Atmospheric calculation

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.
ChLaAsked:
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d-glitchCommented:
Just trying to get some context.

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?
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ChLaAuthor Commented:
To answer d-glitch, I have written a burst height calculator, in Delphi. It uses the speed of sound and the time delay between the light of a pyrotechnic shell burst, and the moment the sound arrives at the camera, which occurs somewhere between 150 and 250 feet up. I have a weather station at the site and so have the temperature, barometric pressure, humidity and of course the site elevation above sea level, for each test. I have the shells on video and so can easily count the fields (1080 60i), between flash and burst. I am standing nearly directly under the shells as I video tape them. Of course the accuracy of my measurements is limited to slightly less than 1/60 second because that is the temporal resolution of the fields. My site is nearly 3000 feet above sea level and so I don't like to use the typical calculations found that assume sea level and standard conditions. I feel I should get a s close as possible with the speed of sound to achieve the greatest accuracy. Knowing the burst height is useful in many ways. It is a safety consideration. It can help me to learn how much small variations in the lift charge will effect the burst height, and how to best compensate for under and overweight shells.
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.
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ozoCommented:
elevation would be a substitute for estimates of temperature and pressure, so given those, elevation should be irrelevant,
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.
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d-glitchCommented:
ozo is certainly correct, on several counts.

 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.Camera Trig
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.
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d-glitchCommented:
If this is an important measurement, and you have a little money to spend, you should be able to get better than 1 ft resolution electronically.

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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ozoCommented:
You could also get a 1.7% speed change from a 13mph wind.
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 difference in times from 3 or more microphones.
https://en.wikipedia.org/wiki/Multilateration
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d-glitchCommented:
Good advice on sources of error and alternate techniques.
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