Compilation: Resonance Labs
Compilation: Resonance Labs
Date: Fri, 10 Mar 2006
From: Charity Clough
Subject: Resonance Lab Question
My honors class just completed a lab on resonance. They were given an open tube with a plunger (to close one end and to change the tube length) and an assortment of tuning forks. Most the groups came up with wonderful data in which the slope of 1/frequency vs. tube length was exactly 1/4 the speed of sound (86.25m/S). Since the tube is open at one end and closed at the other, this is great. However, I found one rogue group with a near perfect correlation with a slope of 40.5m/S, roughly 1/8 the speed of sound. Since the fundamental is 1/4 wave, this is a problem. If they had found overtones, the slope would have been larger, not smaller. I checked their technique and could find nothing wrong. They used metric, they measured from the correct end of the tube, and they were just outside my room so it was very quiet. I even had them simulate a data point where they hummed and measured for me :) I don't know what could have caused this. Any thoughts?
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Date: Sat, 11 Mar 2006
From: Michael Crofton
The explanation for this is that they had tuning forks that also vibrated at the first overtone. If you do a FFT analysis of tuning forks you will often find, for example, that a 256 Hz fork also produces some vibrations at 512 Hz. I have never had kids find this for every tuning fork they used, but I guess you have proven it is possible. Generally the resonance of this overtone is much softer than the true resonance as the amplitude of the overtone is usually quite small. Have them make sure their resonance is very loud.
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Date: Thu, 16 Mar 2006
From: Jean Oostens
The overtone explanation makes sense if one remembers that the tuning fork does not have a harmonic spectrum. The first overtone is at about 6.2… times the marked frequency. This sound is heard when the tuning fork is hit on a hard surface. It dissipates faster than the fundamental due to higher losses at higher frequency. The other part of the explanation assumes that the students used the next resonance of the tube, = (4/3) L. If f is the marked frequency, then one has the relation:
vsound = ( 6.2... f ) * ()
340 m/S = (6.2... f) * ( 4/3 ) L
Solving for f:
f = ( 340 m/S ) * [ 3 / ( 4 * 6.2..) ] * (1 / L)
f = 41.13 Hz.m * ( 1 / L )
in good agreement with the slope found, i.e. 40.5 Hz.m or 1.0 % off. This should be tested using the same equipment, as the effective length of the tube is affected by its diameter.
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Date: Fri, 17 Mar 2006
From: "Nicole Y. Murawski"
I thought it might be interesting to share what I found just this morning, in light of Jean's comment that: "The overtone explanation makes sense if one remembers that the tuning fork does not have a harmonic spectrum. The first overtone is about at 6.2... times the marked frequency. This sound is heard when the tuning fork is hit on a hard surface. It dissipates faster than the fundamental due to higher losses at higher frequency." This morning - using Vernier FFT graph, I struck a '320 Hz' tuning fork on a "hard surface." What I found was that there was a: very intense sound at 322.266 Hz (must be the true fundamental), moderate intensity at 644.531 Hz (a perfect second harmonic!), very weak intensity at 1660.156 Hz (nearly a 5th harmonic), strong intensity at 1972.656 Hz (nearly a 6th harmonic - like what Jean was mentioning), very weak intensity at 2294.922 Hz (near a 7th harmonic). I thought that was really cool. I used it to do some last minute reviewing of timbre and overtones with a few of my physics kids this morning prior to their sound test today.
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Date: Mon, 20 Mar 2006
From: Jean Oostens
Here is additional information on tuning forks. A tuning fork is modeled as a beam clamped at one end and free at the other end (one tine only). One deals with a flexion wave, so the wave equation applies to the curvature: 1/R = y''/ (1 + Y') approximated as y'', the second derivative; the eigenvalue wave equation becomes: d(y'') / dx + wk4 = 0 where w is the angular frequency of the normal mode and k depends on the material and the moment of inertia of the beam. The fourth order differential equation has a general solution which involves the sine & cos, but also their hyperbolic counterparts, sinh and cosh:
y = A sin kx + B cos kx +C sinh kx + D cosh kx
with the boundary conditions, at the clamped end, being x = 0, yo= 0, and yo' = 0. The tangent is not moving.
