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Illustration Plane wave in a complex plane

Plane wave in complex plane
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A one-dimensional plane wave with amplitude \(A\), wave number \(k\) and angular frequency \(\omega\) represented as a complex exponential function:$$ \mathit{\Psi}(x,t) ~=~ A \, e^{\mathrm{i}\,(k\,x - \omega\,t)} $$

This is represented in a complex plane as a vector. Its length corresponds to the ampltude \(A\) and the angle \(\varphi\) between the \(\mathit{\Psi}\)-vector and the real axis corresponds to the phase \( k\,x - \omega\,t \). As time passes, the phase changes and the vector rotates (here: clockwise).