A sound propagation model in the air
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An insulated pipe of cross section is divided into an infinite number of compartments ( ) by insulated pistons and of cross section and mass .
In each compartment is a mole of air, compared to a perfect gas evolving isentropically according to the law of Laplace, .
At equilibrium, the abscissa of the piston ( ) is :
and pressure has the same value in each compartment.
Besides the equilibrium position, the abscissa of the piston ( ) is :
With and the pressure in the compartment is .
Establish the expression of the pressure as the function of , , , and and the linearized using :
Deduce the linear differential equation determining the motion of the piston number ( ).
Is Laplace checked ?
Do not forget that .
The approximation of a continuous medium is done by defining a function varying little in the scale of , as .
Establish the partial differential equation whose solution is .
Define a speed and comment expression.
The Newton's second law applied to the piston number gives:
The approximation of continuous medium resulting in :
increases if the environment is more rigid ( increases) and less inert ( decreases), which is natural for mechanical waves.
Evaluate the speed of sound in air assuming that the piston mass of the model are in fact constituted by the volume of air between two pistons in the model.
We give :
We find : (in good agreement with the expected value)