## 2.5. Impedance of a closed tube using the inhomogeneous wave equation

### Boundary conditions

We have already found the impedance of a closed tube by taking the solution to the following Helmholtz wave equation

$\left(\frac{{\partial}^{2}}{\partial {x}^{2}}+{k}^{2}\right)\tilde{p}\left(x\right)=0$

(2.79)

and applying boundary conditions to the solution. It is known as a homogeneous wave equation because there are no sound sources explicit in the equation. These are included in the boundary conditions that are applied to the solution. Here we shall consider the inhomogeneous wave equation

$\left(\frac{{\partial}^{2}}{\partial {x}^{2}}+{k}^{2}\right)\tilde{p}\left(x\right)=-\delta (x-$

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