Mathematical description of a plane wave

2.15.1 Mathematical description of a plane wave

The mathematical description of a plane wave combines the properties of the scalar product, the vector product, and the complex exponential function; so in can serve as a very good example to illustrate the geometrical properties of this three concepts.
The most illustrative way to define the set of vectors pointing to a plane wave is given by

\[\vec{r}=\vec{r}_0+\alpha \vec{a}+\beta \vec{b};\]

here \(\vec{r}_0\) is a vector from the origin to a point of the plane, \(\vec{a}\) and \(\vec{b}\) are two non parallel vectors lying in the plane and \(\alpha\) and \(\beta\) are two real numbers.
A normal vector of the plane can be calculated by

\[\vec{N}= \vec{a}\times \vec{b}.\]

This normal vector can be used to define the set of vectors pointing to the plane by

\[\left\langle \vec{N}\right|\left. \vec{r}\right\rangle = \left\langle \vec{N}\right|\left. \vec{r}_0\right\rangle .\]

i.e. all vectors pointing to the plane have the same projection onto the normal vector of the plane.
A moving plane thus is represented by

\[\left\langle \vec{N}\right|\left. \vec{r}\right\rangle = \left\langle \vec{N}\right|\left. \vec{r}_0\right\rangle + v\; t\quad;\]

here \(v\) is the speed of the plane in the direction of the normal vector.
A plane wave typically is written as

\[\exp i \;\left(\left\langle \vec{k}\right|\left. \vec{r}\right\rangle - \omega\; t \right)=\exp i \;\left(\vec{k} \; \vec{r} - \omega\; t \right)\quad;\]

here \(\vec{k}\) is again the normal vector to the plane (the length is \(k = \left|\vec{k}\right|=\frac{2\pi}{\lambda}\), \(\lambda\): wave lenght), \(\omega=\frac{2\pi}{T}\) (\(T\): period), and the velocity of the plane wave is \(c = \frac{\omega}{k}\).

Real part and imaginary part of the exponential function are solutions as well. Examples are shown in the following animation.

With frame

Vector product