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Do Maxwell's equations alone allow for propagating waves in charge/current (\phi,s[/B]J)?

I was rather struck dumb by this question out of the blue. I've never seen it addressed, denyed or confirmed.

Schematically the electric and magnetic fields are first derivatives of the vector and electric potential, and can propagate as waves under first derivative constraints of the currents and charges.

Schematically the charge/current is a second derivative of the potentials. It seems it should have waving solutions under fourth derivative constrains of the potentials.

To make thing even better, there is no other velocity in maxwells equations other than c,

so should Maxwell's equations admit charged fields that propagate at the speed of light?

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# Charged Massless Waves in Maxwells Equations?

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