The change in the electric potential due to lightning is evaluated.
The potential along the lightning channel is a constant which is
the projection of the pre-flash potential along a piecewise harmonic
eigenfunction which is constant along the lightning channel.
The change in the potential outside the lightning channel is a harmonic
function whose boundary conditions
are expressed in terms of the pre-flash potential and
the post-flash potential along the lightning channel.
The expression for the lightning induced electric potential change is
derived both for the continuous equations, and for a spatially discretized
formulation of the continuous equations.
The results for the continuous equations are based on the properties of
the eigenvalues and eigenfunctions of the following generalized eigenproblem:
Find $u \in H_0^1 (\Omega)$, $u \ne 0$,
and $\lambda \in \mathbb{R}$ such that
$
\langle \nabla u, \nabla v \rangle_{\mathcal{L}} =
\lambda \langle \nabla u, \nabla v \rangle_{\Omega}
$
for all $v \in H_0^1 (\Omega)$, where $\Omega \subset \mathbb{R}^n$
is a bounded domain (a box containing the thunderstorm),
$\mathcal{L}$ is a subdomain (the lightning channel),
and $\langle \cdot, \cdot \rangle_{\Omega}$ is
the inner product
$
\langle \nabla u,\nabla v\rangle_\Omega =\int_{\Omega}
\nabla u\cdot\nabla v \; {{\rm d}x}.
$