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But your question was this: does the Characteristic Impedance of a coax
vary with frequency?
I suppose, in answer, I could cite a voltage wave that rises at 0.1 volt
per minute
into a coax. You will understand that the coax looks quite resistive, at
a rather high value in this case. And at the other end, if you shine a
flash light into a coax, it will be reflected and absorbed like a
(relative) short circuit.