At the free end, x = L, yL'' = 0, the curvature is zero (as there is no local torque), and yL''' = 0.
The derivative of the curvature is 0, as no wave can propagate beyond the end of the beam. When the various derivatives are calculated and the four conditions are set up, we obtain a 4th degree homogeneous equation in A, B, C and D that needs a no-trivial solution (not just A = B = C = D = 0). After a painful calculation (I failed to reconstruct it from scratch) one obtains a transcendental equation that determines an infinite number of values for the product k = w. The value of " 6.2... " I quoted is the ratio of the first two solutions for w. It is a transcendental number, of the same kind as and e; it is not the solution to any algebraic equation of any degree with rational coefficients. Next week, I will be back home and will give more details. If interested: send me your e-mail, so that I can send you dat files and jpg pictures.
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Date: Wed, 22 Mar 2006
From: David Groth
As I was catching up on the Modeling digest submissions, I came across an entry by Andy Edington that contained the thought "While students found the lab interesting, I'm not convinced that it increased their understanding". Immediately, I thought of the tuning fork lab that determines the speed of sound. I have used this lab many times in both a physical science course as well as in physics courses. I myself struggle with the large amount of wave theory that lies at the roots of the calculations. Why does a closed end tube display resonance at every 1/4 of a wavelength? How does a pressure wave reflect off of the sides of the tube? Is the crest of a pressure wave high pressure? What would it be like at 1/4 of a wavelength when the waves interfere? Do students really understand how resonance is involved? The lab works and the students love banging tuning forks and putting them up to their ears and noses, but I struggle with whether they are really able to understand why we need to correct for the diameter of the tube, and why we need to multiply the length of the tube by four to get the corrected wavelength. Is a lab that is thought of as "fun" by the kids okay to do even if it does not significantly increase their knowledge of the subject?
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Date: Thu, 23 Mar 2006
From: Bob.Allison
Sure. Physics is fun, so if you have time in the course, why not let them just have fun once in a while? But here is how I do a simpler experiment, which I have found they can understand, and that as a bonus is fun too. I am sure this idea came from someone else, but I do not know who, so thanks to all those who gave me all my lab ideas:
- Get a pair of hardwood blocks that make a sharp and loud sound. (The band will have some in the percussion section.)
- Find a large and flat wall with a large open space along side it. (Maybe a playing field next to a 2 story gym wall.)
- Take the class outside and have a musician who can keep a steady beat clap the blocks. Find a place 100 to 300 feet away from the wall where there is a good echo. Stand sideways with one ear towards the wall and clap at different speeds until the echo overlaps the next clap. The clapper must hear the claps and echo as one sound and keep a steady beat. - One group measures the distance to the wall. One group measures the temperature (and humidity if you like). Everyone else starts their stopwatch and counts claps, stopping after as long a time as possible without losing count. They each record their time and number of claps. - Back inside you can get the average time between claps. Question them until they see that this is the time to get to the wall and back. From here they will get the speed of sound on their own.
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Date: Thursday, 23 Mar 2006
From: Don Yost
Just to agree with comments made about measuring speed of sound with waves. Unless this is the culmination of a unit of waves, I think it's too removed from conceptual level. If looking for concept, I would suggest first going out with flag, starter gun and two groups a quarter mile apart and measure sound directly. That could be followed by clapping boards near gym wall and "timing" reflections. I would think that measuring interference patterns from two speakers on frequency generators would be more conceptually accessible than the resonance pipes.
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Date: Friday, 24 Mar 2006
From: Bob Allison
FYI - I have had good results by using two family radios instead of flags. The far group can
hear the start and stop sounds. The response delays and error seems to be reduced over flags.
