- #1
Cathr
- 67
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Thread moved from the technical forums, so no Homework Template is shown
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!
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!