THEORY OF WAVES OE FINITE LONGITUDINAL DISTURBANCE. 
287 
Supplement to a Paper “ On the Thermodynamic Theory of Wa ves of Finite Longitudinal 
Disturbance by W. J. Macquorn Rankine, C.E., LL.D ., F.P.SS. Lond. & Edin. 
Received October 1 , — Read December 16, 1S6U. 
Note as to previous investigations. — Four previous investigations on the subject of the 
transmission of waves of finite longitudinal disturbance may be referred to, in order to 
show in what respects the present investigation was anticipated by them, and in what 
respects its results are new. 
The first is that of Poisson, in the Journal de l’Ecole Poly technique, vol. vii. Cahier 14, 
p. 319. The author arrives at the following general equations for a gas fulfilling 
Mariotte’s law : — 
dt ' C ax ' o ' dx z ’ 
in which <p is the velocity-function ; ^ the velocity of disturbance, at the time t , of a 
particle whose distance from the origin is w; a is the limit to which the velocity of 
propagation of the wave approximates when ^ becomes indefinitely small, viz. /\J 
p 0 being the undisturbed pressure and § 0 the undisturbed density ; and f denotes an 
arbitrary function. This equation obviously indicates the quicker propagation of the 
parts of the wave where the disturbance is forward (that is, the compressed parts) and 
the slower propagation of the parts where the disturbance is backward (that is, the 
dilated parts). 
The second is that of Mr. Stokes, in the Philosophical Magazine for November 1848,. 
3rd series, vol. xxxiii. p. 349, in which that author shows how the type of a series of 
waves of finite longitudinal disturbance in a perfect gas alters as it advances, and tends 
ultimately to become a series of sudden compressions followed by gradual dilatations. 
The third is that of Mr. Airy, Astronomer Royal, in the Philosophical Magazine for 
June 1849, 3rd series, vol. xxxiv. p. 401, in which is pointed out the analogy between 
the above-mentioned change of type in waves of sound, and that which takes place in 
sea-waves when they roll into shallow water. 
The fourth, and most complete, is that of the Rev. Samuel Earnsiiaw, received by the 
Royal Society in November 1858, read in January 1859, and published in the Philoso- 
phical Transactions for 1860, page 133. That author obtains exact equations for the 
propagation of waves of finite longitudinal disturbance in a medium in which the pressure 
is any function of the density ; he shows what changes of type, of the kind already men- 
tioned, must go on in such waves ; and he points out, finally, that in order that the type 
cId Old 
may be permanent § 2 -p- ( = — ^ in the notation of the present paper) must be a constant 
