What is the General Form of a Wave Equation in a Medium?

[Killtech]

Hi,

i am looking for a general form of a wave equation in a medium. i am not looking for a concrete physical equation but rather a generalized form (preferably in n dimension) of such under the simplest assumptions (it's of course a little equivocal what 'simplest' means but, well).

so for a scalar field $\rho$ a free wave equation is just
$\frac {\partial^2 \rho} {\partial t^2} - c^2 \nabla^2 \rho = 0$

but how does it look like in an inhomogeneous (but for now static) medium?
$\frac {\partial^2 \rho} {\partial t^2} - c(\vec x)^2 \nabla^2 \rho = 0$
like this?

and how does it look in the general case that the medium is time dependent with a flow $\vec f(t, \vec x)$? basically where the speed of the wave becomes direction dependent. $c$ and $\vec f$ would have to be related via a continuity equation for the medium.

 

[Killtech]

hmm, i hoped for some more answers.

well, in cases of physical equation like those of sound waves, the waves uses same quantities as are used to describe the medium itself, e.g. air density., thus technically the wave changes the underlying medium and since the speed of sound depends on it it probably is a non-linear effect. i do not want to consider such things and i am looking for a simplified model where the waves don't affect medium in any way - i.e. the wave-medium interaction is completely one sided.

in any case the problem of the flow kept bothering me and occupied my mind all day. call it a mathematicians thinking but i tried reduce it to a know case. i started with just a simple case of a homogeneous medium (i.e. $c = const$) and a constant flow with velocity $\vec v = const$. i though that this case should be equivalent to doing a Galilean transformation on the free wave equation - which form is well know. but is the assumed equivalence actually valid?

if so that idea could work for a more general case of a velocity density $\vec v(\vec x)$ of the medium that is divergence free but otherwise not restricted. i could try to find a coordinate trafo such that within the new coordinates the $\vec v(\vec q) = 0$ for all $\vec q$. the question is then if the wave equation in such coordinates would take the form of a free wave equation again?? for example if $\vec v$ is a curl around $\vec 0$, e.g. like in a tornado or a water vortex, would sound waves behave anything like this, e.g. free waves that just get dragged along with the air current??

i suppouse such coordiantes could be constructed by letting them get 'carried by the flow', as well. this would mean that an arbitrary point $\vec q$ - which is just constant in canonical coordinates - would instead have to satisfy an equation itself. i think $\dot {\vec q} = \vec v(\vec q)$ and $\vec q(t=0) = \vec x$ should do the trick? and would transforming back to Cartesian coordinates yield the equation i am looking for??

does this approach make any sense?

 

[Killtech]

can no one help me?

 

[Dale]

Sorry, I like the question, but don't know the answer.

 

[olivermsun]

Will look at it more later, but some quick comment:

For the inhomogeneous medium — the equation you have with $c(\vec{x})$ that you have is right if for example, you have a string with non-constant density but fixed spring constant. However, you have another term if the restoring force is also a function of $\vec{x}$.

Next suppose you have a wave propagating going across a shear flow. There are some well known cases in geophysics, e.g., mountain waves propagating upward where the wind speed changes aloft. A few things have to remain constant as the wave travels — the frequency in the fixed frame as usual, but also the energy density. So if the wave gets "stretched out" in the along-flow direction then the amplitude has to change as well.

 

[Killtech]

For the inhomogeneous medium — the equation you have with $c(\vec{x})$ that you have is right if for example, you have a string with non-constant density but fixed spring constant. However, you have another term if the restoring force is also a function of $\vec{x}$.

i want to understand the simplest case first before going into more complex scenarios - which i would say is the one with the least terms. it's interesting though what the implicit assumptions are for this easiest model.

Next suppose you have a wave propagating going across a shear flow. There are some well known cases in geophysics, e.g., mountain waves propagating upward where the wind speed changes aloft. A few things have to remain constant as the wave travels — the frequency in the fixed frame as usual, but also the energy density. So if the wave gets "stretched out" in the along-flow direction then the amplitude has to change as well.

so how does the sound wave equation look like in a fluid with a shear flow? let's just take the most simplest case i can think with $v(\vec x) = y \vec {e_x}$ being the velocity of the fluid at $\vec x$. and does that wave equation take the form of the free wave after making the coordinate transformation $x' \rightarrow x + v_{x}(\vec x)t = x + yt$ such that within these coordinates fluid flow vanishes? since the wave amplitude is a scalar it wouldn't be affected by this transformation so this would mean i get additional terms handling the lowering of the amplitude due to the 'stretching out'? so does it get a form similar to a dampened wave equation?

