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I am looking for a single mathematical formula that takes in elevation, temperature, humidity and barometric pressure, and gives the speed of sound as an answer. I have found many simplified formulae which use only one variable to give an approximate answer but can't find one that provides a more accurate answer. I realize this may be out of the scope of experts exchange, but thought it can't help to ask.

http://resource.npl.co.uk/acoustics/techguides/speedair/

http://www.sengpielaudio.com/calculator-airpressure.htm

http://asa.scitation.org/doi/10.1121/1.405827

https://en.wikipedia.org/wiki/Speed_of_sound#Details

http://gsd.ime.usp.br/acmus/software/yili/SpeedOfSound/Speed.html

Is there a practical purpose for this calculation?

How much accuracy do you think you have and how much do you think you need?

How accurately do you expect to measure elevation and environmental variables?

Do you have a particular acoustic waveform and power level in mind?

Are you aware of non-linear and dispersion effects?

I would want to have the calculation inside my program rather than needing to use online web sites to perform the calculation. As far as how much accuracy I have or need, I don't know how much accuracy the four measurements can provide me with, or which ones are the most important or if one or more of these is redundant.

except that it seems like you have measurements at one elevation and want speed of sound as it varies at different elevations above that.

But 1/60s already represents about 10% of your distance, so greater accuracy in estimating speed shouldn't matter much.

Maybe you could get better height estimates by triangulating the position with two cameras.

Consider the sources of error and their magnitudes.

Here is a chart for the speed of sound at a variety of altitudes and temperatures: https://www.fighter-planes.com/jetmach1.htm

761.1 mph at 0 ft and 15 ºC

747.9 mph at 5000 ft and 5.1 ºC <== This is 1.7% change.

1/60th of a second at 761.1 mph gives you a resolution of 18.6 ft, which is approx 10% of your expected value.

Doing some trig with a second camera is also a great idea, if you can handle the infrastructure.

This is cell phone photo of a yardstick from 6 ft away. The camera lens was even with the 10 in mark.

I marked the pixel coordinates of the 10 and 35 in marks.

To scale this up to 250 ft altitude, multiply the distances by 120, so the camera would need to be 720 ft from the launch pad.

The nominal resolution here would be 902 pixels for 250 feet, or about 3.3 inches.

You would need a light detector that clicks when it sees the flash and two microphones. One microphone would be at ground level and the other at 6 or 10 feet.

The delay between the flash/click and the ground microphone would represent the altitude. The delay between the two microphone signals would give you a local calibration for the speed of sound.

Picking events of an audio recording is feasible, but ugly. You could use the same three sensors and two digital timers channels as well. This sort of project would fit easily into a PIC or Arduino for a junior-level EE project.

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But it also occurs to me that you may be able to time the burst better than you can time the flash,

in which case you may be able to get a height estimate from the

differencein times from 3 or more microphones.https://en.wikipedia.org/wiki/Multilateration