3 December 2004
Why is My Cup Two-Tone?
John W. Dooley, Physics Department
Millersville University
Conrad Miziumski waved me into his office one day and
tapped his coffee cup with a pencil. I heard the usual
tone. He tapped again, with a grin that said, "Can
you explain it?" . I heard a new tone, at a different
pitch. He did it by tapping the cup at two different
places, as in Figs. 1 and 2. (The effect sends warning
flags to those doing the "hot cocoa experiment."
There, the pitch changes as time goes by, and care must
be taken to tap in the same place on the cup.)
Before going on, can you guess which tap had the lower
pitch? After a guess, it's easy to try the experiment.
Figure 1. The cup is struck near the handle.
Note the microphone at right.
Click on image to hear the sound.
Figure 2. The cup is struck away from the handle.
Note the change in the position of the microphone.
Click on image to hear the sound.
If you click on the picture of the cup, you can hear
a .wav file (about 1.5 Mb worth) of the sound made by
the arrangement shown in the picture. To return to this
page, don't close the window on your sound software.
Instead, use the back button. A file with both sounds,
low pitch followed by high pitch, can be downloaded
here.
It's fairly easy to believe that when
you strike the cup rim at the handle, you generate a
low pitch. The vibration must include the handle, and
that increases the amount of mass that is in motion.
For a simple mass vibrating on a spring, the frequency
of vibration (i.e. sound pitch) decreases as the mass
increases. It's plausible that the handle vibrates something
like a mass on a spring, and that its motion has a relatively
low frequency.
That argument does not have anything to say about the
tap in Fig. 2 producing a higher pitch. To think about
the two tones, we consider a model of how the cup might
be vibrating. Figure 3 shows a model for the vibration
when the pen strikes the cup at the handle. An important
assumption is that the center of mass of the cup does
not move. This means that when the left side moves outward
(left), the right side must move outward (right) at
the same time.
In Fig. 3, the solid line represents the outside of
the cup when the handle is furthest left in its vibration,
and the dotted line represents the outside of the cup
when the handle is furthest to the right. We expect
the cup to distort in the same way that a cardboard
tube does when it is squeezed or stretched: Stretching
left and right pulls the top and bottom together. Squeezing
left and right makes the top and bottom bulge out.
This motion may be thought of as a standing wave on
the circumference of the cup. It is usually called a
"normal mode" of the motion.
Note the four places in the diagram at which there is
no motion. These points are the places where the solid
and dotted lines cross. As with standing waves, the
places which do not move are called "nodes"
of the motion.
Figure 3. Click on image to enlarge.
Figure 4 shows a model of the motion when the pen strikes
the cup 45 degrees around the rim from the handle. We
still assume that the center of mass of the cup does
not move. Opposite sides bulge out at the same time
that the neighboring sides bulge in, just as in Fig.
3. The dotted and solid lines show the extremes of the
motion, as in Fig. 3.
Figure 3 differs in that the nodes
of the motion have shifted. Most importantly, there
is a node at the handle. In this normal mode, the handle
does not move.
It is plausible that, without the mass
of the handle involved in the motion, the frequency
is higher, just as the vibration of a light mass on
a simple spring has higher frequency.
Figure 4. Click on image to enlarge.
Experiment corroborates this model in two ways. First,
the low tone can be produced by tapping the cup at any
of four points: The side opposite to the handle and
at points half way between the handle and the opposite
side (90 degrees around the circular rim from the handle).
Similarly, the high tone is produced by tapping any
of the four "45 degree" points where Fig.
4 shows the largest excursions.
Second, the microphone picks up the low tone best when
positioned at one of the four "good points."
That is the reason for the microphone placement in Fig.
1 and Fig. 2.
Figure 5. Click on image to enlarge.
In practice, it is difficult to isolate the tones. The
ear clearly hears the difference, but spectrum analysis
shows that both tones are usually present. Figure 5 shows
the spectrum (using GoldWave software) for the low frequency
tone when the cup is tapped at the handle. The lower pitch
dominates, with a frequency of about 2.3 kHz, but the
higher pitch is clearly present.
Figure 6. Click on image to enlarge.
Figure 6 shows a typical spectrum for the high pitched
tap. Its frequency is about 2.45 kHz, and in this case
the low pitch tone is almost absent.  |
