  We show explicitly that the Hertz-form Maxwell's equations and their
extensions can be obtained from the non-relativistic expansion of Lorentz
transformation of Maxwell's equations. The explicit expression for the
parameter $\alpha$ in the extended Hertz-form equations can be derived from
such a non-relativistic expansion. The extended Hertz-form equations, which do
not preserve Galilean invariance, origin from Lorentz transformation of
Maxwell's equations and differ from the Galilean-transformed Maxwell equations
(the original Hertz equations) by the relative sign differences between the two
$\alpha$ terms etc. Especially, the $\alpha$ parameter is of relativistic
origin. The superluminal behavior illustrated by the D'Alembert equation from
the extended Hertz-form equations should be removed by including all subleading
contributions in the $v/c$ expansion, although such a superluminal behavior
will not occur in the vacuum because $\alpha=0$. We should note that in the
Hertz form and extended Hertz form equations, the electromagnetic fields should
take the forms $ \vec{\mathcal{E}}(x)=\vec{E}(\Lambda^{-1}x)$ and $
\vec{\mathcal{B}}(x)=\vec{B}(\Lambda^{-1}x)$. Such a choice of description for
the fields is different from the ordinary one with $\vec{E}(x)$ and
$\vec{B}(x)$, which are well known to satisfy the ordinary Maxwell's equations.
The descriptions of electromagnetic phenomena using the function set
$\{\vec{\mathcal{E}}(x),\vec{\mathcal{B}}(x)\}$ and the function set
$(\vec{E}(x),\vec{B}(x))$ are equivalent, with the
$\{\vec{\mathcal{E}}(x),\vec{\mathcal{B}}(x)\}$ description satisfying the
extended Hertz-form Maxwell's equations in the low speed approximation. The
solution of (extended) Hertz-form Maxwell's equations describe the traveling
wave form electromagnetic field.
