This was really exciting reading, to me. I was really startled in class when Prof. Lyzenga suggested that complex wave-vectors would yield physically meaningful waves that we were missing with just our P- and S-waves. I have to admit that I was just as startled once everything worked out with the algebra.
Other than that, I consider this reading a real treat in terms of a nifty mathematical treatment of a problem of physical interest. My last thought, then, is in line with Andrew's question:
- Is there a way to exhaust the solutions to our equations for elastic medium? I have heard the terms inverse scattering and integrable thrown around with regard to systems to which we know everything...is this or some simplified version one of them? It seems that even if they are a simpler version, that they can afford the Navier-Stokes equations means that it should probably be more difficult than integrable.
Showing posts with label excitement. Show all posts
Showing posts with label excitement. Show all posts
Monday, February 25, 2008
Sunday, February 17, 2008
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 Vi, then (2.147) is equivalent to saying that for this new quantity, Vi, the divergence is 0. We can also note that for this new quantity the curl is zero.
Now what we can say is that Vi 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 Vi vanishes everywhere.
Lastly, this Vi 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 Vi 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.
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 Vi, then (2.147) is equivalent to saying that for this new quantity, Vi, the divergence is 0. We can also note that for this new quantity the curl is zero.
Now what we can say is that Vi 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 Vi vanishes everywhere.
Lastly, this Vi 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 Vi 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.
Subscribe to:
Posts (Atom)