Wednesday, July 25, 2012

M and I: Speed of Sound from First Principles

I'm excited about using the Matter and Interactions curriculum to connect the micro-world to the macro-world. Two of the big candidates are speed of sound and specific heat. Right now, I'm looking at the speed of sound.

Using the interatomic bond length and stiffness to determine an expression for the speed of sound (longitudinal) in a material, the authors arrive at:
Here, k is the interatomic bond stiffness and d is the interatomic bond length (modeling the solid as a cubic lattice of atoms connected by Hooke springs), and m is the mass of one atom.

What I really want is to be able to put the physics students (who cover sound and know the standing wave stuff and can experimentally determine the frequency of a standing wave and calculate the speed of sound) together with the AP Physics students (who can calculate the speed of sound from experimental measurements, if I can get a Young's Modulus apparatus) to see if the approaches agree.

While I don't teach it, this made me think of the speed of sound in a wire expression:
Here, T is the tension and mu is the mass per unit length.  This has the obvious dependence of speed on tension explicitly and made me think about the M and I expression - tension's not (obviously) there.  Is this an approximation or will the approach only work under no tension (as the value was calculated)?

Actually, the two (pretty much) reduce to each other. Starting at the speed in a wire expression:
Comparing the tension in the wire to the tension in an individual "strand" of atoms:
Substituting, simplifying, rearranging:

The approximation was that one fourth equalled one sixth, from the volume/area substitution. The problem is that this really only works if we call the tension force between two atoms kd, which isn't really what it should be.  I'm not sure if we're dealing with a limitation of the M and I model, some unstated assumptions in the wire model, or both.  ...or a longitudinal/transverse mismatch. Can anybody help?

I'm encouraged about the possible agreement, though it's not really necessary for the expression to agree for me to test the prediction made by the M and I model.  I tried to test it out a bit using a 17 cm long 2 cm thick copper cylinder.  I dropped it on its end on the floor and recorded the high-pitched ringing while it was in the air.  The frequency spectrum:
The fundamental frequency for that bar (using all of the available simplifications) should be around 15 kHz, but it's quite a bit lower (less than 3 kHz).  The spacing's not even here, so I'm thinking that I have an issue with the cylinder having some odd vibrational modes, but I can't find a reference on what to expect.  In any event, I'd like something with well-behaved harmonics that my physics students can analyze successfully. A copper wire seems great, but the waves are the wrong mode, and definitely wouldn't agree.

Another idea that I had: a long aluminum rod (one of those Arbor Scientific dealies).  It's 74.5 cm long, and I recorded this spectrum from the demo video on the Arbor site (I can't find the rosin):
These peaks are nice and harmonic, with a fundamental right around 3346 Hz. This means a speed of sound of 4986 m/s, which should agree nicely with the value that we'll get from using the bond strength of aluminum (OK - I need to get some aluminum wire.  Hopefully that's a thing.).  It looks like this is the way to go.

Other suggestions, anyone?  I'm going to try thumping a marshmallow, too, since the modulus is easy to determine, but I'd have to do some voodoo with the calculations, since it's not an elemental substance.

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