In summary, the characteristic impedance of a cable is its high frequency impedance and is determined by the ratio of voltage to current in the cable. The wave's velocity is also dependent on the medium's permitivity, which is in turn influenced by the frequency of the wave.
Why does a wave reflect when it meets an impedance discontinuity? And why is there a 180 degrees phase shift of the reflected wave when the transmission line finished with short circuit and no phase shift when the far end of the line is opened?
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Like, how do you define characteristic impedance? What is the relationship between the wave and the impedance of the medium that it is traveling through?
The characteristic impedance of a cable (it was coaxial cable I was interested in) is, as far as i know, its high frequency impedance (if we supply the line with some signal, the line draws some current from the source and the ratio voltage/current defines the characteristic impedance of the cable. (the cable acts as a series of LC cells, and is lossy because of its resistance)
As regards the relation betwen wave and medium impedance, I don't know if there is a general formula to characterize this. I know that the wave's velocity depends on medium permitivity which in turn is dependendent on waves' frequency.