 

[Nidum]

For sound waves traveling through a uniformly flowing medium don't you just get simple Doppler frequency shift ?

And through a shearing medium don't you just get refraction (or something very like refraction) - basically deflection of the axis that the sound wave are traveling along ?

 

[olivermsun]

For sound waves traveling through a uniformly flowing medium don't you just get simple Doppler frequency shift ?

And through a shearing medium don't you just get refraction (or something very like refraction) - basically deflection of the axis that the sound wave are traveling along ?

Yup, and yup. The ray tracing approach also gives an intuitive picture of why the wave amplitude has to change to conserve energy (flux).

 

[olivermsun]

i want to understand the simplest case first before going into more complex scenarios - which i would say is the one with the least terms. it's interesting though what the implicit assumptions are for this easiest model.so how does the sound wave equation look like in a fluid with a shear flow? let's just take the most simplest case i can think with $v(\vec x) = y \vec {e_x}$ being the velocity of the fluid at $\vec x$. and does that wave equation take the form of the free wave after making the coordinate transformation $x' \rightarrow x + v_{x}(\vec x)t = x + yt$ such that within these coordinates fluid flow vanishes?

For things like acoustic and water waves, the "free wave" (along the phase speed $c$) are usually defined in the frame moving with the medium. Thus you can freely change to and from this frame if "everything" is translating together. (The energy is frame dependent, however).

since the wave amplitude is a scalar it wouldn't be affected by this transformation so this would mean i get additional terms handling the lowering of the amplitude due to the 'stretching out'? so does it get a form similar to a dampened wave equation?

The amplitude won't be affected by the above transformation. However, if the source or receiver are not moving with the medium or there is a shear flow then you need more than a simple translation to accommodate "passing off" the wave between parts of the system which are moving at different velocities. In the limit where ray tracing is valid, you can just integrate along the ray path and you have all the information you need.

 

[Killtech]

For things like acoustic and water waves, the "free wave" (along the phase speed $c$) are usually defined in the frame moving with the medium. Thus you can freely change to and from this frame if "everything" is translating together. (The energy is frame dependent, however).

thanks for the answer. does your statement refer to just a global movement of the medium (its center of mass) or does is really work locally as well (for any medium with constant density and current with $\nabla \vec v(\vec x) = 0$)? as in the example i presented the current was location dependent - to my understanding it was a flow with a shear. but googling it does make me uncertain what physicists understand as a shear flow. could you maybe elaborate? in any case the coordinate transformation i suggested would build up quite a massive shear over time so it's no simple translation.

and what do you mean with frame in the current context? the general transformations to annul the mediums current everywhere (in the case it's location dependent) are curvilinear and even locally time dependent (time dependency can be different at each location) and correspond to no kind of observer frame (that i could think of). in the general case the transformation should be $\vec x' \rightarrow \vec x + \int_0^t \vec v(\vec x) dt$.

the approach i was trying here is kind of similar to generalized coordinates in Hamiltion-Jacobi formalism of classical mechanics that intends to find coordinates where the problems becomes 'as trivial as possible' - however weird the resulting coordinates may be.

as for the energy being frame dependent - googling sound wave density i found that it explicitly depends on the particle speed. this is kind of not what i want to do because it intertwines the medium and the wave. i would like to assume (for simplicity) that the effect on the medium caused by the waves are negligible in order to keep the equations linear (otherwise a wave passing though another will notice the change in medium density which again would change its speed of propagation locally differently causing some non-linear interaction between two plane waves). using this simplification the energy density of the medium could be separated from the wave energy.

The amplitude won't be affected by the above transformation. However, if the source or receiver are not moving with the medium or there is a shear flow then you need more than a simple translation to accommodate "passing off" the wave between parts of the system which are moving at different velocities. In the limit where ray tracing is valid, you can just integrate along the ray path and you have all the information you need.

if there is a source or receiver the flow has a divergence due to continuity equation. i thought since any vector field (in this context the medium flow) can be split up in a part with zero divergence and one with zero curl and a constant vector it might be easier to get each problem sorted out independently for now - since it seems my original question is more difficult then i anticipated.

in any case do you know any (freely available) sources where i could read up on sound wave mechanics within medium currents?

