Timeline for Derivation of Bessel functions
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| May 23, 2013 at 2:01 | vote | accept | user34091 | ||
| May 19, 2013 at 15:06 | history | edited | user34091 | CC BY-SA 3.0 |
added 468 characters in body
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| May 19, 2013 at 2:46 | comment | added | Carlo Beenakker | yes, $u\ll 1$ means that the quadratic (convective) terms are neglected; and indeed, the Bessel equation, which has Bessel functions as solutions, is a linear equation, so it cannot include the convective term; see my full answer below. | |
| May 18, 2013 at 21:15 | comment | added | user34091 | @Carlo Beenakker, thank you for the observation. Could that neglect originate from the author's search for solutions of low limited amplitude? You might have deduced the assumption from the solution given, that is extremely skillful. Or are there more consequences if the assumption (from text): "At low amplitude this reduces to the wave equation: $\partial^2 \rho - c^2\nabla^2\rho = 0$ (2)" from page 1. Is this sentence correctly written as: $u \ll 1$? If the assumption is the cause I will update that into my question. And is it the wave equation the source of the Bessel function solution? | |
| May 18, 2013 at 21:05 | comment | added | user34091 | @S. Carnahan , thank you for the resources suggestions, I will take to heart. Do you mean the Bessel function is an approximate solution because they require extra assumptions to inviscid flow? Like small amplitudes? Or do you mean the Bessel function itself cannot be calculated with much accuracy? | |
| May 18, 2013 at 14:56 | answer | added | Carlo Beenakker | timeline score: 3 | |
| May 18, 2013 at 9:22 | comment | added | Carlo Beenakker | as far as I can see, the solution you have written down for the density $\rho$ neglects the nonlinear term $u\cdot\nabla u$, so any effect of convection is neglected and it only applies in the limit of small velocities $u$; in that limit, indeed, the differential equation for $\rho$ is just the Poisson equation, which in cylindrical coordinates has the Bessel function solution; all the complications of hydrodynamics (shock waves, turbulence, etc.) come from the nonlinear convective term $u\cdot\nabla u$ which is neglected; once you add that term, there is no closed form solution. | |
| May 18, 2013 at 0:20 | comment | added | S. Carnahan♦ | As far as I can tell, the Bessel function only gives an approximate solution to the equation, so it is unlikely to pop out of a first-principles attempt at an exact solution. It might be best if you broke your question down into simpler pieces, and asked them at math.stackexchange.com or one of the other sites listed in the FAQ. There are nice free materials on fluid mechanics and differential equations at MIT's OpenCourseware, but they require a serious commitment and investment of time to work well. | |
| May 17, 2013 at 17:35 | history | edited | user34091 | CC BY-SA 3.0 |
Update 1; added 4 characters in body
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| May 17, 2013 at 17:20 | comment | added | user34091 | See what I mean? Just because of the connection to QM someone made an idiotic vote that the question was not clear or useful. Or am I missing something here? | |
| May 17, 2013 at 17:12 | history | edited | user34091 | CC BY-SA 3.0 |
no qm references
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| May 17, 2013 at 17:07 | comment | added | user34091 | @The User Would it be better if I just erase part of the background to the question so these kind of comment stops? | |
| May 17, 2013 at 17:04 | comment | added | user34091 | @The User I thought he was being a little sarcastic by the way he worded that if I "want to be taken seriously rather than viewed as a crank", I should fully understand Quantum Mechanics like him. But I have been arguing this topic a lot lately to know that my question doesn't entail me being "viewed as a crank". He is the one thats being a crank by bashing on a legitimate mathematical question because he thinks the long-term goals are not achievable. Or the philosophy of it is inconsistent in his view. I didn't delve in such topic in my question exactly because of this. | |
| May 17, 2013 at 16:55 | comment | added | The User | @d12 Jeff gave you a friendly and serious comment. There is nothing bad about that. | |
| May 17, 2013 at 16:33 | history | edited | user34091 | CC BY-SA 3.0 |
don't raise QM polemics
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| May 17, 2013 at 16:27 | comment | added | user34091 | @Jeff, if you want to help with the math I will appreciate. | |
| May 17, 2013 at 16:26 | comment | added | user34091 | Not useful at all! Your comment is completely off topic. And not original at all. Really. This is not the place for this discussion. If you want to have it, send an email for Robert Brady. If you just want to bash him, join every one at: scottaaronson.com/blog/?p=1255 (Get cred with Scott Aahoronson while you do your rants.) Or if you want to know why this approach is worth it and probably right: vzn1.wordpress.com/2013/02/20/… | |
| May 17, 2013 at 16:09 | comment | added | Jeff Harvey | Not a comment on the math, but if you really want to reformulate Quantum Mechanics in terms of classical fluid dynamics and want to be taken seriously rather than viewed as a crank then you are first obligated to understand how Quantum Mechanics is currently formulated and used in some detail. | |
| May 17, 2013 at 14:45 | history | asked | user34091 | CC BY-SA 3.0 |