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Traveling Plane Wave

Overview

From Wikipedia:

A traveling plane wave is a special case of plane wave, namely a field whose evolution in time can be described as simple translation of its values at a constant wave speed \(c\), along a fixed direction of propagation \(\vec{n}\).

The general form of the traveling plane wave can be represented as

\[ F( \vec{r}, t ) = G ( \vec{r} \cdot \vec{n} - ct ) \]

where

  • \(\vec{r} \in \mathbb{R}^N\) is a position in the N-dimensional space
  • \(\vec{n} \in \mathbb{R}^N\) is the direction vector of the wave propagation
    • Note: by definition, the direction vector is of unit length.
  • \(d = \vec{r} \cdot \vec{n} \in \mathbb{R}\) is the "displacement" of the wave, computed as the vector dot product of \(\vec{r}\) and \(\vec{n}\)
  • \(G(u): \mathbb{R} \to \mathbb{Z}\) is a function describing the profile of the wave, namely the value of the field at time \(t = 0\), for each displacement \(d\).
    • \(u = d - ct \in \mathbb{R}\)
    • In most real-world cases such as a sinusoidal plane wave, the output of \(G(u)\) is real (i.e., \(G(u): \mathbb{R} \to \mathbb{R}\))

Sinusoidal Plane Wave

A sinusoidal plane wave is a special case of traveling plane wave. It can be represented as

\[ F( \vec{r}, t ) = A \cos( 2 \pi \nu \left( \vec{r} \cdot \vec{n} - ct \right) + \varphi ) \]

where

  • \(A \in \mathbb{R}\) is the amplitude of the wave
  • \(\nu \in \mathbb{R}^+\) is the spatial frequency of the wave
    • The spatial frequency is the inverse of the wave length (i.e., \(\nu = \frac{1}{\lambda}\))
  • \(\varphi\) is the initial phase of the sinusoidal wave.

Wave Vector of Sinusoidal Plane Wave in 3D space

In the 3D world, if a sinusoidal plane wave travels in a direction described by the spherical coordinates \((\theta, \phi)\), where \(\theta\) is the polar angle and \(\phi\) is the azimuth angle, then the direction vector \(\vec{n}\), in the Cartesian coordinates, will be

\[ \vec{n} = \left[ \begin{array}{c} \sin\theta \cos\phi \\ \sin\theta \sin\phi \\ \cos\theta \end{array} \right] \]

And therefore the wave vector \(\mathbf{k}\) of such sinusoidal plane wave is

\[ \mathbf{k} = 2 \pi \nu \vec{n} = 2 \pi \nu \left[ \begin{array}{c} \sin\theta \cos\phi \\ \sin\theta \sin\phi \\ \cos\theta \end{array} \right] \]

Reference