# IllustrationPlane wave in a complex plane

Level 3
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

Share — copy and redistribute the material in any medium or format

Adapt — remix, transform, and build upon the material for any purpose, even commercially.

Sharing and adapting of the illustration is allowed with indication of the link to the illustration.

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).