Reference solution
=====================


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 :math:`(\omega L/c\ll 1)`. The problem is therefore one-dimensional along the :math:`x` axis.

We show [:ref:`bib1 <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:

.. math::
 :label: eq-1

    \ mathrm {\ omega} =\ sqrt {\ frac {k} {m}}

With:

.. math::
 :label: eq-2

    k= {k} _ {\ text {air}}} = {\ mathrm {\ rho}} _ {a} {c} _ {a} ^ {2}\ frac {S} {S} {{L} _ {a}}

So we find:

.. math::
 :label: eq-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* :math:`\omega` *verified* :math:`(\omega L/c\ll 1)` *.*


Bibliographical references
---------------------------


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