B Compressible potential flow equation

The governing equations are (6.1), equivalent to

$$
\begin{align}
& \frac{1}{\rho} \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{u} + \frac{1}{\rho} \mathbf{u} \cdot \nabla \rho = 0 \nonumber \\
& \frac{\partial \mathbf{u}}{\partial t} + \frac{1}{2} \nabla (\mathbf{u} \cdot \mathbf{u}) = - \frac{\nabla P}{\rho} \tag{B.1}
\end{align}
$$

With **u** = **∇**φ, the momentum equation becomes:

$$
\begin{align*}
& \frac{\partial \nabla \phi}{\partial t} + \frac{1}{2} \nabla (\nabla \phi \cdot \nabla \phi) = - \frac{\nabla P}{\rho} = - \nabla \int_{P_0}^{P} \frac{d\tilde{P}}{\rho} \tag{B.2} \\
& \Rightarrow \frac{\partial \phi}{\partial t} + \frac{1}{2} (\nabla \phi \cdot \nabla \phi) + \int_{P_0}^{P} \frac{d\tilde{P}}{\rho} = C \\
& \Rightarrow \frac{\partial^2 \phi}{\partial t^2} + \frac{1}{2} \frac{\partial}{\partial t} (\nabla \phi \cdot \nabla \phi) + \frac{\partial}{\partial t} \int_{P_0}^{P} \frac{d\tilde{P}}{\rho} = 0 \\
& \Rightarrow \frac{\partial^2 \phi}{\partial t^2} + \frac{1}{2} \frac{\partial}{\partial t} (\nabla \phi \cdot \nabla \phi) + \frac{1}{\rho} \frac{\partial P}{\partial t} = 0
\end{align*}
$$

The definition of the speed of the material wave (here the speed of light) yields $c^2 = \frac{dP}{d\rho}$. Assuming constant the speed of wave, $c^2 \frac{\partial \rho}{\partial t} = \frac{\partial P}{\partial t}$. The momentum equation therefore becomes

$$
\frac{\partial^2 \phi}{\partial t^2} + \frac{1}{2} \frac{\partial}{\partial t} (\nabla \phi \cdot \nabla \phi) + \frac{c^2}{\rho} \frac{\partial \rho}{\partial t} = 0 \Rightarrow \frac{1}{\rho} \frac{\partial \rho}{\partial t} = - \frac{1}{c^2} \frac{\partial}{\partial t} \left( \frac{\partial \phi}{\partial t} + \frac{1}{2} (\nabla \phi \cdot \nabla \phi) \right)
$$

From equation (B.2),

$$
\frac{1}{\rho} \mathbf{u} \cdot \nabla \rho = \frac{1}{\rho c^2} \mathbf{u} \cdot \nabla P = - \frac{1}{c^2} \nabla \phi \cdot \nabla \left( \frac{\partial \phi}{\partial t} + \frac{1}{2} (\nabla \phi \cdot \nabla \phi) \right)
$$

As $\nabla \cdot \mathbf{u} = \Delta\phi$, the continuity equation of (B.1) yields

$$
\begin{align*}
& \Delta\phi - \frac{1}{c^2} \frac{\partial}{\partial t} \left( \frac{\partial\phi}{\partial t} + \frac{1}{2} (\nabla\phi \cdot \nabla\phi) \right) - \frac{1}{c^2} \nabla\phi \cdot \nabla \left( \frac{\partial\phi}{\partial t} + \frac{1}{2} (\nabla\phi \cdot \nabla\phi) \right) = 0 \\
\Rightarrow \Delta\phi & - \frac{1}{c^2} \left( \frac{\partial^2\phi}{\partial t^2} + \frac{\partial}{\partial t} (\nabla\phi \cdot \nabla\phi) + \frac{1}{2} \nabla\phi \cdot \nabla (\nabla\phi \cdot \nabla\phi) \right) = 0
\end{align*}
$$

Let us note $x_i$ with $i \in [0, 2]$ representing coordinates $x, y$ and $z$ and $j \equiv i + 1 \pmod 3$. Then the general form of the equation is

$$
\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = \sum_{i=0}^{2} \left[ \left( 1 - \left( \frac{\partial \phi}{\partial x_i} \right)^2 \right) \frac{\partial^2 \phi}{\partial x_i^2} - \frac{2}{c^2} \left( \frac{\partial \phi}{\partial x_i} \frac{\partial^2 \phi}{\partial x_i \partial t} + \frac{\partial \phi}{\partial x_i} \frac{\partial \phi}{\partial x_j} \frac{\partial^2 \phi}{\partial x_i \partial x_j} \right) \right] \quad (B.3)
$$

Let us assume small perturbations in the flow, neglecting third order terms. By introducing the Mach number in each direction $Ma_i = \frac{u_i}{c} = \frac{\frac{\partial \phi}{\partial x_i}}{c}$, equation (B.3) looks like

$$
\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} - \sum_{i=0}^{2} (1 - Ma_i^2)^2 \frac{\partial^2 \phi}{\partial x_i^2} + \frac{2}{c} \sum_{i=0}^{2} Ma_i \frac{\partial^2 \phi}{\partial x_i \partial t} = 0 \quad (B.4)
$$

Finally, if the fluid velocity is assumed to be mainly along one direction such that **u** = (V∞ + u<sub>x</sub>)e<sub>x</sub> + u<sub>y</sub>e<sub>y</sub> + u<sub>z</sub>e<sub>z</sub> with u<sub>x</sub>, u<sub>y</sub> and u<sub>z</sub> small velocity disturbances of the flow far smaller than V∞, then the second-order terms in u<sub>x</sub> can be neglected in equation (B.4):

$$
\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} - (1 - Ma^2)^2 \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial y^2} - \frac{\partial^2 \phi}{\partial z^2} + \frac{2}{c} Ma \frac{\partial^2 \phi}{\partial x \partial t} = 0 \quad (B.5)
$$