High School Mathematics-Physics SMILE Meeting
1997-2006 Academic Years
Acoustics

26 October 1999: Bill Shanks (full time music student now; Joliet Central HS, ret.)
showed us a free ISPP "giveaway": a film canister with a 5/8 in hole in base and cap. Put plastic "drumhead" over canister and snap cap on to hold. A 3/8 in hole is on the side of the canister. Blowing in the hole generates a sound of a particular pitch. A PVC pipe to fit into the 5/8 uncovered hole enables one to effectively double the length of the canister, reducing the pitch by one octave! Innovative!

23 November 1999: Bill Shanks [Music Student at Joliet Junior College; formerly at Joliet Central HS]
The Physics of Music is hard to cover in Physics classes these days, because very few students know much about MUSIC, it seems. What is music, anyway? If a mass is attached to an elastic object [spring or whatever], that mass can be made to vibrate, and such vibrations produce sound. For example, a stretched string or a vibrating column of air may produce sound. One example of such a vibrating object, used both in Physics Labs and by piano tuners, is the tuning fork. The tuning fork is built to oscillate at a specific frequency [along with overtones]. One can amplify the sound by touching the vibrating fork to a flat surface, such as a desk.

A vibrating string with fixed ends held under tension can have standing waves that have an integral number of half wavelengths on the string. If the string is of length L, then the wavelengths will be ln, where L = n ln/2; or ln = 2 L/ n. The resonance frequencies are multiples of the fundamental frequency n0; i.e., n0; 2 n0; 3 n0; 4 n0; ... In a plucked string, several harmonics are excited, depending upon how and where you pluck it. The difference in between various stringed or wind instruments is determined by the excitations of different distributions of harmonics. See the website http://sprott.physics.wisc.edu/demobook/chapter3.htm. You can achieve "vibrato" in stringed instrument by changing the tension in the strings by moving your fingers.

How do you tuned musical instruments, such as pianos? Well, you can set one string frequency absolutely with a tuning fork, by listening to the beat of the tuning fork with the string. Also, take into account that an octave is a factor of two in frequency, and that the standard note is "A", corresponding to a frequency of 440 Hz. There are 12 half tone notes on a piano in an octave, and according to the generally accepted chromatic scale [http://people.bu.edu/scott/chromatic.html] or equal temperament scale, the half tone notes are taken a factor of 21/12 = 1.059463094 apart in frequency.

Here are the ratios of frequencies for notes that are n half-tones apart, for various n.

[I have given them all, so that you can see the approximate correspondences upon which primitive western harmony is based -PJ]

Factor			  n

1.05946309 1
1.12246295 2
1.18920712 * 3
1.25992105 ** 4
1.33483985 *** 5
1.41421356 6
1.49830708 **** 7
1.58740105 8
1.68179283 9
1.78179744 10
1.88774862 11
2.00000000 12
Note that, for n = 3, the Factor is fairly close to * 6/5, for n = 4 it is fairly close to ** 5/4, for n = 5 it is fairly close to *** 4/3, and for n = 7 it is fairly close to **** 3/2.

Alternate scales exist [http://www.bartleby.com/65/sc/scale2.html], such as the Pythagorean scale, the Mean Tone 1/4 scale, or the Just Scale, in which some of these ratios are set exactly at the "harmonic values" for some notes, but you cannot set up all the notes in that fashion, since that would be equivalent to making powers of the twelfth root of two to be rational numbers, and they aren't.  The chromatic scale is the one to use, in spite of the fact that the harmonics are approximate, but not precise, in it, because the ear is a forgiving detector!

23 November 1999: Roy Coleman and Lee Slick [Morgan Park High School]
They supervised our preparation of a "turkey caller" (originally developed by Bob Grimm), for special use just before Thanksgiving. The ingredients were a plastic cup, cotton string, a toothpick, and a rag wet with water. Punch a small hole in the plastic cup, push the string through it, and tie the string around the toothpick. You should break the toothpick in half before tying it. Then, pull the string so that the toothpick is stuck at the bottom of the cup. While holding the cup by the edge, pull the string, using the wet rag as protection for your hand. You should be able to get the most plaintive and distressing sounds out of this device, which is a sound generator. It works through a "slip-stick" mechanism, in which the string alternately slips and sticks as it goes through your hand, producing sounds associated with a driven vibrating string.

