Definition of the Speed of Sound

Deriving the speed of sound is a main component of any undergraduate textbook in gas dynamics and is very important to understanding more complicated concepts! Here, I will give an honest attempt to explain and provide definitions of what it is and why it is important .I will go through the derivation of the speed of sound mathematically.

First, I’ll attempt to explain what the speed of sound is. Most simply put, it is the speed at which sound propagates in a medium. But in fact, it is not just sound. All disturbances and motions are communicated throughout a medium at the speed of sound. For example, ever notice how streamlines around a vehicle begin to bend far upstream from the vehicle itself? This occurs because the presence of the vehicle has been communicated upstream (at the speed of sound) and the gas is forced to move out of the way. The speed of sound is dependent on the properties of the fluid being described, and the fluid itself. For instance the speed of sound in air at standard conditions is ~343 m/s whereas the speed of sound in water is ~1482 m/s. Additionally, and very importantly, the speed of sound represents the coupling of the pressure and density fields in a gas. In an incompressible fluid, the density is constant and therefore not a function of pressure. So we can call these fields uncoupled. But in real flows this is not the case. In fact, it is this couple of the pressure and density fields that make important and interesting changes to the conservation of momentum equation! (More on that in another post)

So why is the speed of sound important? Well let’s think about the vehicle example again. The streamlines upstream are able to bend in response to the oncoming vehicle. This is because the vehicle is travelling less than the speed of sound. But what if it were travelling faster than the speed of sound? Now the streamlines do not see what is coming and are suddenly struck by the car. This results in a shock-wave, and a very different aerodynamic scenario. In my related post regarding (subsonic vs. supersonic flow) I will re-enforce this discussion mathematically.

Now let’s make use of an example to define the speed of sound mathematically. The most straight forward approach to defining the speed of sound is with a piston example:

Piston
Fig: Schematic of the piston example

In the scenario depicted above, a piston is suddenly accelerated to a velocity of dV. This then causes a pressure increase of dp, and sends a compression wave (at the speed of sound) down the channel. The gas in the channel that has not yet been reached by the wave sits at the initial conditions of zero velocity, and initial pressure and density p_o and \rho_o respectively. As the wave moves down stream it causes the velocity of the gas to increase to dV and the pressure and density to increase by dp and dρ, respectively.

Remember: In an incompressible fluid, a piston motion would instantaneously cause all of the fluid in the tube to change velocity to dV. But recall that in a compressible fluid the motion of the piston does NOT do this. It sends a message downstream at the speed of sound telling the gas that the piston has moved. The wave then causes the fluid downstream to respond so as to ensure that the governing laws of fluid mechanics are held.

We are going to use this scenario and solve for the speed of sound, a. In order to analyze this problem we are going to perform a control volume analysis of the compression wave itself. To accomplish this, we are using a control volume that is in the frame of reference of the wave (ie moving to the right at velocity a). This is shown in the following figure:

CV
Fig: Control Volume Schematic

Notice that when we put the control volume in the reference frame of the wave how the problem is simplified? We now have a simple control volume with an inflow and outflow and we need only apply conservation of mass and momentum to solve the problem.

First we start with conservation of mass (Note: A-> cross sectional area of channel):

\dot{m}_{out}-\dot{m}_{in}=0

\left(\rho_o+d\rho\right)\left(a-dV\right)A-\rho_o a A = 0

If we expand, and simplify by removing the high order terms we get:

a d\rho-\rho_o dV=0

Now we can add the momentum equation:

\Sigma F = \dot{m}V_{out}-\dot{m}V_{in}

p_oA-\left(p_o+dp\right)A = \rho_o a A \left[\left(a-dV\right)-a\right]

We can simplify this to:

dp=\rho_oa dV

This equation, and the simplified conservation of mass are then easily combined to achieve:

a^2=\left(\frac{dp}{d\rho}\right)

This is the definition of the speed of sound! However, we must recognize that a sound wave is an isentropic process. And this is not always the case. So the correct definition of the speed of sound uses partial derivatives and a subscript s to denote that it is for an isentropic wave.

a^2=\left(\frac{\partial p}{\partial \rho}\right)_s

So then how do we calculate the speed of sound for a medium! Well, this comes from simply recalling that in an isentropic process the relation between pressure and density is:

\frac{p}{\rho^\gamma} = \textnormal{constant}

From this we can show that:

\frac{\partial p}{\partial \rho} = \frac{\gamma p}{\rho}

And then finally by using the ideal gas law we end up at

a= \sqrt{\gamma R T}

The above relation is the most commonly used formula for the speed of sound.

Conclusion

Thanks for reading! Hopefully this provided some clarification and assistance to those who needed it! As always, please comment if you notice any mistakes, or even little semantic errors. It is important to me to be completely correct!

Some Useful References

… Any Undergraduate Gas Dynamics textbook but my preferred ones are:

[1] Anderson, J. D. (1990). Modern compressible flow: with historical perspective (Vol. 12). McGraw-Hill.

[2] John, J. E. A., & Keith, T, G. (2006) Gas Dynamics, 3rd Edition, Pearson Prentice Hall

[4] Zucrow, M. J., & Hoffman, J. D. (1976). Gas dynamics. New York: Wiley, 1976, 1.

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