A very common misconception is that nothing can propagate at a speed greater than the speed of light in vacuum c. In fact the phase velocity of a wave can be significantly larger than c. As we will see here ,we experience such behaviour on a daily basis. I will particularly discuss the phase propagation of a wave pattern that result from reflection from a shiny surface.

 

Consider the situation shown in the figure below. A wave incident on a conducting surface (shown by the solid blue line) reflects at the conducting surface (solid red arrow). The wavefronts of the incident and reflected waves (dashed lines) interface with one another causing an interference pattern.

 

We will assume that the electric field is polarized in the E_z (i.e. it is parallel to the interface). The incident and reflected waves can be written as

(1)   \begin{equation*} E_{inc}(x,y)=E_0(x,y)e^{i\left(\omega x\sin\theta-\omega y\cos\theta\right)}\mathbf{\hat{z}}, \end{equation*}

(2)   \begin{equation*} E_{r}(x,y)=E_{0r}(x,y)e^{i\left(\omega x\sin\theta+\omega y\cos\theta\right)}\mathbf{\hat{z}}, \end{equation*}

where we have used a normalized units such that c=1. Additionally the e^{-i\omega t} convention was adopted.

 

At the interface, y=0, the tangential field must be zero. Therefore,

    \[E_{0r}(x,y)=-E_0(x,y).\]

The total electric field E(x,y)=E_{inc}(x,y)+E_r(x,y),  is found to be

 

(3)   \begin{equation*} E(x,y)=E_0(x,y)e^{i(\omega x\sin\theta-\omega t)}\sin(\omega y\cos\theta) \end{equation*}

 

At fixed y=y_0 as the  gray horizontal line shows, the field is a function of x only. In this case

(4)   \begin{equation*} E(x,y_0)=E_0(x,y_0)\sin(\omega y_0\cos\theta)e^{i\omega\left(x\sin\theta-t\right)}, \end{equation*}

which is a one dimensional traveling wave with a phase speed that is equal to (normalized)

(5)   \begin{equation*} v_{ph}=\frac{1}{\sin\theta} \end{equation*}

or in terms of c as

(6)   \begin{equation*} v_{ph}=\frac{c}{\sin\theta}. \end{equation*}

Since |\sin\theta|\leq 1, the propagation speed is always \geq c.

 

 

To demonstrate such behaviour, we will consider the numerical computation shown here. In this case, a Gaussian beam illuminates a conducting surface. As long as the beam waist is sufficiently large, the wavefront can be considered to be flat (please check the wikipedia page on Gaussian beams). The equivalence principle is used to represent the incident wave as equivalent electric and magnetic sources over a surface (here taken to be a circle of radius = 6 m).

The white solid lines represent the directions of the incident and reflected waves. In this case, the incident wave is making an angle of \approx, 30 degrees with the normal. This means that the phase velocity is approximately c/\sin\pi/6\approx 2c. The conducting surface is represented by the solid white block. Additionally, I added solid markers along both the incident wave and the section where the incident and reflected waves interfere. It is clear from the animation that the wavelength of the interference pattern is larger than the wavelength of the incident (or reflected) wave. In fact the wavelength has doubled, which results in doubling the phase speed.