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
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.
how hard is it to actually measure
the speed of
sound? If you don't
care too much about precision, it's pretty simple.
1 acoustic impulse generator (er,
that's a pair of hands, actually, struck forcefully together).
"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.
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.
oscilloscope, acquired second-hand on Ebay.
-- Microphones and amplifier:|
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
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
Sources of Error
experiment is complete without an attempt at error analysis; otherwise
we have no idea what, if anything, the result actually means.
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 %.
-- Error in clap location: |
- 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).
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
- 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 %.
response times between the (mismatched) microphones.
This was compensated for by running the experiment from both
ends and averaging. We can, therefore, disregard it.
time base inaccuracy. The time base of the
oscilloscope has inaccuracy on the order of a small fraction of a
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.
first two sources of error could be reduced substantially by using a
larger separation between the microphones. The third source
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.
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
and figure 4
we see shots taken in each direction. In both cases we
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
screen (labeled "Delta" in the screen shots) were 2.960 and 2.820 mS,
respectively; averaging them we obtain 2.89 mSec.
was 1.016 meters, which leads to a speed of 351.6 meters per second.
Rounding to 3 digits and applying our previously computed
bar of +4.2% to -7.2%, we obtain the value and error range of:
Speed of Sound = 352 +15/-25 meters/second
an absolute range of:
Speed of Sound = 327 to 367 meters per second
"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
meters per second, or just 1.9 %.
-- First shot:|
-- Second shot:|
created on 02/04/07