THEORY OE AY A YES OE FINITE LONGITUDINAL DISTURBANCE. 
285 
tions (8) and (18) combined, we find, for the absolute temperature of a particle at which 
the pressure is^>, in a wave of permanent type, the following value: 
PS (y + t)(^+P >- 2 ^ 2 . m.A 
J(c -O* (y+i)(/h +^)P-2P 2 ’ * - 
PS 
in which the first factor - is obviously the undisturbed value of the absolute tem- 
J [Cp—C s ) 
perature. For brevity’s sake, let this be denoted by T. 
The following particular cases may be noted. In a wave of oscillation, as defined in 
§ 10, we have^ 1 -f-i> 2 =2P ; and consequently 
(7 + l)Pp p~ 
yP 2 
(32 a) 
In a wave of permanent condensation or rarefaction, as described in § 11, letj.q=P, 
^? 2 =P; then the final temperature is 
T (r + 1 )P p + (y—i)p* 
' (y + i)P^ + (y— i)P 2 ’ 
§ 13. Types of Disturbance capable of Permanence . — In order that a particular type 
of disturbance may be capable of permanence during its propagation, a relation must 
exist between the temperatures of the particles and their relative positions, such that 
the conduction of heat between the particles may effect the transfers of heat required by 
the thermodynamic conditions of permanence of type stated in § 6. 
During the time occupied by a given phase of the disturbance in traversing a unit 
of mass of the cylindrical body of area unity in which the wave is travelling, the quan- 
tity of heat received by that mass, as determined by the thermodynamic conditions, is 
expressed in dynamical units by ^ 
The time during which that transfer of heat takes place is the reciprocal ~ of the mass- 
(It 
velocity of the wave. Let ^ be the rate at which temperature varies with longitudinal 
distance, and k the conductivity of the substance, in dynamical units ; then the same 
quantity of heat, as determined by the laws of conduction, is expressed by 
The equality of these two expressions gives the following general differential equation 
for the determination of the types of disturbance that are capable of permanence : 
mrd . <p = d . 
(33) 
The following are the results of two successive integrations of that differential equation : — 
dx k 
d r A + m^rdf 
x=b+ S 
kdr 
A + rnfdp ’ 
(33 a) 
(33 b) 
