L 277 ] 
XV. On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance. 
By W. J. Macquorn Rankine, C.E., LL.D., F.R.SS. Lend. & Fdin., &c. 
Received August 13, — Read December 16, 1869. 
§ 1. The object of the present investigation is to determine the relations which must 
exist between the laws of the elasticity of any substance, whether gaseous, liquid, or 
solid, and those of the wave-like propagation of a finite longitudinal disturbance in that 
substance ; in other words, of a disturbance consisting in displacements of particles along 
the direction of propagation, the velocity of displacement of the particles being so great 
that it is not to be neglected in comparison with the velocity of propagation. In par- 
ticular, the investigation aims at ascertaining what conditions as to the transfer of heat 
from particle to particle must be fulfilled in order that a finite longitudinal disturbance 
may be propagated along a prismatic or cylindrical mass without loss of energy or change 
of type : the word type being used to denote the relation between the extent of disturbance 
at a given instant of a set of particles, and their respective undisturbed positions. The 
disturbed matter in these inquiries may be conceived to be contained in a straight 
tube of uniform cross-section and indefinite length. 
§ 2. Mass-Velocity. — A convenient quantity in the present investigation is what may. 
be termed the mass-velocity or somatic velocity — that is to say, the mass of matter through 
which a disturbance is propagated in a unit of time while advancing along a prism of 
the sectional area unity. That mass-velocity will be denoted by m. 
Let S denote the bulkiness, or the space filled by unity of mass, of the substance in 
the undisturbed state, and a the linear velocity of advance of the wave ; then we have 
evidently 
«=mS (1) 
§ 3. Cinematical Condition of Permanency of Type. — If it be possible for a wave of 
disturbance to be propagated in a uniform tube without change of type, that possibility 
is expressed by the uniformity of the mass-velocity m for all parts of the wave. 
Conceive a space in the supposed tube, of an invariable length Aa, to be contained 
between a pair of transverse planes, and let those planes advance with the linear velocity 
a in the direction of propagation. Let the values of the bulkiness of the matter at the 
foremost and aftermost planes respectively be denoted by s, and s. 2 , and those of the 
velocity of longitudinal disturbance by u x and u 2 . Then the linear velocities with which 
the particles traverse the two planes respectively are as follows : for the foremost plane 
u \ — lor the aftermost plane u 2 — a. The uniformity of type of the disturbance involves 
as a condition, that equal masses of matter traverse the two planes respectively in a given 
