Speed of Sound

OK, so we're finally there in AP - the speed of sound calculation. Matter and Interactions makes a hand-waving argument based on dimensional analysis to get the longitudinal speed of sound from the interatomic bond length and spring constant and the atomic mass. That's fine, but I wanted to get to the real derivation, which involves a very interesting couple of linear approximations and some possibly bogus calculus, but it's doable.  The transverse one is basically the same story.

It's more guided and less inquiry, but it's a tough bit of stuff. I'm OK with there being less discovery here, because it's going to be a gigantically powerful result - predicting the frequency of a sound using the molar mass, density and Young's modulus? Awesome.

We're applying both of the expressions today in lab.  I'm spitballing the work flow here:

• Wave intro, animations, etc. Define wavelength, frequency, v relationship by analogy to traffic
• Longitudinal wave derivation:
• With known molar mass, modulus, and density, they determine k, d, the atomic mass, and v (using the dimensional analysis result)
• The real derivation - super-guided by me
• Demo singing rod, have them calculate its frequency
• Sing, record, FFT
• Party
• Transverse wave derivation:
• Animations - difference between longitudinal and transverse
• Derivation
• Give them sample of wire - they determine the tension needed to have the sound be an octave below a resonance box (128 Hz); help them with the standing wave's length
• Use force wire strung between two poles (one with a force probe on it) to measure tension. Other end of wire is still on the spool, which is on the rod. Kids twist spool so that the sound is in tune; reveal tension (or vice versa?)
• Party

The setup with the spool (yellow string added so that you can see the path of the wire):

Here's my drawing of the whole process: