2. Benchmark solution

2.1. Calculation method used for the reference solution

The pressure field \(p\) verifies the Helmholtz equation along the \(x\) axis:

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

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

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

This gives in terms of imposed admittance:

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

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

\(\frac{-\frac{\mathit{dp}}{\mathit{dx}}}{-i\rho \omega }={A}_{n}p\)

The pressure in the waveguide has the following form:

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

Thus, we can express the derivative of pressure:

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

The boundary conditions in \(x=0\) allow you to write:

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

The boundary condition in \(x=L\) allows you to write:

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

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

For the sake of simplicity, the analytical expression for the coefficients \({C}_{1}\) and \({C}_{2}\) is not given here.

2.2. Benchmark results

We calculate the pressure at points \(A\), \(B\),, \(C\), and \(D\).

2.3. Uncertainty about the solution

• Analytical solution.