Reading 2/18/2008: Section 2.11

Sorry this is late, I've been rather sick. Anyway, this section is a bit of a beast, but whew - good stuff.

So we can use the symmetry of &xi to plug one equation in the acoustic approximation into the other, and we get something nice (2.137). Fantastic.

What exactly does Helmholtz's theorem state? Wikipedia gives a good enough answer, and I assume this was discussed in class, but I'll let the question stand, as it seems to be an important, if intuitive, result.

Equation (2.144) is awesome. With some fairly basic steps, we've jumped from our general equation for isotropic acoustic-approximated media to the wave equation. BAM. There are huge assumptions on smoothness throughout all the derivations here, but the mathematician in me is throwing up the white flag - it ain't worth the battle.

As far as the derivation of necessity goes (2.147-148), my only question is: is there a cute proof that zero curl and divergence imply constancy? I feel like there must be, but it's not coming to be offhand and I'm too busy to try and find it.

The derivation of the dispersion relations is a little unclear, but I think it's mostly a notational issue. We have wave propagating in direction k_i. For one, &phi and A have fixed values on the plane normal to k_i only in constant time, as I gather, though this isn't said. If we denote the speed by c and let r be a position coordinate in the direction of k_i (presumably with magnitude k), then by definition dr/dt = c. If we assume plane of constant phase, we end up with the dispersion relation. It makes sense after several readings, but the statement feels unclear.

The notion that P- and S-waves represent the two modes of propagation in an isotropic medium is new to me, and actually very cool, though it makes perfect sense now.

The stuff on pages 99-101 is standard stuff, having seen Physics 52 and 116, but the whole tossing in of an incident and a reflected wave out of this air has always seemed a little hand-wavy to me. That said, I'll accept it. (Not like I have a choice ;-)) All that said, why does symmetry of the medium imply an incident and a reflected wave must lie in the same plane? Intuitively this is clear, but mathematically...?

That was mostly it for the chapter. The connection to earthquakes was a nice touch, too.

Until next time...