An introduction to whirled music
Playing the whirly
Introduction
A whirly is a corrugated plastic tube about as long as my arm. Whirl it overhead, and it will sing loud, pure tones. Most people regard whirlies as toys because they are such simple instruments to play; they require no complex breath control, and no tuning. Yet whirlies make pleasant musical sounds and the incongruity of twirling a tube to make music never fails to bring smiles to the faces of firsttime whirly players. Let's use simple physics activities to explore how whirlies sing.
Stories of Paul Playing the Whirly
 Playing the Whirly in Kansas City, Paul competes for the title of Grand Illuminator.
 Playing the Whirly in San Francisco, Paul plays the whirly in front of a paying audience.
 Playing the Whirly at Technorama in Switzerland, illustrated.
Material
Explorations:
To Do and Notice
Begin with a children's toy whirly sold in many
toy stores.
We investigated whirlies sold as "5 tone twirlers" with a length of
about 0.8 m (they vary), an outside diameter of 3.5 cm, and a
corrugation wavelength of 6.5 mm.
There is a bell on one end of these whirlies. To simplify the whirly
you may choose to cut the bell off the end.
Hold one end in your hand.
Twirl the other end in a circle.
Listen to the sound made by the tube.
Vary the speed of the tube.
Notice that the pitch of the whirly jumps from one note to another
and increases as the speed of the twirling is increased.
That is, high speed twirling creates high pitch notes.
If there are musically trained people available,
have them identify the musical intervals between the notes.
Notice that the interval between the lowest note of the standard toy
whirly and the second note is a musical fifth. In an interval of a
fifth, the ratio between the frequencies of two notes is 3/2.
The interval between the second highest note and the third is a ratio
of 4/3.
The interval between the third note and the fourth note is a ratio of
5/4.
(optional, if you have a frequency meter or an oscilloscope you can
directly measure the frequencies of the notes made by the whirly. You
can then calculate the ratios of frequencies and see how close my
prediction was.)
Resonances of the tube
Hold one end of the whirly near your ear and the
other end near your mouth.
Hum into the whirly. Vary the pitch of your humming while you try to
keep the loudness of the humming (heard by the ear not next to the
whirly) the same. Notice that at some frequencies the tone heard in
the whirly is louder. When you hum at a resonant frequency of the
tube your humming is amplified and sounds louder. (You can also drive
the whirly with a frequency generator and a speaker.) Notice that the
resonances occur at the same frequencies the whirly sings. Except for
the fundamental. There is a resonant amplification of your humming at
the fundamental frequency yet the whirly will not play the
fundamental.
What's going on?
A tube open at both ends has resonant frequencies
called harmonics.
The lowest frequency resonance is called the first harmonic or the
fundamental, the next highest frequency, the second harmonic, is
double the fundamental. The third harmonic has three times the
frequency of the fundamental and so on.
The maximum backandforth motion in the second
harmonic.
(See the Ringing Aluminum Rod activity for a graphical description of these types of drawings.)
The musical interval between the fundamental and
the second harmonic is an octave, (named after the eight white keys
on a piano spanning these two notes) the frequencies have a ratio of
2/1.
This is not the interval between the lowest two notes of the whirly,
musicians report that this interval is a musical fifth.
The only two adjacent harmonics which are separated by a musical
fifth are the second and third harmonics.
Thus the lowest note played by the whirly must be the second
harmonic.
This whirly does not play the fundamental frequency.
(Why the fundamental frequency of resonance of the tube is missing
from the frequencies played by the whirly is a question we will
investigate later.)
The length of the tube adjusts the fundamental frequency of the
whirly and all of the harmonics which are multiples of the
fundamental.
Which harmonic is played is selected by controlling the speed of
airflow through the whirly.
One of my whirlies had a length of 0.78 m, looking at the fundamental
drawn above, its wavelength is twice the length of the tube or 1.56
m. The frequency of the fundamental is 220 Hz, the frequency of the
lowest note played by my whirly was twice this or 440 Hz.
To Do and Notice
Does the air moving across the rapidly moving end
of the whirly make it sing?
Blow across the end of the whirly with your mouth or a blower.
Notice that no matter how fast or slow you blow across the end
of the whirly it does not sing.
Blow through the whirly.
Notice that it sings.
So air flowing through the whirly makes it sing, not air
moving across the end.
(See the Etc on Bernoulli below.)
While playing the whirly:
1. Cover the stationary end with your hand. Notice that the sound
stops immediately.
