velocities of wave displace- 

 ment in the successive strips be u^, u^, . . . Ug. 

 The component velocities of translation of the 

 gas molecules normal to the surface are U^, 

 U., . . . Un- The two other components con- 

 tribute nothing to the pressure on the surface. 

 The resultant velocity of the molecules having 

 a velocity of translation V^ in the first strip 

 will be fJj H- M,. As they are reflected with 

 the same velocity, the change of momentum 

 of each molecule is 



%m{TJ^-\-u,)=.f.dt, 



where m is the mass of each molecule and 

 f.dt the impulse of the force during collision. 

 If N^ is the number of molecules per unit 

 volume having the velocity U^, the number in 

 the strip of thickness x is N^x and if i, is the 

 time required for the strip to move a distance x, 



N,x = N,(U, + u,)t,. 



Taking accoimt of the fact that half the 

 molecules of this class will be moving away 

 from the surface, the total change of momen- 



