Displacement and Pressure in acoustics

[zpatenaude37]

So my question deals with open-open tubes. When there is a displacement antinode there is a pressure node. I can somewhat conceptualize this because of the boundary condition where the pressure at the ends needs to be atmospheric. But I am confused about the ideal gas laws relation to this. If a maximum displacement is at a compression of the air, then won't pressure increase since PV=nRT, PM/RT=ρ. At a compression, does the density of particles not increase? Wouldn't this result in an increase in pressure? I can't find anything answering this specifically.

 

[Simon Bridge]

So my question deals with open-open tubes. When there is a displacement antinode there is a pressure node. ... If a maximum displacement is at a compression of the air, then won't pressure increase ...

This is a standing wave. The pressure node is a zone where the pressure does not vary much, while the pressure antinode is where it varies a lot ... so it has both compression and rarefaction periods. You can relate that to the motion of the air particles.

So the max displacement occurs when there is a rarefaction, and minimum displacement when there is compression... at the displacement antinodes.

 

[zpatenaude37]

So is the pressure antinode at a displacement node because the energy of the particles at a displacement node is kinetic?

 

[Cutter Ketch]

At the displacement antinode the molecules are all moving together at the same time neither compressing or expanding so the pressure doesn't vary. In the crudest sense these regions of moving back and forth together have to run into each other somewhere. At the displacement nodes the molecules are always being pushed or pulled equally from both sides. They can't move either direction and are being alternately squeezed and pulled so the pressure varies the maximum.

 

[zpatenaude37]

Thanks for your replies it has helped me to visualize the process. I'm still having trouble thinking of it as an ideal Gas process .

For example. during a compression does the temperature begin to rise,which provides kinetic energy to the particles. does this increase the mean free path between particles which drives it to a rarefaction? does this process then become self sustaining and periodic?

 

[Cutter Ketch]

Thanks for your replies it has helped me to visualize the process. I'm still having trouble thinking of it as an ideal Gas process .

For example. during a compression does the temperature begin to rise,which provides kinetic energy to the particles. does this increase the mean free path between particles which drives it to a rarefaction? does this process then become self sustaining and periodic?

Under compression and rarefaction the gas does change temperature as you suggest. That change does perturb the process making it less linear. In particular the speed of the wave depends on temperature so at high amplitude the propagation is less simple. Look up nonlinear acoustics.

As for self-sustaining and periodic, I think that may be confusing the issue. A wave (or a standing wave) is already self sustaining in that it will continue to store and pass on energy at least until the energy dissipates. That's true whether or not we get into high enough amplitude that the heating and cooling become significant. So when you mention the heating and ask if it's self sustaining it sounds like you are implying something more than just propagation, like perhaps there is some free energy keeping the wave going. There is nothing like that. The oscillation is the response of the air to an external driving force and while the external energy is stored and released there is no extra energy taken from the air. There can and will be energy lost to heating the air, but that is different from the reversible adiabatic heating from the pressure change which comes right back when the pressure drops again. The spring just doesn't quite follow Hooke's law.

 

[Simon Bridge]

A musical instrument making a sound is a driven-damped system.
Remember you have to blow into it to keep it going.

Have you seen:
http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html
...Acoustics and vibration animations page from Penn state.

 

[zpatenaude37]

Okay thanks. that link helped a lot too I just wasn't picturing it properly. so the compression occurs at the displacement node following a max displacement and a rarefaction is at the node following a min. I was picturing the compression at the max displacement, not a quarter period after. it all makes sense.