28 March 2000: Fred Schaal (Lane Tech HS)
first mentioned Weird Science (Lee Marek http://www.chem.uic.edu/marek/clips/marek_chem_demos.htm) on one of Letterman's shows, in which two cans of soda pop are shaken in an apparatus like a paint shaker. The cans are then simultaneously pierced, and streams of pop shoot out to the ceiling in opposite directions. Spectacular - but not done live because of the messiness. Also one may use an ultrasonic cleaner with high frequency vibrations in place of a paint shaker.

26 September 2000 Ann Brandon (Joliet West HS)
blew on a Stadium Horn she had bought at K-Mart, and pointed out salient features of its construction. Then she showed us how one could make the same thing from an old film can and some PVC pipe. She blew on a version of that, and sure enough! - Ann showed us "sound" physics!

29 February 2000: Larry Alofs (Kenwood Academy)
placed two tuning forks [http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/waves/tfl.html] on the table, each mounted on its own sounding box. They appeared to be identical except one was shorter than the other by about 1 cm.

Larry asked us to predict how their frequencies (pitches) would compare. And then he tapped one, damped it, and then the other. The longer fork had a lower pitch, which is what one would expect, because its mass would be greater; the stiffness of each would be comparable, being of same material and cross-section. But then Larry attached two identical small masses, one to each of the tines of the shorter fork, and positioned them carefully to a pre-marked location. Once again he sounded the forks, and now they appeared to have the same pitch! The added mass on shorter fork had lowered its pitch to match that of the longer fork. Neat!

But Larry had more. He placed the forks so the open ends of their sounding boxes faced each other, and then struck one fork. We heard it sound, and then Larry damped it; but we then could hear the faint sound coming from the other fork. Resonance! But the sound was weak, and to make it more obvious that the second fork was set into resonance by sound from the first, he suspended a ping pong ball from a light string so that the ball very lightly touched the upper end of one of the tines of the second fork. Then he repeated the experiment. The ping pong ball bounced away from the tine of the second fork as it vibrated from the resonant transfer of sound energy from the first fork. Beautiful!

Were the forks at exactly the same frequency? Someone suggested sounding them both at once; we would hear beats if the frequencies were slightly different. And we did! By adjusting the location of the small masses, Larry controlled the beat frequency that we heard. He set up a frequency meter with microphone, and displayed the frequency of each fork. The beat frequency would be the difference of the two (262 - 256 Hz). See the website http://hyperphysics.phy-astr.gsu.edu/hbase/sound/beat.html. Very nice! 

24 March 2001 Don Kanner (Lane Tech HS, Physics)
began by taking a plastic straw, flattening one end to make a double reed, and blowing into it to make a sound.  He cut pieces off the other end  and blew to demonstrate that, as the straw becomes shorter, the pitch of the sound goes up.  He made a very interesting presentation at the Elementary SMILE Meeting on 06 March [Section B] , which is described in detail on the website em030601.htm.

Don then announced that he would give us a surprise test -- which turned out to be a hearing test. To that end, he put a chart on the board that looked like this one:


Average Sensitivity of Human Hearing at Various Frequencies
Source: http://www.bcm.tmc.edu/oto/studs/aud.html
He started to give us the hearing test, but we were saved because there was too much hum in his audio oscillator [perhaps because of a problem in the amplifier driven by  it]. The hearing test was deferred, and Don concluded by making the following general comments:

Porter Johnson commented that our region of maximum hearing sensitivity, 2 KHz, corresponds to a wavelength of about 10 cm---the distance between our ears. This rough correspondence also works pretty well for animals.  He also commented on transmission of sound in the oceans, which enables marine mammals to communicate over great distances.  "Acoustic thermometry, however, capitalizes on the presence of sound channels present in the deep sea capable of trapping and transmitting sound over very long distances. The channels are created by the variation of pressure and temperature with depth. Located at a depth of about 3,000 feet, these deep sea super-highways act almost like a lens in focusing the sound and guiding it over thousands of miles." Source: http://www.sio.ucsd.edu/scripps_news/pressreleases/ATOC98.html See also http://www.silentoceans.org.

11 September 2001: Bill Blunk (Joliet Central HS, Physics)
made another pilgrimage this past summer to his favorite*** toy store, namely

Amazing Toys
319 Central Ave
Great Falls MT 59401
[406] 727-5557 [Bob Pechlin]
http://www.amazingtoys.net/

There he discovered a new plaything, called a Sonic Lightning Ball, which he showed to us.  When you drop the plastic ball [with electronic components clearly visible inside], it makes different sounds and colorful light flashes when it bounces up from the floor.  The sounds and lights change with the orientation of the ball, as well as the drop height. Thus, the magnitude and direction of the impact force are relevant for the effects that follow the rebound.  The device was passed around the room, and some of us got sounds by squeezing on the ball.
***In the interest of full disclosure, Bill indicated that he had no commercial interest in the store.

