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In a text on application of electromagnetism in transmission line, there introduces a phasor for the voltage (in frequency domain)

$$\tilde{V}(x) = V^+e^{-i\beta x} + V^-e^{i\beta x.}$$

Here $V^+$ and $V^-$ are the amplitude of the incoming wave and reflected wave. My question is the exponential $e^{-i\beta x}$ and $e^{i\beta x}$ are already complex, so should $V^+$ and $V^-$ be real or complex? and why?

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up vote 0 down vote accepted

Complex $V^+$ and $V^-$ (or in practice, just the reflected wave amplitude) represent the relative phase shift between the two signals.

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Thanks. So the phase shift will be reflected in $V^+$ and $V^-$ instead of $\beta x$? – user1285419 May 5 '13 at 7:39
@user1285419: You're welcome. Yes, $\beta x$ gives the phase characteristic of the wave as a function of position, but it has a fixed value of 0 at $x=0$. A complex amplitude permits that value to be different. – Art Brown May 5 '13 at 15:37
I got it. Thanks. – user1285419 May 5 '13 at 22:31

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