  We describe solvable models that capture how impurity scattering in certain
fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge
mode that does not carry electric charge. These models consist of two
counter-propagating chiral Luttinger liquids together with a collection of
discrete impurity scatterers. Our main result is an exact solution of these
models in the limit of infinitely strong impurity scattering. From this
solution, we explicitly derive the existence of a neutral mode and we determine
all of its microscopic properties including its velocity. We also study the
stability of the neutral mode and show that it survives at finite but
sufficiently strong scattering. Our results are applicable to a family of
Abelian fractional quantum Hall states of which the $\nu = 2/3$ state is the
most prominent example.