11 October 2001 Fred Schaal (Lane Tech HS, Mathematics) Conversion from Bat to Human Frequencies
Fred handed out and discussed the following exercise:

The range of a human's voice frequency, h (all in Hz), is 85 £ H £ 1100. The range of a human's hearing frequency, H, is about 20 £H £20,000.
  1. The relationship between a human's voice frequency h and a bat's voice frequency, b, is given as
    h = 85 + (b - 10000) ´ 203 ¸ 22000.
    Find the range of the bat's voice frequency.
  2. The relationship between a human's hearing frequency and a bat's hearing frequency is given to be
    H = 20 + (B - 1000) ´ 999 ¸ 5950.
    Find the range of the bat's hearing frequency.
  3. If a bat flies into your room and you scream, will it hear you? If you scare it and it screams, will you hear it? Explain.
The numbers for bats come out to be
Voice:      10000 £ b £ 110,000
Hearing:        1400 £ B £ 1,200,000  

It seems that you should hear the bats (if you are young enough), but the bats cannot hear you.  Alas, your screams (blood-curdling though they may happen to be) are wasted on them!

Porter Johnson remarked that bats and other small animals tend to produce and detect sounds of higher frequency, because their vocal chambers and  hearing receptors are smaller.  The lowest sound produced by humans, about 85 Hz, corresponds to a wavelength of about 4 meters.  For quarter-wave resonance, the length of the vocal cavity should be about 1 meter.  This sound of a Basso Profundo must come from deep within the lungs! The ratio of maximum to minimum frequency for human voice is 1100 ¸85 » 13, corresponding to 3.7 octaves.  The ratio for human hearing is 22000 ¸20 »1100, corresponding to 9.7 octaves.  Thus, we can hear every note on the piano [88 ¸ 12 » 7.3 octaves], but we cannot sing that full range!  Trained voices can roughly be categorized by the following types, in order of increasing frequency:

Although the vocal range can be extended through extensive training, to a great extent it is determined naturally. It is said that the popular singer Whitney Houston  [http://www.infoplease.com/ipea/A0154789.html] and Mariah Carey [http://mrshowbiz.go.com/celebrities/people/mariahcarey/bio.html] are said to have vocal ranges of 5 octaves, achieved without classical voice training.  See also http://www.snopes.com/music/artists/carey.htm.

For bats the ratios of maximum to minimum frequencies are similar:

Voice:      110,000 ¸10,000 » 11
Hearing:        1,200,000 ¸1400 » 860

Bats are rather intelligent mammals that use ultra-sound sonar (as well as their eyes; they are not blind) to guide their flight, and to track the flight of other things in their air space. Because of the webbed wings attached to their limbs, their pattern of flight is distinctively jerky, rather than smooth as in birds with feathered wings. Bats are nocturnal animals, and eat vast quantities of  insects. They hang by their limbs and sleep upside down  in caves, high in trees, or inside abandoned buildings. They make a high-pitched, audible sound.

20 November 2001: Larry Alofs (Kenwood HS, Physics)  [Reveille]
Larry
passed out copies of  music for the standard bugle calls, as obtained from the website of the Boy Scouts of America, http://www.usscouts.org/mb/bugle_calls.html. (Sound files of the various calls can be obtained at http://www.kmialumni.org/bugle_calls.html.)

Larry asked why only certain notes appear in these calls --- only those notes that can be played on a bugle.  What is special about those notes?  To address the issue, he took an electric shaver, with a fairly loud characteristic 120 Hz hum, and placed it against his cheek with his mouth open.  By changing the position of his mouth, he was able to excite those same notes, and thus to play the familiar bugle calls.  How come?

As an additional exercise, he put a plastic tube inside a water-filled cylinder, and moved the tube up to produce a column of air about 50 cm long.  Then, he played the bugle calls using the hum from the electric shaver, varying the size of the tube opening with his hands.  Next, he took a tuning fork, and showed that the loudest resonance of the fork [440 Hz] corresponded to a column of air of about a quarter wavelength, or about 20 cm.  The other resonances of the column of air, at wavelengths l = 4 L/(2n+1), are those excited by the bugle. The mouth is more of a broad Helmholtz resonator [http://www.phys.unsw.edu.au/~jw/Helmholtz.html], which does have such a simple spectrum of resonances, but it is recognizably close to the linear resonator.

