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


Calculation method used for the reference solution
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The pressure field :math:`p` verifies the Helmholtz equation along the :math:`x` axis:

:math:`\frac{{d}^{2}p}{{\mathit{dx}}^{2}}+{k}^{2}p=0`, :math:`x\in [\mathrm{0,}L]`

In :math:`x=0`, we find conditions with the imposed speed limits :math:`{V}_{n}^{s}` and impedance :math:`{Z}_{\mathit{n0}}` limits:

:math:`p(x=0)={Z}_{\mathit{n0}}\left({V}_{n}(x=0)-{V}_{n}^{s}\right)`

This gives in terms of imposed admittance:

:math:`{V}_{n}(x=0)={V}_{n}^{s}+{A}_{\mathit{n0}}p(x=0)`

In :math:`x=L`, the impedance :math:`{Z}_{\mathit{nL}}` (or its opposite, the admittance :math:`{A}_{\mathit{nL}}`) is imposed:

:math:`\frac{-\frac{\mathit{dp}}{\mathit{dx}}}{-i\rho \omega }={A}_{n}p`

The pressure in the waveguide has the following form:

:math:`p(x)={C}_{1}{\mathrm{e}}^{-\mathit{ikx}}+{C}_{2}{\mathrm{e}}^{\mathit{ikx}}`

Thus, we can express the derivative of pressure:

:math:`\frac{\mathit{dp}(x)}{\mathit{dx}}=-{\mathit{ikC}}_{1}{\mathrm{e}}^{-\mathit{ikx}}+{\mathit{ikC}}_{2}{\mathrm{e}}^{\mathit{ikx}}`

The boundary conditions in :math:`x=0` allow you to write:

:math:`-{\mathit{ikC}}_{1}+{\mathit{ikC}}_{2}=-i\rho \omega {V}_{n}^{s}-i\rho \omega {A}_{\mathit{n0}}({C}_{1}+{C}_{2})`

The boundary condition in :math:`x=L` allows you to write:

:math:`{\mathit{ikC}}_{1}{\mathrm{e}}^{-\mathit{ikL}}-{\mathit{ikC}}_{2}{\mathrm{e}}^{\mathit{ikL}}=-i\rho \omega {A}_{\mathit{nL}}({C}_{1}{\mathrm{e}}^{-\mathit{ikL}}+{C}_{2}{\mathrm{e}}^{\mathit{ikL}})`

These last two relationships make it possible to write a system of equations whose resolution makes it possible to determine the constants :math:`{C}_{1}` and :math:`{C}_{2}` and therefore the particle pressure and speed at any point in the waveguide.

For the sake of simplicity, the analytical expression for the coefficients :math:`{C}_{1}` and :math:`{C}_{2}` is not given here.


Benchmark results
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We calculate the pressure at points :math:`A`, :math:`B`,, :math:`C`, and :math:`D`.


Uncertainty about the solution
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* Analytical solution.