Open JohnDenker opened 9 years ago
ericmazur
commented
9 years ago I'll think about this. In Tony French's book on Waves (in the MIT series), he derives this statement using two pages of math. I was quite proud of finding a completely visual/conceptual way of reasoning!
JohnDenker
commented
9 years ago It's a nifty argument, and a nifty result ... but even visual/conceptual reasoning has to be done carefully.
I've been thinking some more about this. The claimed result is overstated. It can be repaired in at least two ways:
As Pólya was fond of saying, when you have obtained a result, see if you can generalize it. This particular result generalizes rather spectacularly. We end up with a virial theorem, with a whole zoo of examples.
Obviously all of these require choosing a suitable gauge for the PE.
This, like so many other big ideas, is annoyingly difficult to fit into the structure of an ordinary textbook. It doesn't really fit in the wave chapter, or the harmonic oscillator chapter, or the orbit chapter, or the particle-in-a-box chapter ... but still, as Henry James was fond of pointing out, real understanding, useful understanding comes from seeing the connections between ideas.
Possibly semi-constructive suggestion: Have a section "X" somewhere that discusses the general idea, and then put in lots of cross-references to-and-from the places where the idea crops up. Somebody who starts at "A" can follow the link to "X" and then find all the other occurrences:
A <--> X
B <--> X
C <--> X
D <--> X
et cetera
In chapter 16 on page 12 it says emphatically:
This is an important idea. For one thing, it serves as a counterweight to the oft-abused example of an ideal gas, where all the energy is kinetic. In particular, for an ordinary solid, half the heat capacity is associated with kinetic energy, and half with potential energy.
Alas, however, the claim as stated cannot possibly be true, for multiple reasons. For starters, it's not gauge invariant. We can shift the potential energy by an arbitrary constant and get the same equation of motion. However such a shift would break the claimed equality. You could fix this by specifying the straight, unexcited string as the zero-reference. Alternatively, you could fix it by talking about changes in energy, e.g. as we see in the heat-capacity situation.
Secondly, the claim seems to apply to all waves at all times, which cannot possibly be true. In fact the Gedankenexperiment used to "prove" the result in fact proves the exact opposite. When the two waves overlap, neither one of them has any potential energy. To fix this requires quite a bit of attention to detail: If we have an isolated pulse, moving strictly leftward, not interacting with any rightward-moving waves, then it has equal amounts of KE and PE.
Furthermore, the derivation requires the medium to be linear and non-dispersive. Otherwise the clever destructive-interference argument doesn't work.
Combining this with item #74, we have to assume:
It is "traditional" to make such assumptions in the interests of simplicity ... but it is quite easy to find significant violations in the real world.
Suggestion: Make explicit the various assumptions and provisos that underlie the claims in this chapter.
On the other hand .... this is one of those situations where the result is more robust than the derivation. The idea of half KE, half PE is more robust than the foregoing would suggest, if we take suitable averages, assuming only linearity. I don't immediately see a simple way of proving this, not without dragging in the heavy machinery of classical mechanics. Something to think about....