Brass Instruments:  "The pitch of a brass instrument depends on the volume of air that is vibrating, as well as the speed at which the player's lips vibrate. The volume of air depends on the length of the tube; a longer tube means a larger volume of air, hence lower pitch. By buzzing her lips faster or slower, the player can cause the air in the tube to resonate at different harmonics (see the discussion of harmonics and overtones in the physics section). With a single-length tube this yields only the notes found in bugle calls. To get all 12 notes of the chromatic scale, the player needs to change the length of the tube, as on the trombone, or play through different lengths of tubing, as on the brass instruments with valves."
Source:  http://exhibits.pacsci.org/music/Instruments.html.

Natural Trumpet: "The natural trumpet differs from the modern trumpet in two crucial ways: firstly, it is twice the length and secondly, it has no valves and is therefore unable to play more than a limited number of notes. These properties make the natural trumpet very distinctive. The military connotations of the trumpet made it one of the most prized instruments at court, often the trumpet corps outnumbered all the other courtly musicians put together; the Charamela Real corps in Lisbon called for 24 trumpets for one of its fanfares." Source:  http://www.iquint.co.uk/instruments.html.

It was pointed out that Rudy Keil had once used an electric shaver in a SMILE class to produce vibrations in a string, which were studied with a strobe light.

05 March 2002: Don Kanner (Lane Tech HS Physics) -- Reed Instruments
Don
showed us how to play reed instruments made from soda straws. When he shortened the straw with scissors, we heard the pitch go up, in agreement with our expectations.  We would expect that the shorter the straw the higher the pitch, and the longer the straw the lower the pitch.  We measured the length of the straw to be L = 0.205 m, and for the fundamental mode [with open end boundary conditions, neglecting end effects] , the wavelength should be l = 2 L = 0.41 m. The frequency can be calculated by dividing the speed v of sound (in air at 25 °C) by this wavelength; that is f = v / l = 340 m/s / 0.41 m = 830 Hz. That seemed consistent with our observations of a "shrill" sound, nearly an octave above middle A (440 Hz).

Don then sucked in some Helium gas from a balloon, and spoke a few reassuring words, sounding like Donald Duck, an erstwhile namesake. Next he took Helium into his lungs again, and blew into the straw.  We expected a very high pitched sound, since the velocity of sound in Helium is about 970 m/sec, so that f = v / l = 970 m/s / 0.41 m = 2400 Hz.  (This frequency could be achieved in air with a straw of length 0.07 m!)  The pitch we heard with Helium was definitely higher than that obtained for the same straw with air, but by no means as high as expected. How come?  One possibility was that Don was still exhaling Helium mixed with residual air in his lungs.  You took our breath away, Don!

Don also gave Earl some articles and a video demonstration concerning the "dimple" in the belt mentioned by Earl in the SMILE meeting of 05 February 2002.   Earl will look at them, to see if the effect is the one he mentioned, and whether the explanation is reasonable.

02 April 2002: Larry Alofs (Kenwood HS Physics) -- Vibrating Pipes
Larry
showed us a cylindrical whistle with a sliding plunger.  He blew on the whistle, and as he moved the plunger back and forth, we heard the pitch of its sound vary smoothly higher and lower, not unlike a siren. He then pulled the plunger all the way out, leaving its bottom end open, and we heard the sound reach its lowest pitch.  Next, he covered the open end with his finger, and we heard the pitch go up by about one octave.  After repeating this a few times so that we could be certain of what we observed, he excited the higher harmonics by "over-blowing" (blowing very hard) on the open-ended whistle, showing that the harmonics were about one and two octaves higher than the fundamental.  By contrast, the first harmonic of the closed end pipe had three times the frequency of the fundamental, corresponding to and octave and a fifth [i.e. » 2**(1 + 7/12)]; for details see the 23 November 1999 SMILE lesson, ph112399.html.

Larry "played" the open pipe, first tapping and bouncing one finger off the end, and then "closing" the end with each finger tap.  We could certainly see and hear the difference in "open end" and "closed end" modes; the transient sounds were about an octave apart.  Pretty!  Larry next filibustered a bit on the conceptual errors caused by most textbooks, that display longitudinal sound waves in air as transverse waves.  He showed the following alternative display (of his own invention) of the first few modes of a pipe with one end open:

                 Nodes: ©       Antinodes:  <--->      
___________________________
|
|
|© <-----> Fundamental
|
|___________________________
___________________________
|
|
|© <---> © <---> First harmonic
|
|___________________________
___________________________
|
|
|© <--> © <--> © <--> Second Harmonic
|
|___________________________

Using such pictures, you can clearly indicate the resonant sound as a longitudinal displacement wave.  Larry indicated that he was showing nodes and antinodes of the displacement , and that for pressure or density (as measured with a transducer and shown on an oscilloscope), the nodes and antinodes would be reversed.

