Monday, February 25, 2008
Reading 3.1
For all that talk, I kind of miss elastics already. It was really powerful to see the parabolic profile immediately (sort of) fall out of (sort of) our new equations. However, symmetries and elastic coefficients seem so far behind us, already. ;-(
Reading 2.12
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.
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.
Sunday, February 17, 2008
Reading 2.13
This was a great chapter too, but I'm going to save comments and question for later. I'm tired and I already made a mistake about assuming I understood what was going on.
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.
Sunday, February 10, 2008
Reading 2.10
Linearization in sight! Mathematicians delight! Wow, there are way to many syllables to sing that accurately.
To sum up, the acoustic approximation rocks. It makes it so that convection drops out and so that the continuity equation is decoupled from the field equation. Pretty neat. I'd like to explore, eventually, real life situations where this approximation breaks down, but I'm pretty sure that we have only seen the tip of iceberg when it comes to the plethora of real life things that do fall into this approximation.
To sum up, the acoustic approximation rocks. It makes it so that convection drops out and so that the continuity equation is decoupled from the field equation. Pretty neat. I'd like to explore, eventually, real life situations where this approximation breaks down, but I'm pretty sure that we have only seen the tip of iceberg when it comes to the plethora of real life things that do fall into this approximation.
Saturday, February 9, 2008
Reading 2.8-2.9
2.8: I'd like to try and address Max's question about the quasistatic approximation. While reading this section, I too was frustrated by this blatant "physics move" where you assume something is so that doesn't really make sense and then things work out the way you want it to.
For the short story, skip this paragraph: However, just like most blatant physics things, this one ended up having a purpose that I realize if you bash your head against your book enough. This section is all about understanding the relationship between (the) stress (tensor) and (the) strain (tensor), i.e. the cijkl. If we think about the simpler Hooke's Law that we already know, F=-kx, we are making the same quasistatic approximation. We are saying, "irrespective of the way in which we got to displacing a spring by x, what is the force exerted between either ends of the spring?" If we think about "what actually happens" as we push a spring to a new displacement if the acceleration is fast, we see a wave propagate down the axis of the spring. However, in the quasistatic approximation of pushing a spring really slowly (i.e. not very much acceleration) toward the new displacement, we see that Hooke's Law still does exactly what we want it to do (even though it would do exactly what we want it to do once the wave died out, and the displacement of the spring became fixed). Even though the law works all the time (hence "law") we needed to derive it from the quasistatic situation, since it's waaaaay easier to see it in action that doing all of the simulations of a wave along a spring in your head to come up with the F=-kx relationship.
Long story short, in this scenario we want to know, like the spring, what happens once the deformation has occured and not what is happening during it, just to understand the Hookean relationship at work. Once we understand this, I think we can extrapolate to the moving case, saying that at every instantaneous moment the stress is related to strain as such, because every moment is quasistatic...right?
Wow, I went on a lot longer than I had expected to. Does this help, Max? Is it right, professors?
Also, nice work, Higgy, on your 6-choose-2 method of finding the 21 independent constants. That's slick.
2.9: This is a pretty section, except for the means-to-an-end of 2.100. I am a little put-off by the "Now consider the specific case i=k=1, j=l=2." I mean, they're all δ's, so we really only need i=k, j=l,i≠k...right? However, I know really see even where this is something that's recommendable. It is late, and I'm not thinking very hard, so that could also be the problem. Also, I bashed my head against the book for the last section. It hurts now. Someone else can pay it forward and bash their head on section 2.9 for me. Otherwise, Lamé constants are sweet!
For the short story, skip this paragraph: However, just like most blatant physics things, this one ended up having a purpose that I realize if you bash your head against your book enough. This section is all about understanding the relationship between (the) stress (tensor) and (the) strain (tensor), i.e. the cijkl. If we think about the simpler Hooke's Law that we already know, F=-kx, we are making the same quasistatic approximation. We are saying, "irrespective of the way in which we got to displacing a spring by x, what is the force exerted between either ends of the spring?" If we think about "what actually happens" as we push a spring to a new displacement if the acceleration is fast, we see a wave propagate down the axis of the spring. However, in the quasistatic approximation of pushing a spring really slowly (i.e. not very much acceleration) toward the new displacement, we see that Hooke's Law still does exactly what we want it to do (even though it would do exactly what we want it to do once the wave died out, and the displacement of the spring became fixed). Even though the law works all the time (hence "law") we needed to derive it from the quasistatic situation, since it's waaaaay easier to see it in action that doing all of the simulations of a wave along a spring in your head to come up with the F=-kx relationship.
Long story short, in this scenario we want to know, like the spring, what happens once the deformation has occured and not what is happening during it, just to understand the Hookean relationship at work. Once we understand this, I think we can extrapolate to the moving case, saying that at every instantaneous moment the stress is related to strain as such, because every moment is quasistatic...right?
Wow, I went on a lot longer than I had expected to. Does this help, Max? Is it right, professors?
Also, nice work, Higgy, on your 6-choose-2 method of finding the 21 independent constants. That's slick.
2.9: This is a pretty section, except for the means-to-an-end of 2.100. I am a little put-off by the "Now consider the specific case i=k=1, j=l=2." I mean, they're all δ's, so we really only need i=k, j=l,i≠k...right? However, I know really see even where this is something that's recommendable. It is late, and I'm not thinking very hard, so that could also be the problem. Also, I bashed my head against the book for the last section. It hurts now. Someone else can pay it forward and bash their head on section 2.9 for me. Otherwise, Lamé constants are sweet!
Wednesday, February 6, 2008
Reading 2.7
This is an exciting section: now we have all of the equations we will need to do whatever we want with anything!...as long as it has to do with elastic medium...and we are told the elastic spring constants...and we don't have to actually solve the equations...
Darn. I'll have to sit tight for vortex rings, I guess.
Darn. I'll have to sit tight for vortex rings, I guess.
Friday, February 1, 2008
Reading 2.4-2.6
So I'm trying to get ahead a bit since I have my first papers and presentations due next week. Luckily I can choose to get ahead in a course that doesn't give me any credit!
Section 2.4 really was great. Even though I've seen the continuity equation derived before, this was done in such a blatantly analogous way to the derivation of any kind mass conservation equation...I wonder what I must have been paying attention to instead of this the last time I saw it.
I find the argument used to enforce the symmetry of &Piij in section 2.5 very interesting, and it's motivation comes from making sure that the thing that we derived from physical examples makes physical sense. Additionally, the 3D conic section representation of the information in a symmetric rank-2 tensor is awesome, and completely new to me. Sweet!
Strain in 2.6 is great, but it is nothing needed without the equation of state in 2.7. That's some good stuff.
Section 2.4 really was great. Even though I've seen the continuity equation derived before, this was done in such a blatantly analogous way to the derivation of any kind mass conservation equation...I wonder what I must have been paying attention to instead of this the last time I saw it.
I find the argument used to enforce the symmetry of &Piij in section 2.5 very interesting, and it's motivation comes from making sure that the thing that we derived from physical examples makes physical sense. Additionally, the 3D conic section representation of the information in a symmetric rank-2 tensor is awesome, and completely new to me. Sweet!
Strain in 2.6 is great, but it is nothing needed without the equation of state in 2.7. That's some good stuff.
Subscribe to:
Posts (Atom)