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The general formula for the speed of sound in a gas is
where dp/dρ is the derivative of pressure p with respect to density ρ.
The presentation below is a derivation of this relationship. This derivation is more pedestrian than the usual derivation and does not require so many feats of mental agility.
Consider a tube of gas with constant cross-sectional area A. Let u be the fluid velocity, p the pressure and ρ the density. The momentum equation for this flow is:
which says that the force on a parcel is equal to the negative of the pressure gradient. The rate of change of the velocity du/dt is the instantaneous rate of change ∂u/∂t plus the advection term u∂u/∂x. When the left is divided by density and the terms are replaced the result is:
Since mass is conserved the equation of continuity holds, which for this case is:
These may be moved toward linearization by assuming u is small and ignoring products of small terms. The form of the equations is then
It is unnecessary to have both p and ρ in the equations in as much as they are connected through the equation of state. Then
Note that the derivative of p with respect to ρ is a total derivative.
With this change the equations become
To make the equations completely linear the coefficients of the partial derivatives must frozen; i.e. take on constant values equal to their average values. Thus the fully linear equations would be expressed as:
Now we can look for wave-like solutions of the form u=Ae^{k(x-ct)} and ρ=BAe^{k(x-ct)}. When these forms are substituted into the equation and the results divided by e^{k(x-ct)} we obtain the two homogeneous equations in two unknowns
This set of equations will have a nontrivial solution only if the determinant of the coefficient matrix is equal to zero.
The determinant equation is
The solution to this equation is simply
If the ideal gas law holds and temperature is constant then
When adiabatic conditions hold then the Poisson equation can be used for evaluating dp/dρ; i.e.,
where K is a constant and γ = c_{p}/c_{V}.
The derivative is then
One implication of the above formula is that if the absolute temperature increases 1 percent then the speed of sound will increase 1/2 of 1 percent. The formula is
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