23 April 2002: Arlyn VanEk (Illiana Christian HS Physics) -- Standing Waves Using Scroll Saw Apparatus and a Marimba
Arlyn
brought in a scroll saw [form of jigsaw with flat table]. He tied one end of a string to the top end of the blade. He played out 2-4 meters of string, pulled the string taut, and turned on the saw.  By varying the tension in the string, he could produce various standing waves.  These transverse waves, with nodes at the ends, corresponded to N half-wavelengths, with N-1 internal nodes.  That is, the length L of the string and the wavelength l are related by the relation L = N /2 l .  We could see the fundamental mode N = 1, as well as the first two harmonics, N = 2; N = 3. The frequency, n, of the waves is fixed by the scroll saw frequency; presumably, something like 60 Hz. The velocity, v, of waves on the string is given in terms of the tension T and the mass per unit length m as v = Ö (T/m ); in turn the velocity v is given by v = ln. Thus, the tension required to excite the Nth mode is inversely proportional to N2 TN = 4mn2  L2 / N2.  In order to increase the mode number N, one must decrease the tension T

Arlyn next described a Marimba [like a xylophone, except perhaps more so; see http://www.mallet-percussion.com/marimba.html], which typically consists of wooden pieces with supporting members arranged to tune to a pentatonic scale.  The avid Marimba player then plays the instrument by striking the pieces one at a time, as required by the melody.  Arlyn illustrated the operation with two different [old pine construction] boards [2" ´  8" ´ 4'] supported transversely by pieces of (slit) hard rubber hose.  By adjusting the distance between supports; then striking them with a rubber mallet, Arlyn tuned the resonant frequencies of these boards.  By sprinkling sand on top of one of the boards, he showed that nodes occur at the support locations, so that the distance between them was about a half wavelength.  The distance between supports was 0.5 meters, the resonant frequency was be around 400 Hz. This simple "two note marimba" sounded quite nice, Arlyn!  He went on to strike the end of a metal rod against the desk, holding it at various points to excite various normal modes.  The "punch line" is that if you hold the rod at the location of a node of a low-lying resonance of the longitudinal vibrations of the bar, you will excite that mode when striking the bar.

Beautiful Physics, Arlyn [and especially enlightening for scroll saw operators and marimba players].

25 February 2003: Don Kanner [Lane Tech HS, Physics]     The Search for the Missing Jug Equation
Don
began by describing an assignment for his class to find web-based information on the collapse of the Tacoma Narrows Bridge in November 1940.  One student discovered that the frequency of vibration of the bridge was 30 Hz, so that the torsional wave might more appropriately be described as a "hummer", rather than a Galloping Gertie: http://www.ketchum.org/tacomacollapse.html.  Also, at about the same time that Professor Faquharson was running down the nodal line on one side, Leonard Coatsworth, a Tacoma newspaper reporter was running along the edge on the other side --- and hanging on for dear life.  The wind-induced oscillation was described in one place in terms of "negative damping", and an "aerodynamically assisted self-excitation", which they insisted that it was not a "resonance", as such: http://www.ketchum.org/wind.html.  For additional information see the NASA Report From Bridges and Rockets: Lessons for Software Engineering by C Michael Holloway: http://shemesh.larc.nasa.gov/fm/papers/Holloway-Bridges-Rockets.pdf.

Don then took out an empty 5 liter wine jug, blew across the lip of the jug, and measured the frequency f as D#: 75 Hz on his music note audio detector.  He computed the wavelength for this note using the velocity of sound (c    »  350 meters/second) by  l  = c / f = 350 / 75 = 4.68 m. Don asked how this could possibly be correct, in light of the fact that the lowest resonant frequency for a long, thin tube of length L, enclosed at one end, corresponds to a quarter wavelength; i.e. l / 4 = L.  There are higher harmonics in vibrations of, say, an organ pipe, but not here!  Don then opened a half-liter soft drink bottle, drank from it until the fluid level was down to the major diameter of the bottle, and blew across the bottle, obtaining the frequency  A: 440Hz, corresponding to a wavelength  l  = c / f = 350 / 440 = 0.80 m.  Again Don was puzzled, in light of the fact that the height of the air inside the bottle was only 7 cm = 0.07 m.  He continued to experiment with an Erlenmeyer flask (straight sides) and a Florence Flask (curved sides), finding that the frequencies were different, under similar conditions.  How come?  Larry Alofs [Kenwood HS, Physics] wisely suggested that the devices in question were called Helmholtz Resonators, which would provide a key for determining the Jug Equation, or the equation for the vibrational frequency of the jug in terms of its size and shape.

