  In the extension of Maxwell equations based on the Aharonov-Bohm Lagrangian
the e.m. field has an additional degree of freedom, namely a scalar field
generated by charge and currents that are not locally conserved. We analyze the
propagation of this scalar field through two different media (a pure dielectric
and an ohmic conductor) in a range of frequencies such that the properties of
the media are independent from the frequency. We find that an e.m. scalar wave
cannot propagate in a material medium. If a scalar wave in vacuum impinges on a
material medium it is reflected, at most exciting in the medium a pure
"potential" wave (which we also call a "gauge" wave) propagating at $c$, the
speed of light in vacuum, with a vector potential whose Fourier amplitude is
related to that of the scalar potential by $\omega
\mathbf{A}_{0}=\mathbf{k}\phi _{0}$, where $\omega^{2}=c^{2}\left\vert
\mathbf{k}\right\vert^{2}$.