 

[olivermsun]

thanks for the answer. does your statement refer to just a global movement of the medium (its center of mass) or does is really work locally as well (for any medium with constant density and current with $\nabla \vec v(\vec x) = 0$)? as in the example i presented the current was location dependent - to my understanding it was a flow with a shear. but googling it does make me uncertain what physicists understand as a shear flow. could you maybe elaborate? in any case the coordinate transformation i suggested would build up quite a massive shear over time so it's no simple translation.

Sorry, I misunderstood your earlier point. I was viewing on a device that didn't show the equation properly and thought you just meant the simple translation.

I think you will run into difficulty if you just "shear" the coordinates, because the time dependency is changing as you say. I think you might want to try working along the wave characteristics instead.

the approach i was trying here is kind of similar to generalized coordinates in Hamiltion-Jacobi formalism of classical mechanics that intends to find coordinates where the problems becomes 'as trivial as possible' - however weird the resulting coordinates may be.

A classical approach is to work in a frame moving with the wave (groups). See for example Whitham or Andrews and McIntyre, or look up conservation of wave action.

as for the energy being frame dependent - googling sound wave density i found that it explicitly depends on the particle speed. this is kind of not what i want to do because it intertwines the medium and the wave.

For waves propagating in a medium, there is no way around it. The energy is linked to the velocity caused by the wave motion.

i would like to assume (for simplicity) that the effect on the medium caused by the waves are negligible in order to keep the equations linear (otherwise a wave passing though another will notice the change in medium density which again would change its speed of propagation locally differently causing some non-linear interaction between two plane waves). using this simplification the energy density of the medium could be separated from the wave energy.

There are indeed non-linear interactions between waves. A typical assumption is "small" amplitude, so that the interactions at quadratic order or higher can be ignored. You still might want to know how the single wave is changing in energy density or amplitude as it travels through the medium.

if there is a source or receiver the flow has a divergence due to continuity equation. i thought since any vector field (in this context the medium flow) can be split up in a part with zero divergence and one with zero curl and a constant vector it might be easier to get each problem sorted out independently for now - since it seems my original question is more difficult then i anticipated.

in any case do you know any (freely available) sources where i could read up on sound wave mechanics within medium currents?

I found some good Google Scholar search terms to be "acoustic propagation in shear flow." There are parallel examples such as shallow water waves in shear flow or things like that. Some links are about modeling propagation through turbulence!

There are many textbooks and papers that deal with the topic in much more detail. I think you might be interested in particular in Whitham (1974). For sound, Urick (1975) deals with sound propagation in inhomogeneous media (but not a cross flow iirc). Gill (1982) discusses water and atmospheric gravity waves, including mountain lee waves. Durran has some many papers and a primer that can be found online dealing with mountain lee waves (topographically generated internal gravity waves) propagating upward in a shear flow.

Hope some of these resources help. It's an interesting topic but for sure I don't know all the ways people have come up with to deal with particular flows for particular situations.

 

[Killtech]

Sorry, I misunderstood your earlier point. I was viewing on a device that didn't show the equation properly and thought you just meant the simple translation.

I think you will run into difficulty if you just "shear" the coordinates, because the time dependency is changing as you say. I think you might want to try working along the wave characteristics instead.

well, it would be easier to discuss this if we had an equation to work with enabling to check such ideas directly. hence my original post, hehe.

For waves propagating in a medium, there is no way around it. The energy is linked to the velocity caused by the wave motion.

There are indeed non-linear interactions between waves. A typical assumption is "small" amplitude, so that the interactions at quadratic order or higher can be ignored. You still might want to know how the single wave is changing in energy density or amplitude as it travels through the medium.

in terms of simplification/approximation one could simply split the medium density into a sum of two parts where one is considered the actual medium density and the other an 'independent' density of wave equation such that all wave-wave interactions vanish removing the non-linear behavior. this should be the same as a "small" amplitude approach. and thus the energy could be calculated as the sum of the energy of the medium plus that of the wave which would be be something proportional to $\int |\phi(\vec x)|^2dV$ where $\phi$ is to be a wave solution.