2. Hold the stationary end near the burning candle, notice that the
flame bends into the whirly.
3. Hold the stationary end of the whirly near a pile of confetti, or
other small paper pieces. For the best effect hold the confetti in a
strainer so that air can flow around the confetti. Notice that when
the whirly is signing a note the paper pieces flow into the whirly
and are sprayed around.
Notice that, when the whirly sings, air flows through the tube.
What’s Going On?
Why does air flow through a whirly?
Picture a whirly full of marbles. If you twirled such a whirly, the
marbles would fly out of the spinning end. This is what happens to
the air in the whirly. The faster you spin the whirly the faster the
marbles and the air will flow through it. The wall of the whirly
moves in a circle, to make the air inside the whirly move in a circle
a centripetal force is needed. The whirly is open at the end so in
the absence of centripetal force the air inside the whirly
accelerates along the rotating tube. Air flows through the rotating
whirly.
One way to say this is that "You throw the air out of the whirly."
Etc
Bernoulli. Many people first guess that the air blowing across the end of the whirly reduces the pressure at the end of the whirly due to the Bernoulli effect and sucks air through the whirly. The simple experiment of blowing across the end of the whirly shows that this guess is incorrect. While the air does speed up as it flows over the end of the whirly, and while this speeding up is accompanied by a drop in pressure according to the Bernoulli effect, this drop in pressure plays a minor role in making the air flow through the tube.
Math Root
A quantitative investigation of the connection between sound
frequency and the frequency of rotation.
Twirl the whirly so that it plays the lowest
note.
Use a stopwatch to time the rotation of the whirly, find the number
of seconds per rotation. Then use this time to find the frequency of
rotation of the whirly.
The best way to do this is to have one person play the whirly while
another measures the time for ten rotations. Divide this time by ten
to find the time for one rotation, T. The frequency, f, is the
inverse of the time for one rotation. f = 1/T. Have several different
groups do this experiment.
For my whirly playing its lowest note it took 6.5 seconds for 10
cycles, or 0.65 s per cycle, giving a frequency of 1.6 Hz.
Repeat this experiment for the second note. Then
the third, the fourth, and the fifth.
Make a table of your data.
My data was:
harmonic, n Time for 10 cycles, s frequency,f Hz f/n 2 6.5 1.6 0.78 3 4.3 2.3 0.78 4 3.4 3.0 0.74 5 2.7 3.8 0.75
The last column is the frequency divided by the number of the
harmonic. This number is very close to a constant value. This
indicates that the speed of rotation of the whirly is proportional to
the frequency of the sound the whirly makes.
Plot the frequency of rotation of the whirly versus the number of the harmonic. Since the lowest note is the second harmonic plot its rotation frequency versus 2. Plot the frequency of the third harmonic versus 3 and so on. Notice that the points fall near a straight line. The frequency of rotation is proportional to the frequency of the sound and viceversa.
The whirly will sing any given note over a range of rotation frequencies. Repeat the experiment while playing the whirly at the slowest speed which will play a given note. Then at the highest speed. Add these points to the graph.
What's Going on?
When the frequency of whirly rotation is plotted versus the number of the harmonic which it is playing, the data lies near a straight line. This means that the frequency of the note played by the whirly is proportional to the frequency with which it is twirled.
Tape a garbage bag to one end of the whirly.
Inflate the garbage bag with a blower by blowing air through the
whirly. (You can also inflate the bag by blowing into the bag with
your mouth. If you blow into the bag do not press your mouth against
the whirly, hold your mouth a few inches from the end of the whirly
to promote entrainment of surrounding air, this will increase the
flow of air into the bag and decrease the time it takes to fill the
bag.)
Play the lowest note by twirling the whirly. Time how long it takes to empty the air from the bag. (For consistency rotate the whirly as slowly as possible to play this note.) You can also time how long it takes to fill the bag while playing the note with a blower. (It is easier for beginner whirly players to control the note with a blower.)
Play the second note, then the third etc. Plot the time to empty the bag versus the number of the harmonic, i.e. plot the time it takes to empty the bag playing the lowest note at the number 2, the number of its harmonic. Notice that the plot is a straight line. The time it takes to empty the bag is proportional to the frequency of the note being played.
If you measure the volume of the bag using the Math Root below you will be able to compute the speed of the airflow through the bag. See the other Math Root below.
