Irritating Resonance Demonstration
It emits an ear piercing sound which is at once one of the most
irritating noises imaginable and one of the most impressive physics
demonstrations. The trained ear immediately recognizes the sound as
a pure sign wave with a single frequency.
Its highly irritating nature comes not
just from itís loudness but from the fact that we virtually never
hear such a sound in the natural world. We heard it now only because
we have made an aluminum rod resonate at one of its natural
Making a cylindrical metal bar resonate is fairly simple but takes
practice. We use a cylindrical aluminum bar which is normally
inserted into a bushing in a counter and used for supporting
chemistry lab apparatus. The bar is slightly less than a meter long
(98 centimeters) and two centimeters in diameter. While our aluminum
rod seems to work very well, copper or steel rods or tube also work
well. Shorter bar or tubes give a higher pitch. Longer ones a lower
To make the bar resonate first rub a generous amount of rosin on its
surface. The type used on violin bows which is available in any good
music store works well. Next grasp the bar between the thumb and
first finger of your left hand exactly at the rodís center. This is
easy to find since it is the "balance point" of the bar. Finally,
grasp the bar near your left hand with the thumb and first finger of
your right hand. Pull the thumb and first finger of your right hand
along the barís surface while grasping it with a uniform pressure.
At all times maintain your grip on the center of the bar with your
left hand as the right hand slides toward the end of the rod.
The bar vibrates with a standing wave which has a node at its
center. The molecules near a node remain essentially stationary even
though the rod is vibrating. This allows one to grip the rod at this
position without disturbing the sound. The end of the rod is free to
move and forms an anti-node which has a maximum amount of movement
in the bar's molecules. Although the antinodeís movement is too
small to detect visually, its presence can be demonstrated by
touching the end of the rod. When this is done the sound immediately
A standing sound wave is caused by the reflection of sound waves
traveling in the bar. When a wave hits the end of the bar it is
reflected backwards. Since the end of the bar is free to move the
reflected wave is in phase with the original wave and reinforces
it's peaks forming an interference pattern with fixed positions for
nodes and antinodes.
You may have to rub the rod several times before it starts
resonating. This begins as a faint sound. Continue adding energy to
the rod by rubbing it until it reaches an ear piercing level. Do not
attempt this in front of an audience or class until you have fully
mastered the technique! In the beginning almost no one can do it,
but this is part of the fun. Everyone wants to try and it is wicked
entertainment to watch them as they fail. We once did this for an
entire day at a street fair. The local television sports caster was
about the only person to succeed. Not to be bested by a bunch of
nerds, he worked with a determined look on his face for about 20
minutes until he produced the ear splitting noise. Despite our
congratulations and slaps on the back, he didnít, however, ask us to
be on TV.
Impressive as the demonstration is, it raises an equally large
number of questions. First, why does the rod resonate at only one
harmonic frequency? Most musical instruments use a similar principle
but do not sound irritating since they produce a wealth of harmonic
sounds. Second, if the rod can only be gripped in the center and if
touching the end immediately stops the noise, how is it possible to
rub the rod without damping the vibrations?
We start our analysis by determining the rodís natural frequencies.
Figure 1. shows a diagram of the standing waves and the positions of
nodes and anti-nodes for the first three harmonic frequencies.
f1 = v/(2L)
f2 = v/L
f3 = 3v/(2L)
L = the length of the rod
v = the velocity of sound in aluminum
Clearly the second harmonic is unlikely
since it would have an anti-node in the middle of the rod. This
would likely be damped out by grasping the rod.
Figure 1. Diagram of
the First Three Harmonics of a Vibrating Bar
The next logical step is to analyze the
sound output of the rod by making the rod resonate with a high
loudness level. It's best to keep this burst of sound as short as
possible or use ear plugs if done for lengthy periods. Lower
loudness levels donít always yield a pure sine wave.
A short sound segment was recorded into
a microphone connected to a Vernier MPLI system installed in a
desktop computer with MPLI software. This particular system is no
longer available but the same experiment can be done with a Texas
Instruments TI-83 and CBL or with a Vernier LabPro connected to a
When the MPLI software is set for chart recording it gives a
loudness verses time plot. When the MPLI software is set to fast
Fournier transform (FFT) mode it gives an amplitude verses frequency
plot. This is especially useful for detecting resonance frequencies.
They appear as very high peaks on the FFT plot.
The FFT plot gives a single frequency for our rod of 2574 Hz (see
Figure 3). The frequency from the loudness verses time plot is the
reciprocal of the time between two loudness peaks. Figure 2. shows
that the time between peaks is about .004 seconds. This gives a
frequency of 2500 Hz. Generally the FFT and calculated frequencies
are in close agreement.