For additional information, see the Helmholtz Resonance website of Professor Joe Wolfe of the Musical Acoustics Group, University of New South Wales, Sydney Australia: http://www.phys.unsw.edu.au/~jw/Helmholtz.html.  The following information has been extracted from that source:

"A Helmholtz resonator or Helmholtz oscillator is a container of gas (usually air) with an open hole (or neck or port). A volume of air in and near the open hole vibrates because of the 'springiness' of the air inside. A common example is an empty bottle: the air inside vibrates when you blow across the top, as shown in the diagram at left. (It's a fun experiment, because of the surprisingly low and loud sound that results.)

Some small whistles are Helmholtz oscillators. The air in the body of a guitar acts almost like a Helmholtz resonator. An ocarina is a slightly more complicated example. Loudspeaker enclosures often use the Helmholtz resonance of the enclosure to boost the low frequency response.

The vibration here is due to the 'springiness' of air: when you compress it, its pressure increases and it tends to expand back to its original volume. Consider a 'lump' of air at the neck of the bottle. The air jet can force this lump of air a little way down the neck, thereby compressing the air inside. That pressure now drives the 'lump' of air out but, when it gets to its original position, its momentum takes it on outside the body a small distance. This rarifies the air inside the body, which then sucks the 'lump' of air back in. It can thus vibrate like a mass on a spring. The jet of air from your lips is capable of deflecting alternately into the bottle and outside, and that provides the power to keep the oscillation going."

In addition, the Eric W Weisstein / Wolfram Research website on the Helmholtz Resonator (http://scienceworld.wolfram.com/physics/HelmholtzResonator.html) contains the following formula for the angular frequency w= 2 p f of resonance:
w = c [ A / (V ´ L) ]1/2
Here the frequency is expressed in terms of c---  the velocity of sound, A --- the cross-sectional area of the column, L --- the height of the vertical column, and V --- the volume of air inside the container.

Don admitted that he played the jug [and other improvised instruments] in the Windy City Jammers Group, which plays fine music upon request.  Although you played well, you should not quit your day job, Don.  Interesting and Thought-Provoking!

11 March 2003: Don Kanner [Lane Tech HS, Physics]      Update on Jug Investigation
Don
described experiments he had done with several pop bottles of various sizes, which had the same size opening.  By measuring the resonant frequency, f, with various amounts of water in the bottles, he learned that the resonant frequency was dependent only upon the volume, V, of air in the bottle  In fact, he found this approximate formula for the frequency f [Hz] in terms of the speed of sound [c= 35,000 cm/sec] and  V [ in cubic centimeters]:

2 p f = c / ÖV
At this point Don checked the facts on Helmholtz Resonators, which indicated that area, A, and height, H, of the opening column, along with V, were relevant for the resonant frequency: 2 p f = c / Ö(A/HV). For these bottles, the sizes and shapes of the openings are the same, and thus the answer for f depends only on the volume V.  In fact, one might expect to find that  A/H is about 1 cm, for consistency.

11 March 2003: Carl Martikean [Wallace HS, Gary, Physics]  Ding-a-Long:  You Conduit [Con-du-it] Yourself
Carl
had been successful in constructing his own set of one octave chimes, using a 10 foot [3 meter] section of C4 thin wall, galvanized steel conduit, which can be found at any hardware store.  He suspended the chimes, cut at appropriate lengths for the chromatic scale, using 10 pound test fishing line.  Carl played some notes for us by hitting the suspended pieces with a wooden striker, producing an almost recognizable melody. Carl also showed us how to make a closed end pipe, using lengths of hollow PVC tubing, attached at the base on Craft Foam through the use of a Hot Glue GunCarl pointed out that you could make a base consisting entirely of solidified hot glue, without using the craft foam as a substrate at all.  Carl also showed us how to make a mallet, using a wooden ball with a hole in it and a dowel rod.  Materials can be found at Hobby Shop stores, for example.

A serious discussion arose as to whether air vibrating inside the tube, or vibrations of the tube itself, were responsible for the vibration.  We concluded that the tube was vibrating, since the sound was damped when you held it in place, and since the sound did not change very much when one end of the tube was taped over. 

Clever, Carl! You almost had us fooled!

25 March 2003: Larry Alofs [Kenwood HS, Physics]        Totally Tubular!
Larry
placed 8 pieces of 1.25" [3 cm] diameter PVC Tubular Pipe, so they stood vertically on their ends on the table.  He arranged the pipes in a row in order of decreasing height.  He had cut them to length to produce an octave of major notes. He modeled the design of the apparatus on that shown in the Educational Innovations Catalog [http://www.teachersource.com/].

