Question

Given a uniform plane wave in phasor notation and gamma in a lossy nonmagnetic medium; How do I calculate the magnitude of the wave at a certain location and time?

Answer 1

A time harmonic uniform plane wave travelling in the z direction and linearly polarized along the x axis is described by the phasor:

\[E=E _{0}e ^{ - \gamma z}=E_{0}e^{- \alpha z}e^{-j \beta z}\]

The minus signs don't seem to show up but each exponential has a minus sign in front for a wave travelling in the +z direction. Eo is the magnitude of the E field at z=0.

In time harmonic fields, we are dealing with one frequency (at a time -- an arbitrary E field distribution throughout space can be found with Fourier transform) and the phasor notation is shorthand for a time domain sinusoid just like phasors in circuit analysis.

The time domain signal is:

\[E(t)=\Re \left\{ E_{0}e^{- \alpha z}e^{-j \beta z}e^{jwt} \right\}=\Re \left\{E_{0} e^{- \alpha z}e^{j(wt- \beta z)} \right\}\]
\[E(t)=E_{0}e^{-\alpha z}\cos(wt-\beta z)\]