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rst1993 April 24, 2015 05:54
Dispersion
 
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.

robo April 24, 2015 08:57
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.

FMDenaro April 24, 2015 11:14
Quote:
Originally Posted by rst1993 (Post 543548)
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.

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

rst1993 April 24, 2015 13:40
Quote:
Originally Posted by FMDenaro (Post 543610)
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
So is there any information about the medium in that equation?
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?

FMDenaro April 24, 2015 13:46
Quote:
Originally Posted by rst1993 (Post 543629)
So is there any information about the medium in that equation?
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?

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).

robo April 24, 2015 14:48
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
\phi = \frac{gA}{\omega}e^{ky}\sin(kx - \omega t)
while in finite depth
\phi = \frac{gA}{\omega} \frac{\cosh k(y+h)}{\cosh kh} \sin(kx - \omega t)
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.

rst1993 April 24, 2015 15:19
Quote:
Originally Posted by FMDenaro (Post 543630)
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).
It's true that on discretization the dispersion will always be there, but my question is about the governing partial differential equation. Will this governing equation will have modification if two different mediums are considered for sound propagation? Will it be non-dispersive in both say air and water for example?
In short is dispersion a property of governing partial differential equation or the medium?

rst1993 April 24, 2015 15:23
Quote:
Originally Posted by robo (Post 543640)
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
\phi = \frac{gA}{\omega}e^{ky}\sin(kx - \omega t)
while in finite depth
\phi = \frac{gA}{\omega} \frac{\cosh k(y+h)}{\cosh kh} \sin(kx - \omega t)
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.
My question is regarding physics and the governing partial differential equations. If sound wave will be dispersive in some medium say X, then what will the corresponding difference between the governing PDE for acoustic wave in X and some non-dispersive medium such as air?

robo April 24, 2015 15:34
The simplest answer is
\nabla^2 p = \frac{1}{c(\omega)}\frac{\partial^2 p}{\partial t^2}
Dispersion is just the wave speed being dependent on the frequency of the wave. The form of that relationship will vary depending on the medium in question.

FMDenaro April 24, 2015 15:40
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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