Lab -- waves on musical instruments

My general physics class is studying waves. So far, we've just done basic v=λf stuff. We're going to study standing waves, and harmonics in pipes and strings, soon.

For lab tomorrow, I've asked anyone who plays a musical instrument to bring it in. Groups that don't bring in a guitar or trumpet or something will get to use one of the recorders that I borrowed from the music department. (Does every third grader in America still learn to play recorder badly? I have interesting memories of Jingle Bells in the third grade's Arena of tonal Quality.)

Most instruments that I'll see tomorrow can be simplistically modeled as open pipes or fixed strings. And, except for the brass instruments, the wavelength of each note can be approximated as twice the length of the string or pipe. (Brass instruments are rarely playing the fundamental mode... I have to hazard a guess as to which harmonic they're playing. I always hope I get very few brass players in my class.)

So I'll have each group play a chromatic scale for one octave -- that's 12 notes. For each note, they'll approximate the frequency by looking at the chart I've printed on their lab sheet. I'll have a digital frequency generator hooked up in the back of the room so that they can try to differentiate between, say, the 262 Hz C and the 512 Hz C.

For each note, they'll estimate the wavelength by measuring the length of the pipe or string. Strings are straightforward; pipes are not. For the recorder, it's possible to estimate the pipe length as the distance from the mouthpiece to the first open hole, but even that's not quite right. We're using all kinds of crude estimates here.

But, crude or not, I usually see a nice, clear hyperbolic graph of frequency vs. wavelength. That allows me to teach these guys how to turn that into a linear graph, by plotting 1/wavelength on the horizontal axis. I'm happy for students at this level just to be exposed to the process of straightening a graph. AP students do this on a biweekly basis; this is, believe it or not, highly advanced math for the general class.

Anyway, the slope of the straight graph SHOULD be equal to the speed of waves on the string or in the pipe. For woodwinds, this should be the speed of sound in air, and will be unless they chose the wrong octave (i.e. they were playing the 262 Hz C, but they misidentified it as 512 Hz). For strings, they should have a wave speed in the high tens through the hundreds of m/s. It's possible to determine the wave speed by knowing the tension and linear density, but that's not a measurement I'd ask general students to make.

Below is the lab handout I'll give. This will most likely be a two-lab-period experiment -- I'll get the raw data and a start on the linear graph tomorrow, and next week we'll finish.

By the way, I welcome any comments or suggestions from people with better knowledge of the physics of music than I. I recognize how crazy some of these approximations are... but the point is, we get the hyperbolic relationship between frequency and wavelength. That's what I'm after.

Physics Experiment:
Waves in a Musical Instrument

Pre-lab preparation: Bring a musical instrument to class – a guitar, trombone, whatever. If you don’t bring an instrument, you’ll use a recorder.

Now, sketch in your notebook… what should a graph of frequency vs. wavelength look like? Make a sketch and show it to Mr. Jacobs. You will earn an extra point if this sketch turns out to be correct.

Goal of the experiment: You are to demonstrate that v = λf for waves in your instrument.

Instructions:
· You will find the wavelength and frequency for each note in one octave’s worth of a chromatic scale. (That’s 12 notes: C, C#, D, D#, E, F, F#, G, G#, A, A#, B.)

· The wavelength of the note should be twice the length of the pipe or string that produces the note.

· The frequency can be determined using the frequency table on the other side. Be careful – use the frequency generator in the back of the room to figure out what octave you’re playing. Mr. Jacobs will show you what this means.

· Graph frequency on the vertical axis, and the wavelength on the horizontal axis.

· Mr. Jacobs will show you how to convert this graph into a straight line graph.


Analysis: Answer the following questions thoroughly on a clean page in your lab notebook.

(1) Explain why the original f vs. λ graph looked the way it did.

(2) What is the meaning of the slope of the straight line graph?

(3) What was the slope of the straight-line graph?

(4) Comment on the reasonability of the slope you measured. No BS!

Frequencies of musical notes:

Note that multiplying a frequency by two gives the same note but an octave higher.