Math Root, Measuring the volume of the bag
Measure the volume of the bag. Fill the bag with dog food, Styrofoam chips or any other cheap granular material. Weigh the amount of material in the bag, weigh a liter of the material, then compute the volume of the bag.
Weigh the empty bag and whirly, then weigh them full. Subtract the empty mass from the full mass to find the mass of the fill, M.
Measure the mass of a liter of the fill material, m_{l}.
The volume of the bag in liters, V, is then the mass of the bag divided by the mass of one liter of fill material.
V = M/m_{l}
convert the volume to cubic meters. There are 1,000 Liters in a cubic meter.
Alternatively, place the bag into a large body of water and pour a measured amount of water into the bag through the whirly until it is full.
Math Root, The speed of flow through the whirly.
Measure the time, t, it takes to empty a bag of
volume V.
Measure the area of the inside of the whirly tube, A.
The area is A = pi r^{2}
where r is the radius of the inside of the
whirly.
Our whirly had an inside diameter of 2.5 cm and so a radius of 1.25
cm.
Express the area in square meters, ours was 4.9 x 10^{4} m^{2}.
The speed of airflow through the whirly, s, can be found by knowing the time it takes for a volume of air to flow through the cross sectional area of the tube.
s = V/At
The speed will be in meters per second.
Calculate the speed of the airflow for each of the notes played by the whirly.
Measure the frequencies of the notes using a
frequency meter or oscilloscope.
(If you do not have a frequency measuring device you can estimate the
frequency from the length of the tube. For a tube of length L the
fundamental frequency will be
f1 = v/2L
where v is the speed of sound in air (about 350 m/s)
for our tube of length 0.78 m
the fundamental frequency is
350/(2 * 0.78) = 220 Hz
Plot the speed of the airflow versus the frequency of each note.
Note that the speed is proportional to the frequency.
Find the constant of proportionality, the slope of
the plot of speed versus frequency.
The slope will have units of meters.
Measure the distance from the center of one ridge
on the whirly to the center of a neighboring ridge. This is the
wavelength of the ridges in the whirly. Ours was 0.65 cm or 0.0065
m.
At the speed of airflow in the lowest note calculate the number of
ridges the air will hit each second.
What's going on?
Theory says that when the whirly plays a note the
airflow will collide with the ridges as it flows through the whirly.
The air tumbling over each ridge makes a donutshaped vortex. Each
vortex includes a backandforth motion of the air, this back and
forth motion drives the resonant frequency of the tube. This
phenomenon is known as impinging shear flow instability, it is what
makes your tea kettle whistle. In the mouth of a whistling tea kettle
there is a plug with two disks separated by a space, there is a hole
in the middle of each disk. When the air, or steam, flows through the
first hole and then flows through the second hole it exits in
vortices which cause oscillating pressure in the air, heard by the
human ear as a whistle. The ridges in the whirly tube play the same
role. As the air flows first over one ridge then over a second it
tumbles into a vortex. The faster the air flows through the tube the
higher the frequency of the sound produced by the vortex. When the
frequency of the vortex matches one of the natural resonant
frequencies of the tube it is amplified.
To drive the resonant frequencies of the tube, air must tumble over each ridge in the whirly. This means that there must be turbulent flow of air in the tube. At low speeds the air can flow through the whirly tube smoothly, this is called laminar flow, such a smooth and quiet laminar flow will produce no turbulent eddies to drive the resonant frequencies of the tube.
Whether the airflow is laminar or turbulent depends on the Reynolds number of the airflow. Osbourne Reynolds studied the flow of fluids through tubes and found the conditions under which fluids flowed in laminar flow versus turbulent flow. This depended on the density of the fluid, r, the viscosity of the fluid, h, on the speed of fluid flow, v, and on the diameter of the tube, d. For a given fluid such as air through a tube like the whirly, the transition from laminar to turbulent flow happens as the speed of the flow is increased past a critical value at which the Reynolds number exceeds 2,000.
The Reynolds number is
R = r v^{2}r/ h
where v is the velocity of the flow in meters per
second, 4.8 m/1.3 s = 3.7 m/s
r is the
density of air, 0.8 kg/m^{3}
h is the
viscosity of the air 1 centipoise, 0.01 poise
r is the radius of the tube r = 1.25 cm = 0.0125 m
Calculate the Reynolds number for the fundamental note of your tube
and for the second harmonic.
R = 0.8 * 16 * .01/ .01 = 10
way too low for turbulence.
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25 May 2000 