If we assume that the single frequency is the first harmonic we can
calculate the velocity of sound using the following formula:
v = (f1)(2L)
Using the 2574 Hz from the FFT plot
gives a velocity of sound in aluminum of 5045. This compares well
with the textbook value of 5000 m/s. Such close agreement confirms
the hypothesis that we do indeed hear the first harmonic.
Figure 2. Microphone
Output vs Time for a Resonating Rod
Figure 3. FFT Graph of Microphone Output for Resonating Rod
Drums, pipe organs, guitars and just
about all forms of acoustic musical instruments work by vibrating at
natural frequencies but do not sound irritating because they contain
many harmonics. By contrast we hear only one frequency with our rod.
The likely explanation for the single frequency is that the bar is
not being excited at any of its higher harmonics. To test this we
attach the microphone to the end of the bar with a rubber band and
rub the bar while recording the results in the FFT display.
Typically this shows multiple
frequencies similar to white noise (see Figure 4.). However, the
highest exciting frequency is less than the third harmonic of the
bar. Using poor technique does not make the bar resonate because it
does not input enough energy in a frequency corresponding to any of
the barís harmonics (see figure 5).
This explains why most people cannot
perform the trick without a significant amount of practice.
Figure 4. FFT Graph
of microphone Output With Effective Technique
Figure 5. FFT Graph
of microphone Output With Ineffective Technique
The single biggest factor in making the
bar resonate seems to be the amount of rosin on the bar. Rosin
creates a high static friction between the bar's surface and a
person's fingers. Static friction occurs when no sliding is present
and prevents sliding from happening at least up to a point. When the
shear stress between surfaces reaches a critical level they break
lose and slide.
The friction becomes dynamic which is
much lower than the static friction and the high shear stress is
temporarily relieved. The lower shear force allows the sliding to
stop. This cycle is called stick slip and continuously occurs as the
bar is rubbed. The stick slip action cases the vibration with
results in resonace.
The final mystery to be answered is why rubbing the bar does not
damp out the vibrations. If the bar vibrated with a transverse wave
which caused movement perpendicular to the barís axis, the gripping
action of sliding fingers would damp the vibrations enough to
prevent resonance. Since this does not happen, the vibrations must
be longitudinal or in the same direction as the barís axis. In this
case the gripping action would cause far less damping especially at
low levels of vibration.
We spent a long time pondering how to confirm this experimentally.
We wanted to use cool stuff like laser beams, high speed photography
through microscopes etc. but in the end, used a childís xylophone
striker. Holding the center of the bar, we carefully pinged the end
in a longitudinal direction (the direction of the rod's axis) and
analyzed the results with FFT.
We repeated the experiment with a
transverse ping on the rod's side (perpendicular to the axis) and
again recorded the results. Theoretically, a longitudinal ping
should create a longitudinal wave, etc. As seen in figure 6 the
longitudinal waveís dominate peak perfectly matches the resonating
soundís frequency while the transverse waveís sound spectrum is
considerably different. This supports the hypothesis that the
resonating barís vibration is in a longitudinal direction.
This analysis can be done even without FFT equipment. For most
listeners the noise from the longitudinal ping sounds remarkably
like the resonating barís sound although much quieter. The sound of
striking the bar in the transverse direction is completely
The sound spectrum from the longitudinal ping contains not just the
first harmonic but also an additional smaller peak. Calculations
show that it matches the third harmonic of the bar. The action of
the striker hitting the end of the bar excites more of the high
level harmonics than rubbing the bar's side. Indeed, striking an
object and measuring the resulting vibrations is considered by
engineers to be one of the ways to identify potential resonate
Good lab technique with the striker makes a distinct difference in
the analysis of the rod. A light ping seems to work better than a
heavy one. Also care needs to be taken to hit the rod either exactly
perpendicular to its axis to cause the transverse wave or in exactly
the same direction as its axis to cause the longitudinal wave.
Errors in technique will result in simultaneously exciting the bar
in both the transverse and longitudinal directions since the striker
will have a momentum component in both directions.
To further confirm the longitudinal nature of the barís vibrations,
we tried gripping the end of the bar in the transverse direction as
it was resonating. While this does damp the vibrations and reduce
the sound level the effect is far less dramatic than touching the
end of the bar.
Although this series of experiments is not for beginners, it is
dramatic and yields satisfying agreement between theory and measured
results. The fact that broad spectrum sound input to the barís sides
is emitted as intense single frequency noise from the barís ends is
strikingly similar to the way a laser beam works.
Whether done as formal experiments or as
demonstrations the resonating bar always leads to stimulating
physics discussions and a sense of wonder at the power of resonance.
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