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## A Measurement of the Speed of Sound

You can, of course, look up this value in lots of standard texts.  Wikipedia gives the value for dry air at sea-level pressure as

(1)

where T is the temperature in degrees Celsius.  In my location, a few hundred feet above sea level, in the winter when the indoor air is very dry, this formula is probably pretty accurate.  At 23 degrees C. this works out to about 345 meters per second.

But how hard is it to actually measure the speed of sound?  If you don't care too much about precision, it's pretty simple.

### Test Equipment Used

1 acoustic impulse generator (er, that's a pair of hands, actually, struck forcefully together).

1 "Phone Mate" answering machine microphone, saved from a time long ago when people actually used answering machines instead of having the phone company take all their messages for them.

1 random Radio Shack microphone, saved from a brief interest in taping music, some years back.

1 low-end Radio Shack integrated amplifier (about 3 watts per channel, I think), saved from some project or other a few years ago.

1 digital oscilloscope, acquired second-hand on Ebay.

### The Experiment

 Figure 1 -- Microphones and amplifier:
The microphones were placed 40 inches apart, with the bodies arranged parallel to each other (see figure 1). The distance between their centers was measured with a tape measure.  The microphones were connected to the phono inputs on the amplifier.  (The equalization on the phono inputs is grossly incorrect for microphones, but for this experiment it's OK.)  The oscilloscope was connected to the speaker outputs, in single-sweep mode.
I stood on the line connecting the microphones, about twenty inches past one microphone, and clapped.  I repeated this until I got a reasonably clean set of traces, then measured the distance between the first "hump" in the clap sound on each channel.  This was repeated at the "other end", and the results averaged, to cancel out any difference in microphone response time or propagation delay in the the signal paths from the microphones.

### Sources of Error

No experiment is complete without an attempt at error analysis; otherwise we have no idea what, if anything, the result actually means.
•  Figure 2 -- Error in clap location:
Intermicrophone distance.  The microphones themselves are about 1/2 inch in diameter, and positioning them with a tape measure isn't very precise in any case.  There could therefore have been inaccuracy of as much as +/- 1/2 inch in the center of each microphone, or +/- 1 inch total.  That results in a potential error of +/- 1/40, or +/- 2.5 %.
• Clap location.  The clap was intended to be on the line connecting the microphones, but the size of my hands, alone, makes the location somewhat imprecise (see figure 2).  If the location was several inches off the line, and 20 inches from the "near" microphone, then the tangent of the angle formed between the "near" microphone and my hands would have been about 1/5.  That's 20 degrees (shown as θ in figure 2).
The mis-positioning would increase the distance to the near microphone by a ratio of 1/cos(θ), while the distance to the far microphone would be substantially less affected. If the clap was 60 inches from the far microphone, but actually on a line separated by 20 degrees from the line between the microphones, this increases the distance to the near microphone by a factor of 1.06, or 1.2 inches.  Since we're actually measuring the difference between the distances to the microphones, this decreases the actual difference versus what we think we're measuring; the result would be an increase in the measured velocity of about +1.2/40, which means the correct result could be between 0 and -3 % of our measured result.
• Oscilloscope reading errors.  The accuracy with which we can read the results from the oscilloscope screen looks like about +/- 0.05 mS (see screen shots, below).  Since the total delay was about 2.9 mS, this represents an error of about 0.05/2.9, or +/- 1.7 %.
• Differing response times between the (mismatched) microphones.  This was compensated for by running the experiment from both ends and averaging.  We can, therefore, disregard it.
• Oscilloscope time base inaccuracy.  The time base of the oscilloscope has inaccuracy on the order of a small fraction of a percent, which is far better than anything else here; hence, we can disregard this error source.
There may also be errors caused by phase shift in the microphones or other issues we haven't thought of.  However, we believe the ones listed above are likely to dominate all other error sources.  Summing the values given above, we find that the true speed of sound should fall within +4.2% to -7.2% of our measured value.

The first two sources of error could be reduced substantially by using a larger separation between the microphones.  The third source of error -- imprecision reading the oscilloscope -- could be reduced by using a continuous tone (sine wave) for the signal rather than a single impulse -- locating the zero crossing in a sine wave can be done far more easily than finding the "knee" in the curve of an impulse.

### Results

As previously mentioned, the actual experiment consisted of setting the oscilloscope to single-trace mode, standing beside one microphone, and clapping.  This was repeated while standing by the other microphone.  In figure 3 and figure 4, we see shots taken in each direction.  In both cases we measured the time from the peak of the first "hump" in the wave form at the leading microphone, to the time at the peak of the first "hump" in the wave form at the trailing microphone.  The times, as seen on the screen (labeled "Delta" in the screen shots) were 2.960 and 2.820 mS, respectively; averaging them we obtain 2.89 mSec.

Our separation was 1.016 meters, which leads to a speed of 351.6 meters per second.  Rounding to 3 digits and applying our previously computed error bar of +4.2% to -7.2%, we obtain the value and error range of:

Speed of Sound = 352  +15/-25 meters/second

or an absolute range of:

Speed of Sound = 327 to 367 meters per second

The "expected" value, as given at the start of the page, is 345 meters per second -- which is, we are pleased to note, near the middle of our range, and well within our "error bar".  Furthermore, the absolute error in our raw number is 6.6 meters per second, or just 1.9 %.

 Figure 3 -- First shot: Figure 4 -- Second shot:

Page created on 02/04/07