electromagnetic properties:
$E =$ electric field, in $\frac{kg \cdot m}{C \cdot s^2}$
$D = \epsilon E =$ displacement field, in $\frac{C}{m^2}$
$B =$ magnetic field, in $\frac{kg}{C \cdot s}$
$H = \frac{1}{\mu} B = $ magnetizing field, in $\frac{C}{m \cdot s}$

$\rho =$ charge density, in $\frac{C}{m^3}$
$J =$ current density, in $\frac{C}{m^2 \cdot s}$

GLM variables:
$\phi =$ electric divergence potential, in $\frac{C}{m^2}$
$\xi =$ speed of propagation of electric divergence, in $\frac{m}{s}$
$\psi =$ magnetic divergence potential, in $\frac{kg}{C \cdot s}$
$\kappa =$ speed of propagation of magnetic divergence, in $\frac{m}{s}$

Maxwell equations, in vector notation:
$\partial_t D - \frac{1}{\mu} \nabla \times B = -J$
$\partial_t B + \frac{1}{\epsilon} \nabla \times D = 0$

(following 2000 Munz):
$\partial_t \mathcal{D}(\Phi) - c^2 \Delta \Phi = 0$
continuity equation: $\partial_t \rho + \nabla \cdot J = 0$
$\partial_t \mathcal{D}(\Phi) - c^2 \Delta \Phi = \frac{1}{\epsilon_0} (\partial_t \rho + \nabla \cdot J)$
Let $\mathcal{D}(\Phi) = \frac{\Phi}{\xi^2}$
$\partial_t \frac{\Phi}{\xi^2} - c^2 \Delta \Phi = \frac{1}{\epsilon_0} (\partial_t \rho + \nabla \cdot J)$
$\partial_t \Phi - \xi^2 c^2 \Delta \Phi = \frac{\xi^2}{\epsilon_0} (\partial_t \rho + \nabla \cdot J)$

combine this with $\partial_t \Phi + \xi c \frac{\partial \Phi}{\partial \zeta} = 0$, and $\frac{\partial \Phi}{\partial \zeta} = \vec{n} \cdot \nabla \Phi$
so $\partial_t \Phi + \xi c \vec{n} \cdot \nabla \Phi = 0$