Hope some of these resources help. It's an interesting topic but for sure I don't know all the ways people have come up with to deal with particular flows for particular situations.

i am looking through the and trying to learn/extract the information i am looking for. however they are mostly focused on specific phenomenons thus discussing quite special cases whereas i am looking for a generalization. that makes it a bit difficult.

i wonder whether what i am looking for is kind of described via Navier-Stokes equations - exception that i would have to "split" a "small" amplitude wave equation from it. isn't there something along that approach? i think that would be most helpful.but in any case i think i can construct a simplest study case for sound waves that yields the core element i want to know:
consider the space being split into two regions: in one where $y>0$ the medium has no current at all i.e. $\vec v(\vec x) = 0$. the other region with $y\leq 0$ the medium has a constant flow in $x$ direction i.e. $\vec v(\vec x) = 1\vec e_x$. besides that the medium should have constant density everywhere. now how to sound waves behave in this scenario? in the upper an lower region i can simply change to the frame moving with the medium as my base frame such that acoustic wave equations take the free wave form there (but not in the other region). now i could simply combine these two frames and transform differently in each region such that i have a free wave equation everywhere except for the border surface. the question is what happens there? both mathematical (how does my equation look there?) and physically (how would an acoustic wave behave passing through it?). if i knew the answer i could probably construct every other case for the incompressible medium case (i.e. for $\nabla \vec v(\vec x) = 0$) applying the same principle as for the construction of the Lebesque integral.

 

[olivermsun]

well, it would be easier to discuss this if we had an equation to work with enabling to check such ideas directly. hence my original post, hehe.

Sure then, let's work with your equation:
$\frac {\partial^2 \rho} {\partial t^2} - c(\vec x)^2 \nabla^2 \rho = 0$

in terms of simplification/approximation one could simply split the medium density into a sum of two parts where one is considered the actual medium density and the other an 'independent' density of wave equation such that all wave-wave interactions vanish removing the non-linear behavior. this should be the same as a "small" amplitude approach. and thus the energy could be calculated as the sum of the energy of the medium plus that of the wave which would be be something proportional to $\int |\phi(\vec x)|^2dV$ where $\phi$ is to be a wave solution.

That is what's usually done. What I called the "quadratic" terms are the product of two "perturbation" quantities.

i am looking through the and trying to learn/extract the information i am looking for. however they are mostly focused on specific phenomenons thus discussing quite special cases whereas i am looking for a generalization. that makes it a bit difficult.

I guess most of the special cases discussed are ones that have some special physical relevance so they are worth singling out. The approaches should give some insight on how to build up your general formulation, though.

but in any case i think i can construct a simplest study case for sound waves that yields the core element i want to know:
consider the space being split into two regions: in one where $y>0$ the medium has no current at all i.e. $\vec v(\vec x) = 0$. the other region with $y\leq 0$ the medium has a constant flow in $x$ direction i.e. $\vec v(\vec x) = 1\vec e_x$. besides that the medium should have constant density everywhere. now how to sound waves behave in this scenario? in the upper an lower region i can simply change to the frame moving with the medium as my base frame such that acoustic wave equations take the free wave form there (but not in the other region). now i could simply combine these two frames and transform differently in each region such that i have a free wave equation everywhere except for the border surface. the question is what happens there? both mathematical (how does my equation look there?) and physically (how would an acoustic wave behave passing through it?). if i knew the answer i could probably construct every other case for the incompressible medium case (i.e. for $\nabla \vec v(\vec x) = 0$) applying the same principle as for the construction of the Lebesque integral.

So here is where physics might yield some insight as how the interface between layers should be handled. For example, if you are talking about an acoustic wave propagating through a layered medium, then the pressure and velocity (or displacement) have to be matched across each interface. This requires, at the very least, that the phase lines (i.e., characteristics) continue unbroken across the interface. (You should get Snell's law almost immediately.)

 

[Killtech]

Sure then, let's work with your equation:
$\frac {\partial^2 \rho} {\partial t^2} - c(\vec x)^2 \nabla^2 \rho = 0$

well, this equation does not allow for any non-constant flows of the medium. at least i don't see a way how to put the flow in thus i am unsure how it can help with the discussion on wave propagation in a shear flow of the medium (or other non-constant flows).

So here is where physics might yield some insight as how the interface between layers should be handled. For example, if you are talking about an acoustic wave propagating through a layered medium, then the pressure and velocity (or displacement) have to be matched across each interface. This requires, at the very least, that the phase lines (i.e., characteristics) continue unbroken across the interface. (You should get Snell's law almost immediately.)

you are right, most quantities have to be matched at the border of the regions. i have taken a long stare at Navier-Stokes and it is obvious that all differentials appearing there have to exist to begin with - so there cannot be any jumps. in my example however medium velocity field is discontinuous, particularly $\frac {\partial v_x} {\partial y}$ does not exist at the border region and therefore the term $\mu \nabla^2 \vec v$ will explode there. that is however the viscosity term and if the medium is assumed to have no inner friction it should vanish and therefore my example flow should be possible, no? so my example physically refers to sound wave propagation in superfluid helium?

indeed i am quite certain that a generalized 'most simple case' of a wave equation in a medium should be given by Navier-Stokes combined with the continuity equation and a non-ideal-gas-law of the form $p = c(\rho)$ to resolve the pressure term. then of course i need quite a few further assumptions like zero viscosity, no external force and so on to get it into a simple enough form. and finally i would need to do a final approximation by splitting the density into two independent ones of which one is completely unaffected by the other - that is the medium shouldn't be affected by the density of the wave at all.

hmm, that said i guess i can now try to check how Navier Stokes transforms under the coordinates i was proposing...
okay i need to find some time to do the math and i get back when have something.