Specifically, the [BOM-100 Basic Boomwhacker Set - $31.95]:  These eight labeled tubes produce the C-Major Diatonic Scale. The end-caps lower the tones by one octave. Included in package: eight tubes, long with removable end caps.

"These brightly colored, tuned percussion tubes are great for teaching students, of any age, about sound. When whacked against your knee or the floor each produces a particular note. The longer the tube, the lower the note. Each tube is color-coded and labeled with its precise note. When the tube is closed at one end with a cap (available with tubes and separately, see below), the note shifts an octave lower. Boomwhackers were invented by Craig Ramsell when he noticed that cardboard tubes from wrapping paper could be used to produce music. These tubes are amazing, loads of fun and very educational. Put a class set together and compose your next science sound lesson!"

Larry first calculated the length of pipe necessary to make Middle C [f = 261 Hz, corresponding to a wavelength l = v / f =( 345 meters/sec) / 261 Hz = 1.32 meters].  Larry anticipated that the open-ended pipe should be of length L = l /2 = 0.66 meters. However, because of end effects, he found through acoustic tuning that a length of 0.63 meters was needed.  Larry then presented the following table of lengths:
Note Length Calculated Length after Tuning
C (63.0 cm) 63.0 cm
C# - Db - -
D 56.1 cm 56.1 cm
D#- Eb - -
E 50.0 cm 49.8 cm
F 47.2 cm 46.9 cm
F#- Gb - -
G 42.0 cm 41.5 cm
G#- Ab - -
A 37.5 cm 36.9 cm
A#- Bb - -
B 33.4 cm 32.2 cm
C 31.5 cm 30.3 cm

 The calculated lengths were obtained from the first number (63.0 cm) by dividing (once or twice, as appropriate) by the factor 2 1/12 = 1.05946 ... , as required for the Chromatic Scale.  There was a lot of discussion as to how to include end effects.  For a pipe with one end closed, the traditional expression for the effective length of a pipe, Leff,  is given in terms of the pipe length L and pipe diameter D as Leff = L + 0.4 D.  

Larry next pointed out that, for pipes of resonating air with one end closed, the pipe length is given in terms of the wavelength  l  by L = l /4.  In other words, for a given pipe, the wavelength would be reduced by a factor of 2, and the frequency f would double, in going from two open  ends to one open end.  By striking the end of the pipe against his hand, Larry demonstrated this octave shift.  While several of us held the pipes, Larry played the tune Mr Frog, which was the first piece he learned to play on a piano.  Not to be left behind in this musical extravaganza, Don Kanner illustrated the West African Shantu [http://www.billabbie.com/nigeria/music.htm] instrument, hitting the pipe on his thigh.  It produces an interesting sound, but it seems likely to leave bruises. For details see Exploring Music: The Science and Technology of Tones and Tunes by Charles Taylor [Institute of Physics, 1992, ISBN: 0-7503-02135]

Larry, you make "fairly" beautiful music while showing very beautiful physics!

08 April 2003: Karlene Joseph [Lane Tech, Physics]     Wine Glass Resonance
Karlene
brought in a  wide variety of wine goblets, and filled them to different levels with water.  She rubbed with her wetted finger in a circular fashion around the rim of one, while holding its stem with the other hand. To our delight, we  heard a musical sound, or ringing, as the glass resonated.  We made the following remarks, based upon our experiences:

For additional information on wine glass resonance, see the website Crystal Goblets Can Singhttp://www.wonderquest.com/goblets-sing.htm.

It sounds great, Karlene!

29 November 2005: Bud Schultz (Aurora Middle School Academy)              Easy primitive toy
Bud
showed us a "bull roarer" which is an Australian aboriginal toy. It operates to produce a roaring sound when it rotates in two planes. The bull roarer is a piece of wood carved to be about the size and shape of a spear head and suspended from a string of about 2 meters in length. Different shapes can be made; Bud's were carved out of Padauk  -- a reddish wood from Africa: http://www.exotic-wood.com/african_padauk.htm. First Bud spun the bull roarer, --- then he held the string at the other end, and twirled the bull roarer in a vertical circle.  The sound was not produced until the rotation became quite rapid -- it requires both rapid rotation and simultaneous spinning in a second plane.  For additional information see the Virginia Tech Multimedia Music Dictionaryhttp://www.music.vt.edu/musicdictionary/textb/Bull-roarer.html. Wow-wow-wow!  Thanks, Bud.

