# Any general method of determining changes in speed of sound with pressure?

I need to correct speed of sound measurements in various gases for changes in ambient pressure, typically 700-1100 mBar. Is there any general theoretical model available for this? Especially something I could code in C

• Under those conditions the gases will be close to ideal so you could simply use the equation for an ideal gas Apr 14 '15 at 9:55
• Unfortunately close to ideal is not good enough. I need to make corrections to within one part in a million
– user56903
Apr 14 '15 at 10:55
• In that case all I can suggest is to do the curve fitting yourself. The ideal gas formula may be close enough if you apply a fudge factor, and I'd guess that measured speeds of sound are Googlable so you could use these to calculate the fudge factor. Your question is a bit specialised and I suspect the probability of any site members having done this is low (though not zero!). Apr 14 '15 at 11:02
• Yes, this is old fashioned physics with experimental connections. The real world meets theory (or not, as is often the case)
– user56903
Apr 14 '15 at 12:39
• A good start would be the NIST Reference on the known thermodynamic properties of air from 1999 although they admit on page 360 that the estimated uncertainties in the speed of sound are at the 0.2% level. Also of interest are the 309 publications that cite it Apr 23 '15 at 3:03

The very famous Newton-Laplace equation is a relation between the speed of sound and the pressure of an ideal gas. It can be written as: $$v = \sqrt{\gamma P / \rho}$$

where v is the velocity of sound in the given medium, P is the pressure, γ is the ratio of the heat capacities for the medium and ρ is the density of the medium.

The Newton-Laplace was first given by Sir Issac Newton in his most famous work Principia Mathematica as: $$v = \sqrt{ P / \rho}$$

But the not so obvious flaw in this formula was that he assumed that the process of sound propagation was isothermic. The great French Mathematician Pierre-Simon Laplace, in the same century, identified this flaw, made corrections for the influence of heat transfer on speed of sound and gave the above Newton-Laplace formula. Additionally, this formula makes it really simple to code in any language for that matter.

• How accurate is it for, say, air?
– user56903
Apr 17 '15 at 17:38
• 100% accurate! It is the formula used in calculation everywhere, be it acoustics, thermodynamics or turbomachinery Apr 17 '15 at 17:44
• Additionally, this the formula found in all textbooks with chapters dedicated to waves and sound Apr 17 '15 at 17:46
• I need it to make pressure correction to the speed of sound which is being measured by a time of flight accurate to one part per million. Even small changes in the weather affect it and I need a "standardized" reference of 1000 mB value
– user56903
Apr 17 '15 at 18:09
• Of course! This formula will give you the requisite corrections to your speed of sound. The precision depends on your system specifications(computer specs) Apr 18 '15 at 1:16

There is a model described in Main's Vibrations and Waves in Physics dealing with the speed of sound variations you might consider useful. Sorry, I would just comment that, but I don't have enough reputation.

The other way might be to derive the speed of sound not from the ideal gas laws but from van der Waals equation, but to be honest, I've never tried that. You would need to eliminate the density as the function of pressure with the fit parameters and then those parameters should appear in the resulting equation for c0 (using continuity equation and momentum equation from Euler's system).

• "Sorry, I would just comment that, but I don't have enough reputation."...and you think bypassing the system by posting it as an answer is the way to go? Apr 14 '15 at 11:23
• No, I didn't mean to bypass the system, but I knew a good hint and this was a way to share it even though the possible negative reputation. Apr 14 '15 at 11:34
• Hi Victor, you can flag your answer and ask the moderators to convert it to a comment. However I think it's a fair answer so I'd leave it in place. It won't be long before you have enough rep to comment. Apr 14 '15 at 12:50