Wave through a compressbile material

[hoomanya]

Hi,

I need to know what is expected to happen if say a wave (say a sine wave or en exponential wave) passe through an isothermal gas or a liquid (with compressibility). So the governing equations are the Euler equations. what is expected to happen if the initial velocity of the gas or the liquid is zero.

Does the wave split into two, one right and one left moving?

Thanks,

H

 

[Studiot]

Does the wave split into two, one right and one left moving?

What happens to a sound wave?

 

[hoomanya]

not sure, goes both waves and then flattens out, depending on the wave speed relative to the speed of sound ? I would appreciate references and some quick answers please.

 

[Studiot]

Well how many directions does a sound wave spread out in from the source?

And does the wave travel or is it stationary?

 

[hoomanya]

What do u mean? That was my question!

 

[Studiot]

I mean I think you already know (most) of the answer and I am trying to help you understand and work it out for yourself.

That is the purpose of this website.

Your original post showed several misconceptions. It is inappropriate to apply the term isothermal to the medium - that is a term applied to the thermodynamic process.

Sound is in any case an adiabatic process.

Can you understand this?
http://www.chem.arizona.edu/~salzmanr/480a/480ants/kapsadex/kapsadex.html

 

[hoomanya]

I see ... thanks. Well, like I said in my question, I expect the wave to split into a right and left moving wave and flatten out by the end when the initial velocity of the material is zero. I need to know if this is correct asap please?

 

[Studiot]

If you stand in the middle of a room and shout

You can hear the sound at each end of the room.
So the sound travels in opposite directions.

But you can also hear the sound at the sides of the room.
So the sound travels sideways as well.

In fact the sound travels in all directions.

Now a compressibility wave in a material is a sound wave.

Do you understand this so far?

 

[hoomanya]

yes , thanks. Sorry, I was thinking 1D when I said in two directions. ... so far I understand.

 

[Studiot]

Sorry I should have said "if someone stood in the middleof the room.."

You obvoiously can't be in two places at once- but you got my meaning.

To proceed.

I'm not sure what you mean by flatten out but are you perhaps confusing compressibility with dissipation, dispersion and diminution?

That is as the 3D sound wave passes through successive spherical balls or shells on its outward journey away from the source it gets weaker as the surface area of the ball increases so it is more spread out.

This will happen without any loss of energy ie in a conservative system.

Do you understand this?

 

[hoomanya]

thanks I understand this. This might sound stupid but why does the wave become weaker as it travels further?

So if I'm looking at just a specific radius from the source, then the wave vanishes in that region (since it's moving out further from the source).

 

[Studiot]

No it's not stupid it's a good question to ask.

Firstly for conservative systems (such as a lightwave) the intensity (measured by the wave amplitude) diminishes as the wave gets further from the source.

This is a straightforward consequence of the law of conservation of energy.

Consider a sphere or ball of radius r centrered the source.
The wave(s) start of with total energy E joules flowing (transported) per second.
Now this total energy E moves away from the source every second and passes through a sphere of radius r every second. If it did not do this there would be a build up of energy on the sphere.
Thus the surface energy flux is E divided by the surface area of the sphere. That is E/4∏r²

Clearly this same energy must pass through a sphere of radius 2r and surface energy E / 4∏(2r)²

So the energy density diminishes in inverse relation to the square of the distance.

However we are not interested in the energy density directly, but the sound amplitude.

Look here at the inverse square law and in particular at the acoustic example

http://en.wikipedia.org/wiki/Inverse-square_law#Example_2

Note that the energy is proportional to the amplitude squared.


When you have digested that you have understood why, even in conservative systems, the energy spreads out with distance we can address non conservative systems.

Here you must add in terms for

1) Dissipation eg as heat or against friction. Energy here is removed faster than the inverse sq law and converted to something other form.

2) Dispersion.
All the foregoing assumes a linear response by the medium to displacement. That is the restoring force that produces the (simple) harmonic motion of each oscillator in the wave is proportional to the displacement.
If the response in non linear we have additional terms which reduce the amplitude (but not the energy since this is conserved).

 

[hoomanya]

Thanks very much for your help and time! I think understood all off this.