Harmonic plane wave

A harmonic wave (or monochromatic wave) is a single-frequency wave. It varies according to a sinusoid in both time and space. If we fix space and observe oscillation, we get a temporal frequency and period. If we fix time and observe, we get a spatial frequency and period.

A plane wave propagates such that the field is uniform over every plane perpendicular to the direction of propagation. All points with the same ${\vec{k}}_o\cdot \vec{r}$ form a plane perpendicular to ${\vec{k}}_o$. The general form of a harmonic plane wave in Cartesian space is:

$$\vec{E}(\vec{r},t)={\vec{E}}_o\exp \left[i\left(\vec{k}\cdot \vec{r}\pm \omega t+{\phi }_o\right)\right]$$

where we have the parameters:

• $|{\vec{E}}_o|$ is the amplitude of oscillation. Its vector direction is the polarisation direction. Units are in volts per metre (same as regular electric field).
• $\vec{k}$ indicates the propagation direction. For $-\omega t$ (above), this indicates propagation in the positive $\hat{k}$ direction. For $+\omega t$, it indicates the negative $\hat{k}$ direction. Units are radians per metre.
• $\vec{r}$ is the position vector in metres.
• $|\vec{k}|$ indicates the spatial frequency.
• $\omega$ indicates the temporal frequency, in radians per second.
• And ${\phi }_o$ is the initial phase at $t=0,\vec{r}=\vec{0}$, in radians.

In isotropic media, plane waves are transverse: $\vec{k}\perp \vec{E}$ and $\vec{k}\perp \vec{H}$.