07 March 2006: Don Kanner (Lane Tech HS, physics)            The Sound of Physics
Don suggested modifying the simple discussion of open ended and closed ended organ pipes, as made at the last meeting by Larry Alofs. The equations l = 4L (for a pipe with one end closed)  and l = 2 L (for a pipe with both ends open) gives the fundamental frequency only for pipes under certain geometrical restrictions.  In fact, it is an oversimplified way of describing the vibrations. It is not just the length of the pipe that determines the pitch; we tested this for various pipes, across which we blew air to try to make standing waves. Another exercise involves a Florence flask and an Erlenmeyer flask of equal heights and volumes. They produce sounds of rather different pitch when air is blown across  them. The size of the neck and opening of the vessel is also important in determining what tone is made in this way.  For additional information see The Resonance of Common bottles and Jugs by Don Kannerhttp://www.iit.edu/~smart/kanndon/lessonb.htm. Hermann Helmholtz actually found out that the ideal shape for a resonating volume is a sphere.  For additional discussion see the comments at the 25 February 2003 HS Math-Physics SMILE meeting:  mp022503.html.

Sounds good!  Thanks, Don.

02 May 2006: Arlyn VanEk (Illiana Christian HS, physics)                 Impedance Matching
Arlyn
held one end of a piece of rope, and a volunteer held the other end.  Arlyn then shook his end of the rope up and down, while the volunteer held the other end fixed.  By adjusting the tension in the rope, Arlyn was able to set up a standing wave with two nodes.  He reduced the tension, and was able to get standing waves with one internal node, and with no internal nodes at all.  By moving his end in a circular path, Arlyn was able to set up a spiral standing wave. He then tied an equal length of lighter rope to the heavier one, and the process was repeated.  Arlyn was unable to produce any standing waves in this case.  With the double rope held fixed under tension, Arlyn plucked his end, and the resulting pulse traveled toward the other end. But when the pulse got to the point where the two ropes were connected, it was partially reflected and partially transmitted.  How come? Arlyn said that there was a mismatch in the (mechanical) impedance at the junction.

The mechanical impedance of a vibrating system is defined in analogy to the electrical impedance of an electric circuit [for details see T.D. Rossing and N.H. Fletcher, Principles of Vibration and Sound -- Springer Verlag 1994, ISBN 0-387-94336-6], as illustrated in the table below: [Note: i = Ö(-1) and j = -i.]

IMPEDANCE

Item Electrical Mechanical
Driving Term Voltage: V0 e-iwt Force: F0 e-iwt
Response Current: I0 e-iwt Velocity: v0 e-iwt
Impedance Z(w) = V0 / I0 Z(w)= F0 / v0
Dynamics L I ' + RI + Q/C = V  mv' + Rv + kx = F
Form Z = R  + j (wL -1/(wC))     Z0 = R + j (wm - k/w)

In both cases, the relevant impedance must match for smooth transfer of energy from one system to another.

Let us consider the (small) transverse displacement of an ideal flexible string of length L that is held fixed at x = 0 and terminated at x = L. Using the formula  y(x,t) = A sin kx e-iwt, the corresponding transverse velocity is v(x,t) = -i wA sin kx e-iwt. The transverse force exerted by the string at x = L  is equal to  T ( y/ x) evaluated at x = L, which equals   kT cos kL e-iwt. The mechanical impedance is the ratio of the transverse force to the velocity x = L.  We obtain:

Z 0=  k T cos kL / [- i w sin kL ] = -j k T cot kL /w
Let us apply the relation w / k = v = Ö(T/m), where the speed of traveling transverse waves, v, is expressed in terms of the tension T and the mass per unit length m, to obtain
 Z0 =  -j kT cot kL / w = - j Ö(T m) cot kL
This input impedance must match on both sides when two ropes are tied together, for effective transfer of energy across the interface.  For details see Mechanical Impedance:  How It Works and Why You should Care on the Symposium Acoustics website: http://www.symposiumusa.com/tech2.html.

If you taper a rope gently (like a whip) you can transfer the wave all the way down to the tip of the whip. Arlyn used a real bull whip to show this, and it made a loud crack -- a sonic boom. For additional details see The Shape of a Cracking Whiphttp://www.npr.org/programs/wesat/features/2002/june/whip/index.html. The whip is an impedance transformer. A megaphone is another example of an impedance transformer, where the megaphone makes a gradual change for the air within the cone, from warm near the mouth to cooler air near the end of the megaphone.
Cracking phenomenological physics, Arlyn.  Thanks.