<commit_before>\chapter{Acoustic Perturbation Equations Solver}

\newcommand{\dd}[2]{\frac{\partial #1}{\partial #2}}

\section{Synopsis}
The aim of APESolver is to predict aerodynamic sound generation. Through
the application of a splitting technique, the flow-induced acoustic field is
totally decoupled from the underlying incompressible hydrodynamic field. The
acoustic perturbation equations (APE-1/APE-4) proposed by Ewert and Shroeder are employed as
the governing equations of the acoustic field and they assure stable
aeroacoustic simulation due to the suppression of the term related to the
production of perturbed vorticity. These equations are similar to the linearised
perturbed compressible equations, but while in the original formulation the flow
decomposition is based on solenoidal vortical perturbations as well as
irrotational acoustic perturbations, in this case perturbations are assumed to
be exclusively of acoustic nature.
\begin{subequations}
    \begin{align*}
	\frac{\partial p'}{\partial t}
	+ \overline{c}^2 \frac{\partial \overline{\rho} u'_i}{\partial x_i}
	+ \overline{c}^2 \frac{\partial \overline{u}_i p' / \overline{c}^2}{\partial x_i}
	&= \overline{c}^2 q_c
	\\
	\frac{\partial u'_i}{\partial t}
	+ \frac{\partial \overline{u}_j u'_j}{\partial x_i}
	+ \frac{\partial p' / \overline{\rho}}{\partial x_i}
	&= 0
	\end{align*}<commit_msg>APE: forgot to remove my custom command def from the userguide<commit_after>\chapter{Acoustic Perturbation Equations Solver}

\section{Synopsis}
The aim of APESolver is to predict aerodynamic sound generation. Through
the application of a splitting technique, the flow-induced acoustic field is
totally decoupled from the underlying incompressible hydrodynamic field. The
acoustic perturbation equations (APE-1/APE-4) proposed by Ewert and Shroeder are employed as
the governing equations of the acoustic field and they assure stable
aeroacoustic simulation due to the suppression of the term related to the
production of perturbed vorticity. These equations are similar to the linearised
perturbed compressible equations, but while in the original formulation the flow
decomposition is based on solenoidal vortical perturbations as well as
irrotational acoustic perturbations, in this case perturbations are assumed to
be exclusively of acoustic nature.
\begin{subequations}
    \begin{align*}
	\frac{\partial p'}{\partial t}
	+ \overline{c}^2 \frac{\partial \overline{\rho} u'_i}{\partial x_i}
	+ \overline{c}^2 \frac{\partial \overline{u}_i p' / \overline{c}^2}{\partial x_i}
	&= \overline{c}^2 q_c
	\\
	\frac{\partial u'_i}{\partial t}
	+ \frac{\partial \overline{u}_j u'_j}{\partial x_i}
	+ \frac{\partial p' / \overline{\rho}}{\partial x_i}
	&= 0
	\end{align*}
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