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Date: Sat, 25 Mar 2006
From: Tim Erickson
Another fun speed-of-sound lab works like this (large urban school in this case, facing a paved yard): We went outside, to the schoolyard, facing the imposing facade of the back of the school. No other students were out playing. We measured the distance to the building; it was somewhere around 40 meters. If one person clapped, we could hear the echo. It became a rhythm game. You work to make it as even as possible: clap - echo - clap - echo, etc. The interclap interval is then time it takes sound to go 4 times the distance you measured.
A nice thing about this is you can measure many claps to increase precision. Is this do-able? Suppose you measure 10 claps and it takes 5 seconds. (That's clapping at 120 bpm, the tempo of a Sousa march, with the echoes in between, "at the eighth notes"; it's completely reasonable to time something that fast.) If we're at 40 meters, the sound went 1600 meters in those 5 seconds; 1600 / 5 = 320 m/S.
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Date: Sat, 25 Mar 2006
From: Don Yost
"FYI - I have had good results by using two family radios instead of flags. The far group can hear the start and stop sounds. The response delays and error seems to be reduced over flags."
Good suggestion. Then, armed with this data, it would seem to me that if you were interested in resonance, you could go back into the lab, run the resonance lab using the speed of sound you have already found, and then build a model for resonating waves. This concept flow resonates better with me than giving information about resonating waves and then finding the speed. Does this sound like a good idea? Reflect a bit and wave if you agree.
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Date: Sun, 26 Mar 2006
From: Michael Crofton
On the Modeling server there are materials for the study of mechanical waves and sound. The materials have been used successfully for the last 3 years in the ASU summer modeling curriculum. To get to them go to the Modeling site, choose Modeling Instruction in High School Physics, then click on curriculum materials, and then choose participant resources.
The sound unit has a large document called teacher notes that contains the curriculum for the unit. The teacher notes are detailed and contain sample graphs and pictures for the unit. The progression for the sound unit is something like this:
1. Discussion of longitudinal waves.
2. Determine the speed of sound. (On page 4) To do this, use a closed tube with a microphone at the end. Determine the time between the generated sound (a finger snap) and the reflected pulse off a graph produced with Logger Pro or Data Studio software. Using the length of the tube, determine the speed of sound.
If you care to get into it you can take the end cap off the tube and show that sound reflects off an open end of the tube also. It can also be shown that the phase of the reflection of the pressure wave is changed 180 degrees on an open end, and no change off a closed end.
3. Demonstrations and discussion of resonance. (Pages 8-12)
4. A short resonance lab (pages 13-15) where the length of the tube at first resonance is graphed versus the wavelength of the tuning fork. The slope of the graph is 1/4 the speed of sound determined earlier. If the students are careful they will also get an intercept that is close to -.4d showing the displacement antinode is not even with the end of the tube.
5. A practicum can be done where students are given an unknown tuning fork and asked to determine its frequency.
6. A sound activity (pages 16-18) is done using Logger Pro or Data Studio where, among other things, beats and harmonics are introduced. Students can bring in musical instruments and examine the sound.
7. The last topic is the Doppler Effect.
If you download the materials and have any new ideas, comments, or corrections, please contact me directly.
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Date: Fri, 31 Mar 2006
From: Jeff Adkins
I do the clap-echo-clap lab described by Tim Erickson earlier in this thread. I've done it with the timing method Tim describes. I haul a laptop outside with a Vernier microphone attached; upon recording the sound, we return to the classroom, and we can measure timing of the clap and the echo clearly on the graph of the recorded sound. Also, I use a hammer and a board to make a sharper, louder "clap." The computer-based measurement isolates almost all the experimental error to the measurement of the distance of the apparatus to the wall. It is basically a large-scale version of the lab Vernier describes when you snap your fingers at the end of a tube with a microphone. This version is much larger, so you can hear the clap and the echo separately.
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