Undergrad Oscillations and Waves in Fluids

[Tazerfish]

I first wanted to ask a very specific question:
There is something called the Brunt-Vaisala frequency.
It describes the frequency of oscillation in a fluid with a density gradient.
Because if a parcel of fluid is pushed up or down from its stable state it will oscillate around it.

What i don't get is why this can be applied to the oscillation of air behind mountains.


One of the fundimental axioms you derive this from is that your fluid parcel has a constant density.
And with air does not make any sense.
A bit of air would simply expand (violating constant density)
and slightly cool when going up and then pretty much stay where it is.
Vice versa when going down.
Wouldn't it ?

But this got me thinking ...
I actually don't really understand gravity driven waves.(Maybe waves in general )
(We didn't have them in school yet.)
(Although we did have basic mechanical and electric oscillations.)
Even the miraculously simple equations for wave speed in water don't really make sense to me.
If somebody had nice derivations for mathematical desciptions of these things i would really appretiate it:
Water/Air Interface (basically ocean) waves
Atmospheric waves (stratified)
Waves on a String or spring ...

Posting some links with reasonably "easy" derivations would really help me out.
I think this and many other questions about waves would solve themselves if i knew a little more about them in general.
Sorry i know this question is a little dumb and the only person who finds it interesting or would be helped is probably me, but i am going to ask it anyways and hope for some replies.

 

[Simon Bridge]

Waves in air are pressure changes - which means density changes ... since a rarifaction is, by definition, a change in particle density in a volume.
However - the math is about oscillations around the mean ... when the sources are talking about a pressure gradient they usually mean the mean pressure is changing smoothly, and there are oscillations about that. This should makes sense because you can already think of the speed of sound in still air... even though the presence of the sound means the air is not still.

There are no "nice" derivations in the sense of "simple" ... the waves in nature are described by the maths of waves. We can derive wave motion for different situations by applying Newton's Laws to models of the situation. You get how you can have wave motion for a mass on a spring right? Have you seen coupled mass-on-spring or pendulums?

For bulk materials you may want to start here:
http://www.acs.psu.edu/drussell/Demos/waves/wavemotion.html
... and realize that the difference between states of matter is how strongly the particles are coupled to each other ...

For the maths, brace yourself and look up "damped driven harmonic oscillator" and "wave equation" and "helmholtz equation".
It is a common exercise for senior college students to derive the helmholtz equation for simple systems like an infinite line of small masses separated by massless springs.