Reading 2.11

I'd like to start this post with some kind of exclamatory remark like "wow" or "awesome" but I feel like they would all cheapen my actual response to this section. We get at some really cool results from things that are borderline "blatant physics move" and "completely justifiable modeling assumptions" and I am really happy with the content of this reading.

I'd like to, if I can, address Andrew's worry. Looking at (2.147) on p.92 , we note that

[(λ + 2μ) φ_,nn - ρ₀φ'']_,kk = 0 (the primes are equivalent to dots)

which follows from (2.146) on p.91, simply takes the scalar Laplacian (correct me if I'm wrong) of what is inside the brackets.

However, if we take what is inside the brackets and one of the derivatives to be a new quantity, that is V_i, then (2.147) is equivalent to saying that for this new quantity, V_i, the divergence is 0. We can also note that for this new quantity the curl is zero.

Now what we can say is that V_i vanishes everywhere or is constant. It is because we are not interested in constant solutions, we assume that the other situation is true, i.e. that V_i vanishes everywhere.

Lastly, this V_i represents...wait a minute. I really thought that there was a problem with what you were saying, since the equation you wrote seemed a little off. While it still seems off, I don't see a problem with your question.

I am confused too, then, about why assume that the constant solution is trivial. While a constant vector field V_i may not seem interesting, that field is representative of something else, and that something else, i.e.

[(λ + 2μ) φ_,nn - ρ₀φ'']_,i ≠ 0
seems to be more complicated than we are granting it in the treatment on p. 92. Sorry to knock you, Higgy. Damn, what a fool I am.