2. Reference solution

2.1. Calculation method used for the reference solution

At low frequencies, the acoustic wavelengths of the movements envisaged are long compared to the characteristic dimension of the fluid volume \((\omega L/c\ll 1)\). The problem is therefore one-dimensional along the \(x\) axis.

We show [bib1] that a light fluid like air essentially acts as an added stiffness while a heavy fluid behaves only as an added mass. We can therefore calculate the first natural frequency of the system:

(2.1)\[ \ mathrm {\ omega} =\ sqrt {\ frac {k} {m}}\]

With:

(2.2)\[ k= {k} _ {\ text {air}}} = {\ mathrm {\ rho}} _ {a} {c} _ {a} ^ {2}\ frac {S} {S} {{L} _ {a}}\]

So we find:

(2.3)\[ \ mathrm {\ omega} =\ sqrt {\ frac {\ frac {{\ mathrm {\ rho}}} _ {a} ^ {2}} {{L} _ {a}\ left ({\ mathrm {\ rho}}} _ {\ rho}} _ {rho}} _ {rho}} _ {rho}} _ {e}} {L}} _ {e}\ right)}}\]

Note:

The system’s first clean pulse \(\omega\) verified \((\omega L/c\ll 1)\) .

2.2. Bibliographical references

1. GIBERT - Vibrations of Structures. Interactions with fluids. Random sources of excitement - Collection of the Direction des Etudes et Recherches d’EDF.