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If you shout "HELLO!" at the Grand Canyon,
you will hear "HELLO!" return back in a few
seconds. But suppose you shout the entire Gettysburg
Address? You will be shouting "...our forefathers
brought forth on this land..." when "Four
score and seven years ago" returns as the echo.
Your outgoing speech will be interfering with the echo
and will create a jumble of words.
What would happen if you sent out in a tight beam a
continuous wave of single frequency and amplitude. When
it reflects back, it would interfere with the outgoing
wave. In some places, it might interfere "constructively"
producing maximum sound and in others, "destructively"
producing zero sound. Where you get maxima and minima,
and even if you do, depends on the length of the path
the sound takes, and the frequency or wavelength of
the sound. With the right frequency and path length,
you can set up a "standing waves" --- a wave
that appears to be fixed in space.
When a wave reflects from a fixed end, like the end
of a vibrating string, it makes sense that it must have
ZERO amplitude at the fixed end. That's why it's called
"fixed". But when a wave reflects back from
a free end, it forms a maximum amplitude at the end,
and hence it's called a "free" end.
In a tube with two fixed ends, the standing wave patterns
looks just like a vibrating string. You can get half
a wavelength, a whole wavelength, 3/2 wavelength, and
so on, as standing waves. These particular frequencies
are known as the resonant frequencies. They provide
just the right wavelength to completely fit inside the
pipe. Slightly longer or shorter wavelengths don't fit
exactly.
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| You can observe this dramatically with a
demo I call the "Flame Tube". I put a speaker
at one end of a tube, close off the other end and pump
propane into the tube. The propane comes out of the little
metal burners on top. At most frequencies of sound, nothing
happens --- all the flames are the same height. But at
certain special or resonant frequencies, the flames take
on the shape of the standing wave within the tube: |
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"WAIT A MINUTE, Professor!
I THOUGHT YOU SAID THE ENDS OF THE TUBE WERE NODES or
POINTS OF ZERO DISPLACEMENT. HOW COME THE FLAMES ARE HIGHEST
AT THE ENDS????"
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Ummm, well... ok, Einstein.
You're right. There should be Zero amplitude at the
ends, but Zero displacement amplitude. What you are
looking at with the flames is the pressure variation
inside the pipe, not the displacement variation. Remember
that pressure and displacement are 90 degrees out of
phase with each other. Zero displacement means maximum
pressure. Maximum displacement means minimum pressure.
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For my next trick, I have Paul take an ordinary
cardboard tube, open at both ends. If he holds it over
a Bunsen burner, he can create longitudinal waves in the
tube. Those wavelengths that happen to match the length
of the tube in some whole or half wavelength will be made
much stronger than others. That's resonance. So the open
pipe will make a very loud, low tone corresponding, in
this case, to its fundamental resonant frequency.
Or, how about..... |
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| .... if Paul takes an aluminum rod. He can stroke the
rod and excite within it (Hey! Come on, pay attention!)
longitudinal vibrations that correspond to resonant frequencies.
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| If he holds the rod in the center, the center
will be a displacement node, the ends displacement antinodes,
and the rod will contain one half wavelength. |
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If he holds it at the first 1/4 length point, that
becomes a node, the 1/4 length point on the other end
is the other node, and he has a full wavelength in the
rod.
The sound from the rod is a high pitch (high pitch
= high frequency). This is because the wave speed is
very high:
v = f λ, so f = v/ λ
What determines the wave speed is primarily the Bulk
Modulus of the aluminum, or how easily compressed the
aluminum is. v = √
(B/ρ)
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| If you tap the rod on its side, it makes
a low "clang" --- a much lower frequency than
if you stroke it. That's because now you're not compressing
the aluminum but making it vibrate back and forth (related
to something called "Shear Modulus"). It's much
easier to vibrate the rod this way than by trying to compress
the aluminum. Hence, the wave speed will be much lower,
and the resulting sound a lower pitch. |
