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This velocity is referred to as the speed of sound.
To answer this question consider a piston moving from the left to the right at a relatively small velocity (see Figure 3.1). The information that the piston is moving passes thorough a single ``pressure pulse.'' It is assumed that if the velocity of the piston is infinitesimally small, the pulse will be infinitesimally small. Thus, the pressure and density can be assumed to be continuous.
or when the higher term is neglected yields
From the energy equation (Bernoulli's equation), assuming isentropic flow and neglecting the gravity results
neglecting second term ( ) yield
Substituting the expression for from equation (3.2) into equation (3.4) yields
An expression is needed to represent the right hand side of equation (3.5). For an ideal gas, is a function of two independent variables. Here, it is considered that where is the entropy. The full differential of the pressure can be expressed as follows:
In the derivations for the speed of sound it was assumed that the flow is isentropic, therefore it can be written
Note that the equation (3.5) can be obtained
by utilizing the momentum equation instead of the energy equation.
Next: Speed of sound in Up: Speed of Sound Previous: Motivation Index Created by:Genick Bar-Meir, Ph.D.
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