THEOEY OF WAVES OF FINITE LONGITUDINAL DISTUEBANCE. 
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according asp is greater or less than P. The squares of the mass-velocity and of the linear 
velocity of advance are respectively as follows : 
m 2 
_ p — P . 
( 9 ) 
a*=m* S 2 =4 -P .S 2 (10) 
o — s 
The velocity of the disturbed particles is as follows : 
«=»«(S- S )=^A- v / (i— P).(S-s); (11) 
and it is forward or backward according as the wave is one of compression or of rarefaction. 
The energy expended in unity of time, in producing any such wave, is expressed by jyu 
for the wave may be conceived to be produced in a tube closed at one end by a moveable 
piston of inappreciable mass, to which there is applied a pressure p different from the 
undisturbed pressure P, and which consequently moves with the velocity u. The way 
in which that energy is disposed of is as follows : actual energy of the disturbance, 
rruj - ; work done in altering bulkiness, + — — ; and the equation of the conserva- 
tion of energy is 
pw=f K+(p+P)(S— s)} (11a) 
§ 6. Thermodynamic Conditions . — While the equations of the two preceding sections 
impose the constancy of the rate of variation of pressure with bulkiness during the dis- 
turbance — m 2 ^j as an indispensable condition of permanency of type of the wave, 
they leave the limits of pressure and of bulkiness, being four quantities, connected by 
one equation only ( ^ g 2 _ p 1 = — y s ~ • Two only of those quantities can be arbitrary; 
therefore one more equation is required, and that is to be determined by the aid of the 
laws of thermodynamics. 
It is to be observed, in the first place, that no substance yet known fulfils the con- 
dition expressed by the equation ^ =— m 2 = constant, between finite limits of disturb- 
ance, at a constant temperature, nor in a state of non-conduction of heat (called the 
adiabatic state). In order, then, that permanency of type may be possible in a wave 
of longitudinal disturbance, there must be both change of temperature and conduction 
of heat during the disturbance. 
The cylindrical or prismatic tube in which the disturbance is supposed to take place 
being ideal, is to be considered as non-conducting. Also, the foremost and aftermost 
transverse advancing planes, or front and back of the wave, which contain between them 
the particles whose pressure and bulkiness are in the act of varying, are to be considered 
