Understanding the Applicability of the Acoustics Wave Equation

[Radiohannah]

Hello!

When considering the acoustics wave equation

$$\frac{\partial^{2}P}{\partial t^{2}} = c^{2} \nabla^{2} P$$

I don't really understand why you can say that the applicability of this equation varies for different sound pressure levels. I don't see why this shouldn't hold for all pressures? Am I missing the point somewhere?


:-)

Hannah

 

[AlephZero]

The acoustic equation describes small linearised perturbations about the steady state of the fluid.

For large amplitudes of pressure and velocity the fluid behaviour is not linear. For example the velocity of the vibrating fluid as predicted by the acoustic equation might be greater than the speed of sound in the fluid. Or the pressure amplitude might be greater than the static pressure in the fluid, so the minimum pressure (according to the acoustic equation) would be negative. Both of those situations are physically impossible.

 

[Radiohannah]

So when considering the higher pressure waves, you would derive an acoustics wave equation in which you didn't make small amplitude waves approximations? I get it now! Thank you :-)

 

[Studiot]

There is more.

Sound waves in our atmosphere are usually adiabatic processes.
However in some media (liquid nitrogen) the process can switch to isothermal.

Here is a comparison.

http://www.mathpages.com/home/kmath109/kmath109.htm

 

[Radiohannah]

Great, thanks! With that further approximation that the sound waves are adiabatic, PV^{\gamma} = constant, does that also then put limitations on the pressures that can be described by the acoustic equation?

 

[AlephZero]

So when considering the higher pressure waves, you would derive an acoustics wave equation in which you didn't make small amplitude waves approximations?

That's right. For example you find that the wave velocity depends on the amplitude. There are also non-sinusoidal shapes of waves that can propagate.

Great, thanks! With that further approximation that the sound waves are adiabatic, PV^{\gamma} = constant, does that also then put limitations on the pressures that can be described by the acoustic equation?

The bottom line is "can the energy go anywhere fast enough to make a difference". For most applications of acoustics the answer is no, but obviously you could have a situation where would make a difference - for example if the temperature change was large enough that radiation heat transfer was important. Heat conduction in gases is usually very poor, and if it wasn't poor you wouldn't get adabatic behaviour.

There is usually no measurable difference between adiabatic and isothermal behaviour in liquids, because Cp/Cv is very close to 1. But what happens in liquid nitrogen at supercritical temperatures, and frequencies of the order of THz, may well be interesting if you want to make a "cloud chamber" type of detector for sub-atomic particles...
http://arxiv.org/PS_cache/cond-mat/pdf/0512/0512383v1.pdf

 

[Radiohannah]

That's super, thank you!