Accoustic waves notions (propagation in a tube)

[Cathr]

Please help me with this problem I am facing, I am lacking notions of acoustics and I would be very grateful if someone could clarify them:

A tube has a revolution symmetry arounf the $x$ axis and has a section dependent of the value of the abscissa (x), so the profile $S(x)$ is known. The tube is filled with a fluid with the density (mass/volume) at rest = m and the pressure P0. Whe call $p(x,t)$ the overpressure and $e(x,t)$ the elongation in the presence of an accoustic perturbation. The total pressure is $P(x,t)=P0+p(x,t)$.

We admit the by applying the fundamental principle of dynamics for a slice of fluid of width dx at the x abscissa, we can show that the tube profile does not modify the relationship that is obtained for cylindrical tubes:

$\frac{dp}{dx}=-m \frac{d^2e}{dt^2}$

These were the given statemens, now the questions are:

1. Using the linear response approximation, write the relationship for the dilation D, the compressibility X and overpressure p.
2. Making sure to control the coherence of the approximation orders of the different terms, show that:
$D=\frac{1}{S} \frac{d(Se)}{dx}$

There are notions here that I don't quite understand. What is the difference between the elongation e and the dilation D? What do they correspond to?

Why, when writing the fundamental principle of dynamics (F=Ma) we don't use the mass, but the mass over volume for the fluid?

I would really like to know the relationships between these variables, as I searched everywhere I could and didn't find. Thanks a lot in advance!

 

[haruspex]

What is the difference between the elongation e and the dilation D?

Analysing the equations given dimensionally, I see that m is a density, e is a distance, but D is dimensionless. So maybe D is the fractional expansion of a region... but then, it would have a simple relationship to pressure, no?
Not sure how S is defined. From the description, I would have guessed radius, but from the equations it looks like cross sectional area.

when writing the fundamental principle of dynamics (F=Ma) we don't use the mass, but the mass over volume for the fluid?

On the left it is dp/dx, a pressure gradient. If you consider a small volume, the net force acting is pressure gradient multiplied by the volume. So you can go from F=ma to dp/dx=ρa by dividing by the volume of the element.

 

[Cathr]

On the left it is dp/dx, a pressure gradient. If you consider a small volume, the net force acting is pressure gradient multiplied by the volume. So you can go from F=ma to dp/dx=ρa by dividing by the volume of the element.

Thank you!
I found a solution, and, if you don't mind, I have a question that is rather out of personal interest. How can I calculate the amplitude of a sound wave that gets out of the tube, comparing to the one that enters?

Direct application: Suppose I don't like the sound intensity of my phone, and I want to amplify it. Is it possible to do this using a truncated cone out of paper? If so, how can I calculate the maximum intensity that I can obtain? This is, of course, in function of the radiuses of the truncated cones, or their ratio.

 

[haruspex]

Thank you!
I found a solution, and, if you don't mind, I have a question that is rather out of personal interest. How can I calculate the amplitude of a sound wave that gets out of the tube, comparing to the one that enters?

Direct application: Suppose I don't like the sound intensity of my phone, and I want to amplify it. Is it possible to do this using a truncated cone out of paper? If so, how can I calculate the maximum intensity that I can obtain? This is, of course, in function of the radiuses of the truncated cones, or their ratio.

By consrvation of energy, the only way you could increase the amplitude is to narrow the wavefront. But at a given distance from the speaker, the wave will still spread over the same area, so that won't help. You would need to channel the sound or get the outlet closer to your ear.