Benchmark solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------

The general analytical solution for a waveguide is written as:


* for pressure:


.. math::
 :label: eq-1

    p (x, y, z) =A\ mathrm {exp} (\ mathit {ikx}) +B\ mathrm {exp} (-\ mathit {ikx})


* for the vibratory speed:


.. math::
 :label: eq-2

    V=-\ frac {1} {{\ rho} _ {0} {0} {c}} _ {0}}\ left (A\ mathrm {exp}) (\ mathit {ikx}) -\ mathit {Bexp} (-\ mathit {bexp}} (-\ mathit {ikx})\ left (A\ mathrm {exp}) (\ mathit {ikx}) -\ mathit {bexp} (-\ mathit {bexp}} (-\ mathit {ikx}) (-\ mathit {ikx}) (-\ mathit {ikx})


* A and B are determined by the boundary conditions:


.. math::
 :label: eq-3

    \ text {En} x=0\ to {V} _ {n} = {V} _ {n0}


.. math::
: label: eq-4

    \ text {En} x=L\ to p (L, y, z) =ZV (L, y, z)\ cdot {n} _ {L}

This results in:

.. math::
: label: eq-5

    A=B\ left (\ frac {Z- {\ rho} _ {0} {\ rho} _ {0}} {Z+ {\ rho} _ {0}} _ {0}}\ right)\ mathrm {exp} (-2\ mathit {iL})

And:

.. math::
 :label: eq-6

    B=\ frac {{\ rho} _ {0} {c} _ {0} {c} _ {V}}} {\ left (\ frac {Z- {\ rho} _ {0} {c} _ {0} _ {0}} _ {0}} {0}}\ right)\ mathrm {exp} (-2\ mathit {iL}}} {0}}\ right)\ mathrm {exp} (-2\ mathit {iL}}}) -1}

In the case studied, the output of the guide is anechoic, :math:`Z={\rho }_{0}{c}_{0}` and therefore:

.. math::
 :label: eq-7

    p (x, y, z) = {\ rho} _ {0} {c} _ {0} {V} _ {n}\ mathrm {exp} (-\ mathit {ikx})
    V (x, y, z) =- {V} _ {n}\ mathrm {exp} (-\ mathit {ikx}) {e} _ {x}

For acoustic intensity:

.. math::
 :label: eq-8

    I=\ frac {1} {2} p {V} ^ {\ text {*}}} =\ frac {1} {2} {\ rho} _ {0} {c} _ {0} {c} _ {0} {V}} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V} _ {V}

That is, the active acoustic intensity is uniform throughout the guide and parallel to the axis. The natural frequencies are given for the guide closed at both ends by:

.. math::
 :label: eq-9

    {f} _ {m, n, p} =\ frac {{c} _ {0}} {0}} {0}} {2} {\ left (\ frac {{m} ^ {2}}} {{I} _ {x} ^ {2}}} +\ frac {{p} ^ {2}}} +\ frac {{p} ^ {2}}} +\ frac {{p} ^ {2}}} +\ frac {{p} ^ {2}}} {{I} _ {z} ^ {2}}}\ right)} ^ {\ frac {1} {2}}


Benchmark results
----------------------

Pressure at points :math:`A,B,C,D` (for A, B, C, D, E models).

Acoustic intensity at points :math:`A,B,C,D` (for models A and C).

Natural frequencies No. 2 to No. 9.


Uncertainty about the solution
---------------------------

Analytical solution


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


1. BOUIZI A. Solving linear acoustic equations by a mixed finite element method - Thesis (1989).