# Representations of the Wave Equation and its Solutions

*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the masters course Selected Topics in Audio Signal Processing, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*

## Time-Domain

The inhomogeneous wave equation in the time-domain is given as

\begin{equation}
\Delta p(\mathbf{x}, t) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} p(\mathbf{x}, t) = - q(\mathbf{x}, t) .
\end{equation}

The wave equation is a linear partial differential equation (PDE) with constant coefficients.

## Monochromatic

Let's assume a monochromatic real-valued excitation $q(\mathbf{x}, t) = Q(\mathbf{x}, \omega_0) \cos(\omega_0 t)$ with angular frequency $\omega_0 = 2 \pi f_0$. Due to the linearity of the wave equation, the following Ansatz is chosen for the sound pressure $p(\mathbf{x}, t) = P(\mathbf{x}, \omega_0) \cos(\omega_0 t)$. It is common to use complex calculus for PDEs, hence 

\begin{eqnarray}
q(\mathbf{x}, t) &= \Re \{ Q(\mathbf{x}, \omega_0) e^{j \omega_0 t} \} \\
p(\mathbf{x}, t) &= \Re \{ P(\mathbf{x}, \omega_0) e^{j \omega_0 t} \} .
\end{eqnarray}

Introducing the complex quantities into the wave equation yields

\begin{equation}
\Delta P(\mathbf{x}, \omega_0) e^{j \omega_0 t} + \left( \frac{\omega_0}{c} \right)^2 P(\mathbf{x}, \omega_0) e^{j \omega_0 t} = - Q(\mathbf{x}, \omega_0) e^{j \omega_0 t} .
\end{equation}

The complex exponential function can be canceled resulting in

\begin{equation}
\Delta P(\mathbf{x}, \omega_0) + \left( \frac{\omega_0}{c} \right)^2 P(\mathbf{x}, \omega_0) = - Q(\mathbf{x}, \omega_0) .
\end{equation}

For $t=0$ the time-domain solution of the wave equation for $\omega_0$ is given as $p(\mathbf{x}, 0) = \Re \{ P(\mathbf{x}, \omega_0) \}$. It is common to discard the index 0 in the angular frequency. The index was however introduced for the sake of comparison with the temporal Fourier transform of the wave equation discussed in the following.

## Temporal Fourier Domain

Temporal Fourier transform of the inhomogeneity and the pressure yields the Helmholtz equation

\begin{equation}
\Delta P(\mathbf{x}, \omega) + \left( \frac{\omega}{c} \right)^2 P(\mathbf{x}, \omega) = - Q(\mathbf{x}, \omega) ,
\end{equation}

where the differentiation theorem of the Fourier transform has been applied. The link to the monochromatic case can be found by considering a monochromatic excitation 

\begin{equation}
Q(\mathbf{x}, \omega_0) = Q(\mathbf{x}, \omega) \delta(\omega - \omega_0) .
\end{equation}

Due to the linearity of the wave equation the following Ansatz is chosen

\begin{equation}
P(\mathbf{x}, \omega_0) = P(\mathbf{x}, \omega) \delta(\omega - \omega_0) .
\end{equation}

Introducing the right-hand-sides of the latter two equations into the Helmholtz equation yields the same result as for the monochromatic case

\begin{equation}
\Delta P(\mathbf{x}, \omega_0) + \left( \frac{\omega_0}{c} \right)^2 P(\mathbf{x}, \omega_0) = - Q(\mathbf{x}, \omega_0)
\end{equation}

Hence, the time-domain solution for a monochromatic excitation for $t=0$ may also be derived from the temporal Fourier transform of the problem by evaluation of $p(\mathbf{x}, \omega) = \Re \{ P(\mathbf{x}, \omega) \}$ for one particular frequency $\omega_0$.

## Temporal Excitation

Let's consider the solution $P_\delta(\mathbf{x}, \omega)$ of the Helmholtz equation for the excitation with a temporal Dirac impulse $q(\mathbf{x}, t) = q(\mathbf{x}) \delta(t)$

\begin{equation}
\Delta P_\delta(\mathbf{x}, \omega) + \left( \frac{\omega}{c} \right)^2 P_\delta(\mathbf{x}, \omega) = - Q(\mathbf{x})
\end{equation}

Due to the linearity of the wave equation the pressure field for a generic time-domain excitation $q(\mathbf{x}, t) = q(\mathbf{x}) \hat{q}(t)$ is given as

\begin{equation}
P(\mathbf{x}, \omega) = P_\delta(\mathbf{x}, \omega) \hat{Q}(\omega)
\end{equation}

or

\begin{equation}
p(\mathbf{x},t) = p_\delta(\mathbf{x}, t) * \hat{q}(t)
\end{equation}

**Copyright**

This notebook is provided as [Open Educational Resources](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text/images/data are licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Selected Topics in Audio Signal Processing - Supplementary Material*.

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