Ignoring initial conditions for a PDE and considering the solution to be in the form of a simple plane wave you get the following results.

Hyperbolic

u_tt = c²u_xx

u(x,t) = A*cos(wt + kx)

This is just the wave equation with a solution oscillating in time.

Parabolic

u_t = a²u_xx

u(x,t) = Ae^{-a^2*k^2*t}e^ikx

This is the diffusion equation and its solution describes a wave decaying in time.

1. Is there an intuitive way to see why a PDE that is second order in time and second order in space causes oscillation, but one that is first order in time and second order in space causes a time decaying wave?

2. How about the Schrodinger equation how would it behave under the above stipulations? It seems somewhat similar to the diffusion equation but its got that damn i in it.

Is there an intuitive way to see why a PDE that is second order in time and second order in space causes oscillation, but one that is first order in time and second order in space causes a time decaying wave?

Schrodinger's equation is first order in time and second order in space. For a potential free region of space the solutions are plane waves!!

Not all PDE which are second order in time and space produce the solutions you suggest.

Ignoring initial conditions for a PDE and considering the solution to be in the form of a simple plane wave you get the following results.

This makes no sense at all, especially since you go on to show that the second equation has a solution which is in fact not in the form of a simple plane wave.

The first solution is not totally correct either. There are an infinite number of solutions to this equation, the sum of any 2 solutions also being a solution.

Keep working.

Originally posted by Sacroiliac

1. Is there an intuitive way to see why a PDE that is second order in time and second order in space causes oscillation, but one that is first order in time and second order in space causes a time decaying wave?


well, for ODEs, it is easy to see why this happens: the solution to the characteristic polynomial with odd degree always has a real part which gives you an exponential.

i don t know if that qualifies as intuitive.... physically, you can think of the first derivative term as just a friction force that gets in the way of the harmonic oscillation.

i think the picture is roughly the same for PDEs.

and as you say, adding a complex number can change everything. if your characteristic polynomial does not have real valued coefficients, there is no longer a guarantee that an odd degree polynomial will have a real root. so the Schrödinger equation is qualitatively different from the diffusion equation.

that s the best explanation i can give, and i don t find it wholly convincing. take it or leave it.

Originally posted by ryans
Schrodinger's equation is first order in time and second order in space. For a potential free region of space the solutions are plane waves!!

did you even finish reading his post? he said as much in his post, and put forth a possible explanation of why.

WTF?

Not all PDE which are second order in time and space produce the solutions you suggest.



This makes no sense at all, especially since you go on to show that the second equation has a solution which is in fact not in the form of a simple plane wave.
that solution is a plane wave if you allow for complex phase velocity.

The first solution is not totally correct either. There are an infinite number of solutions to this equation, the sum of any 2 solutions also being a solution.
nowhere did he claim an exhaustive list of solutions, he only gave one solution. does that make that one solution incorrect?

Keep working.
why don t you take this opportunity to brag about all the calculations you have done. i mean, god forbid you try to give some constructive input.

Originally posted by lethe
well,

that s the best explanation i can give, and i don t find it wholly convincing. take it or leave it.

I'll take it. And once again thanks for your clear and concise answers. The only sad thing is that I should have been able to figure it out for myself.

Oh and by the way ryans, I'm a division general manager for a very large corporation with over 1000 people reporting to me. But I wouldn't speak to the lowest level floor sweeper in the extremely rude manner you spoke to me.

I wasn't being rude. Read my post without reading lethe's, and the tone will be seen to much more subdued than lethe would care to admit. I was simply pointing to some facts that I assume you misunderstood.
I am sorry that you midunderstood my tone.

BTW I couldn't give a rats what you do or who you are in charge of, the fact that the way your question was structured was very unintelligible and seemingly incorrect, propounded that response from me.

Originally posted by ryans

BTW I couldn't give a rats what you do or who you are in charge of, the fact that the way your question was structured was very unintelligible and seemingly incorrect, propounded that response from me.

Well I see you're not being rude again huh? How old are you...15?

Sorry James R, I don't normally let children's little rants annoy me, but in any case I'm done.

Originally posted by Sacroiliac
How old are you...15?


he is older than i.

he is older than i.

Get the fuck outta here. No way.

**examines pulse** conclusion, normal.

I repeat: Get the fuck outta here. No way.

Lethe, if I am older than you, it is not by much more than a year.

Originally posted by ryans
Lethe, if I am older than you, it is not by much more than a year.

that is correct, less than a year (going by your DOB on your info page)

And what are you up to these days lethe. i.e. trains of thought.

Originally posted by ryans
And what are you up to these days lethe. i.e. trains of thought.

what do you mean? do you want to know what kind of physics i am doing these days?

Well I know that you were completing postgraduate work, contemplating do some work for that crackpot, and drifted of to the IAS for a bit. You were doing something to do with particle physics weren't you?

Originally posted by ryans
Well I know that you were completing postgraduate work, contemplating do some work for that crackpot, and drifted of to the IAS for a bit. You were doing something to do with particle physics weren't you?

well, i went to the IAS for the summer, that was cool as hell. i met some real bigwigs, like e. witten, s. gubser, and d. gross. i think meeting the other upcoming theorists was much more valuable to me than meeting the current giants in the field though.

i never finished the crackpot paper, but i got about halfway through it. it s a real hoot, i ll show it to you, if you want, you will get a real kick out of it. it s way the fuck out in left field.

i guess i m doing particle physics. actually, what i m doing this semester is more like mathematical physics (which suits me just fine). right now, i m preparing a talk on classifying charged d-branes using k-theory, which is a sort of topological way of classifying fibre bundles. the talk is in about 2 weeks, and i regret to say that right now, i feel hopelessly ill-prepared.

I would love to see that paper.

I just gave a talk recently on the recent progress of my research, which I thought was shit. But since most of the people there didn't know what I was talking about anyway, I got positive feedback, which is good. But I go to a small uni in Australia. Where are you at? Big physics Dept.

Originally posted by ryans
I would love to see that paper.
attached.

I just gave a talk recently on the recent progress of my research, which I thought was shit.
yeah, they are never as good as you imagine they will be in your head, i have found.

But since most of the people there didn't know what I was talking about anyway, I got positive feedback, which is good. But I go to a small uni in Australia. Where are you at? Big physics Dept.

i am at a large state university in the US, one which has a bridge program between mathematics and physics. for better or for worse, that means that most of the physics i do has to do with string theory. fortunately, there is enough mathematics in this area that i don t have to worry about it. but sometimes i worry.

I have started reading the paper.

I am going to do the crackpot test to see if it qualifies, i.e. how many terms that are introduced in this paper.
Thanks.

Originally posted by ryans
I have started reading the paper.

I am going to do the crackpot test to see if it qualifies, i.e. how many terms that are introduced in this paper.
Thanks.

yeah, lemme know what you think. i think it is more crackpotty than anything that ever appeared on sciforums, made all the worse because the author has approximately a high school level understanding of nuclear physics and GR.