 

[Killtech]

Okay, i have derived some wave equation from Navier Stokes and would like someone to check if what i did is correct. In the following i will use the lower $t$ index to indicate partial derivative in time direction.

So i found one form of the Navier Stokes equation to be
$\rho \vec {v_t} + \rho (\vec v \cdot \nabla)\vec v = - \nabla p + \mu \Delta \vec v + (\lambda + \mu)\nabla(\nabla \cdot \vec v) + \vec f$

so for the case of no inner friction (a super fluid) and without external forces it simplifies to
$\rho \vec {v_t} + \rho (\vec v \cdot \nabla)\vec v = - \nabla p$

using a general not-so-ideal-gas-law $p = C(\rho)$, then $\nabla p = \nabla \rho C_\rho(\rho) = c(\vec x, t)\nabla \rho$ therefore:
$\rho \vec {v_t} + \rho (\vec v \cdot \nabla)\vec v = - c\nabla \rho$ or in terms of the current $\vec j_t = \rho_t \vec v + \rho \vec v_t = \rho_t \vec v - c\nabla \rho - \rho (\vec v \cdot \nabla)\vec v$

combined with continuity equation $\rho_t =- \nabla \vec j$ i get
$\rho_{tt} = c \Delta \rho + (\nabla c) \nabla \rho - \nabla ((\rho_t - \rho \vec v \cdot \nabla) \vec v)$
which is kind of a wave equation for the medium itself.

and since $\nabla c = \nabla C_\rho(\rho) = \nabla \rho C_{\rho\rho}(\rho)$, then $(\nabla c) \nabla \rho = (\nabla \rho)^2 C_{\rho\rho}(\rho) = (\nabla \rho)^2 c_{\rho}$ which is an unwelcome non-linearity.

assuming the medium has itself no time dependency, then this equation simplifies to
$0 = c \Delta \rho + (\nabla c) \nabla \rho + \nabla (( \rho \vec v \cdot \nabla) \vec v)$
or taking undoing the spatial derivative
$0 = - c\nabla \rho - \rho (\vec v \cdot \nabla)\vec v = \vec j_t$ or alternatively $c\nabla \rho = -\rho (\vec v \cdot \nabla)\vec v$

furthermore i want to make the approximation that waves within the medium don't affect the medium itself (small amplitudes?). so for this purpose i split $\rho = \rho_m + \rho_w$ in a medium and wave part and likewise for the velocity $\vec v$. as the medium is to be independent of the waves in it all terms with the wave index are just annulled. the equation for the waves on the other hand would just be the prior derived wave form of Navier Stokes with both terms for the medium and wave parts. that's a lot of terms. but for simple waves the velocity $\vec v_w$ can probably be assumed irrelevant enough to drop the convection term $(\vec v_w \cdot \nabla) \vec v_w$? furthermore terms like $c \Delta \rho_m + (\nabla c) \nabla \rho_m$ would make the wave equation inhomogeneous and thus act as source for new waves. i don't want to study waves generated by the mediums movement right now so i think its okay to drop these terms as well. and finally i just assume the medium isn't time dependent to simplify things further.

so in the end i arrive at this wave equation:
$\rho_{w,tt} = c \Delta \rho_w + (\nabla c) \nabla \rho_w - \nabla ((\rho_{w,t} - \rho_w \vec v_m \cdot \nabla) \vec v_m) + h$
where $h$ is the inhomogeneous term that i would null out for now with $h = 0 = c \Delta \rho_m + (\nabla c) \nabla \rho_m + (\rho_m \vec v_m \cdot \nabla) \vec v_m$

EDIT: but ehmm. do i get i right, that by attempting to split the wave equation from the medium i get inhomogenity terms that de-facto introduce something like 'charge' to my general wave equation (for the lack of a better word i use the same as maxwell-equations for these inhomogenities)??