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April 24, 2015, 06:54 |
Dispersion
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#1 |
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Rahul Singh
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Is it the medium or the governing equation is dispersive as far as acoustic waves are considered??? Sound is non-dispersive in air, but if we consider linearized euler equation in a uniform flow which governs the acoustic wave, the equations are dispersive???
For e.g., the convection equation, Ut+cUx=0, the dispersion relation is w-ck=0, which is non-dispersive. Pls explain this difference if it is there. |
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April 24, 2015, 09:57 |
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#2 |
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robo
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The governing equation needs to represent the medium. If a medium is dispersive or not, then the equation you use should reflect that. If the equation doesn't, then it is in some ways a poor representation of the medium. The linearized Euler equations are just a model for reality; introducing dispersion where it isn't observed to exist is a failure of the model. Whether this is a problem or not depends on what you're trying to study.
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April 24, 2015, 12:14 |
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#3 | |
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Filippo Maria Denaro
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Quote:
The convection equation you cited has no dispersion, any initial function U(x,0) will be advected at velocity c. However, this is the exact PDE, in the numerical solution you after the adopted discretization method have to see the modified differential equation that can actually have a dispersion term due to the local truncation error |
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April 24, 2015, 14:40 |
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#4 | |
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Rahul Singh
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Quote:
Is it possible for a wave say acoustic being non-dispersive in one medium becomes dispersive in another? If so, then what about the difference in governing equations of these two medium for the same wave? |
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April 24, 2015, 14:46 |
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#5 | |
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Filippo Maria Denaro
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Quote:
No, the physical medium is the same, what happens in the modified equation is the presence of the artificial dispersion that has coefficents that depend on the type of scheme (of course is a combination of the integration step). |
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April 24, 2015, 15:48 |
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#6 |
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robo
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I'm unclear if you're asking about physics or numerics. It is physically possible for a wave to be non-dispersive in one medium but dispersive in another. Free surface waves in the deep ocean will be dispersive, however when they move to shallow water they will be non dispersive. The deep water solution for the potential is
while in finite depth I believe that acoustic waves become dispersive in the nonlinear regime; I can't think of any examples of a medium that would cause acoustic dispersion though. |
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April 24, 2015, 16:19 |
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#7 | |
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Rahul Singh
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In short is dispersion a property of governing partial differential equation or the medium? |
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April 24, 2015, 16:23 |
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#8 |
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Rahul Singh
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Quote:
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April 24, 2015, 16:34 |
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#9 |
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robo
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April 24, 2015, 16:40 |
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#10 |
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Filippo Maria Denaro
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well, if you consider only the physics you can also consider analytical solution for Euler equation in case of homoentropic flows, using Riemann invariants and considering two different medium at sound velocity a1 and a1 (Zucrow is a good textbook). You will see that the Riemann invariants remain the same across the two medium whilst the slope of the characteristic curve changes.
To tell the truth, I don't know if that can be denoted as dispersion. |
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