7.Plane Waves

One of the simple examples of a three-dimensional wave is the plane wave. In order to derive the mathematical expression of a plane wave moving in direction vector $\vec{k}$ perpendicular to the plane, we first write the position vector in Cartesian coordinates in terms of unit vectors $\hat{\vec{i}},\hat{\vec{j}},\hat{\vec{k}}$  (see Figure 7.1.a):

$\vec{r}=x\hat{\vec{i}}+y\hat{\vec{j}}+z\hat{\vec{k}}$                                                          (7.1)

For a plane anywhere in space, that begins at some arbitrary origin O and ends at the point (x,y,z) on that plane (see Figure 7.1.b):

$(\vec{r}-{{\vec{r}}_{o}})=(x-{{x}_{o}})\hat{\vec{i}}+(y-{{y}_{o}})\hat{\vec{j}}+(z-{{z}_{o}})\hat{\vec{k}}$                                        (7.2)

Figure 7.1. (a) Cartesian unit vectors. (b) A plane wave moving in the $\vec{k}$-direction.

We would set the vector $\left( \vec{r}-{{{\vec{r}}}_{o}} \right)$ to sweep out a plane perpendicular to $\vec{k}$ therefore,

$\left( \vec{r}-{{{\vec{r}}}_{o}} \right)\bullet \vec{k}=0$                                                              (7.3)

Rewriting directional vector $\vec{k}$ in term of unit vectors:

$\vec{k}={{k}_{x}}\hat{\vec{i}}+{{k}_{y}}\hat{\vec{j}}+{{k}_{z}}\hat{\vec{k}}$                                                       (7.4)

And Equation (7.3) becomes:

${{k}_{x}}\left( x-{{x}_{o}} \right)+{{k}_{y}}\left( y-{{y}_{o}} \right)+{{k}_{z}}\left( z-{{z}_{o}} \right)=0$                                       (7.5)

Thus, $\vec{k}\bullet \vec{r}=$ constant.

A set of planes over which the wave function $\psi (\vec{r})$ varies sinusoidally for constant $\vec{k}\bullet \vec{r}$  can be constructed in term of complex representation:

$\psi (\vec{r})=A{{e}^{i\vec{k}\bullet \vec{r}}}$                                                         (7.6)

Naturally, these harmonic functions are repetitive, therefore, they can be expressed by:

$\psi (\vec{r})=\psi (\vec{r}+\frac{\lambda \vec{k}}{k})$                                                    (7.7)

In this equation, k is the magnitude of $\vec{k}$ and ${}^{{\vec{k}}}/{}_{k}$ is a unit vector parallel to it (see Figure 7.2).

Figure 7.2. Plane waves.

Applying Equation (7.7) to Equation (7.6), we obtain the exponential form of the harmonic wave function:

$\psi (\vec{r})=A{{e}^{i\vec{k}\bullet \vec{r}}}=A{{e}^{i\vec{k}\bullet (\vec{r}+{}^{\lambda \vec{k}}/{}_{k})}}=A{{e}^{i\vec{k}\bullet \vec{r}}}{{e}^{i\lambda k}}$                                         (7.8)

To satisfy Equation (7.6), we must have:

${{e}^{i\lambda k}}=1={{e}^{i2\pi }}$                                                           (7.9)

Therefore $\lambda k=2\pi $ or $k={}^{2\pi }/{}_{\lambda }$. The vector $\vec{k}$ is called the propagation vector, whose magnitude is the propagation number k.

Since $\psi (\vec{r})$ is moving i.e. is varying in time, we introduce the time dependence ωt in analogy to that of the one-dimensional wave equation:

$\psi \left( \vec{r},t \right)=A{{e}^{i(\vec{k}\bullet \vec{r}\mp \omega t)}}$                                                    (7.10)

With A, ω, and k constant. The wave travels in the direction of propagation vector $\vec{k}$ such that we can assign a phase corresponding to it at each point in space and time. The surfaces connecting every points of equal phase at any given time are known as the wavefronts (see Figure 7.3). 

Figure 7.3. Wavefronts for a harmonic plane waves.

The propagation velocity of the wavefront is equal to the phase velocity of a plane wave given by Equation (7.10). Following from Figure 7.2, r_k is the scalar component of $\vec{r}$ in the direction of $\vec{k}$. Since we assume that the wave is homogeneous and the disturbance on a wavefront is constant, so that after a time dt and after the wavefront moves along $\vec{k}$ by a distance dr_k, Equation (7.10) should be:

$\psi (\vec{r},t)=\psi (r+d{{r}_{k}},t+dt)=\psi ({{r}_{k}},t)$                                             (7.11)

Which can be rewritten in exponential form as follows:

$A{{e}^{i(\vec{k}\bullet \vec{r}\mp \omega t)}}=A{{e}^{i(k{{r}_{k}}+kd{{r}_{k}}\mp \omega t\mp \omega dt)}}=A{{e}^{i(k{{r}_{k}}\mp \omega t)}}$                                      (7.12)

So that $kd{{r}_{k}}=\pm \omega dt$ and the wave velocity ${}^{d{{r}_{k}}}/{}_{dt}$ is then:

$\frac{d{{r}_{k}}}{dt}=\pm \frac{\omega }{k}=\pm v$                                                         (7.13)