Killtech said:
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.