218 DR. LOUIS VESSOT KING ON THE PROPAGATION OF SOUND IN THE FREE 
The discussion of the mode of propagation of waves according to the above 
equation has been dealt with by Riemann* * * § (i860), RankineI (1870), HugoniotJ 
(1887, 1889). In an important memoir, Rayleigh§ (1910) gives a critical and 
historical account of the subject under discussion, and discusses in further detail 
special solutions of the problem of finite waves and the influence of viscosity and 
thermal conductivity on their mode of propagation. It is there shown that in the 
case of adiabatic propagation the velocity u in the medium due to the wave-motion 
expressed by equation (18) is rigorously given by 
u =f[x—(a + eu)t\, .(19) 
where e = 2 (y+ l) and /[•••] is an arbitrary function depending on the circumstances 
of the motion at the instant t = 0. 
The above solution reduces to the case of isothermal propagation (y = l) previously 
considered by Poisson, 
u = f[x— (a + u)t\ . (20) 
The discussion of the propagation of a wave according to equation (19) in the 
particular harmonic form 
u — U sin 27r [x/X — ( l+eu/a) nt ] .(21) 
has been discussed by Stokes|| ; it was pointed out that a wave thus represented 
undergoes a gradual change of form, the condensation overtaking the rarefaction 
until the motion becomes physically impossible for the least value of t for which 
dx/du — 0. At this stage other phenomena due to physical causes not included in 
the above statement of the problem come into play.IF Applying this condition to 
(21) we obtain for the interval t the relation 
< 2.Tre\nt\= a (l/U) (l— tr/U 2 ) -i = (l/U) sec 2?r [x/X— (l +eu/a) nt]. 
from which it follows that discontinuity must occur after an interval not less than 
that given by 
t\ = ct/(2irUen) .(22) 
The distance x 1 travelled in this interval lies between two limits assigned by the 
inequality 
a 2 /(27rUen) < x 1 < a 2 (l+eU/a)/(27rUm),.(23) 
* Riemann, ‘ Gott. Abh.,’ t. VIII., p. 43 (1858-9); ‘Werke,’ 2nd Ed., Leipzig, 1892, p. 157. 
t Rankine, ‘Phil. Trans.,’ vol. 160, p. 277, 1870; ‘Misc. Sc. Papers,’ p. 530. 
I Hugoniot, ‘ Journ. de l’Ecole Polytechnique,’ 1887, 1889. 
§ Rayleigh, “Aerial Plane Waves of Finite Amplitude,” ‘Roy. Soc. Proc.,’ A, vol. 84, pp. 247-284, 
1910; ‘Scientific Papers,’ vol. v., pp. 573-619. 
|| Stokes, loc . cit . 
IF For a discussion of the problem under these conditions see a paper by Taylor, G. I., “ The Conditions 
Necessary for Discontinuous Motion in Gases,” ‘ Roy. Soc. Proc,,’ 84, A, 1910, pp. 371-377. 
