We do know that that transladed $\mathcal{C}^2$ functions are solutions to the wave equation, namely: \begin{equation} \ddot{f}(k\hat{z}-\omega t)=c^2\nabla^2f(k\hat{z}-\omega t) \end{equation} Which gives clear reasoning behind the use of complex exponentials as solutions for the EM wave equations: \begin{align} &\dfrac{\partial^2\mathbf{E}}{\partial t^2}=c^2\nabla^2\mathbf{E}\\ &\dfrac{\partial^2\mathbf{B}}{\partial t^2}=c^2\nabla^2\mathbf{B} \end{align} Actually, one uses the ket notation to define, in terms of polarization angles: \begin{equation} |\psi\rangle\stackrel{\mathrm{def}}{=}\begin{pmatrix}\psi_x\\\psi_y\end{pmatrix}=\begin{pmatrix}\cos\theta\ e^{i\alpha_x}\\\sin\theta\ e^{i\alpha_y}\end{pmatrix} \end{equation} And write the classical sinusoidal solution for the eletric field wave equation: \begin{equation} E(r,t)=E_0\ \mathfrak{R}\{|\psi\rangle e^{i(k\hat{z}-\omega t)}\} \end{equation} Well, that I do not fully understand. In Jackson's Classical Electrodynamics, I believe, he uses this form as a quite elementary one. How can I see that this is a solution to the wave equation, more importantly, why do we use this form? Does it bring anything?
1 Answer
The cosine and sine form a basis for the solution to the one-dimensional wave equation. This can be verified by plugging the sine and cosine into the wave equation, and since they form an infinite orthogonal basis any function of $(\omega t - kz)$ and $(\omega t+ kz)$ are a solution. These are of particular interest because classical electrodynamics is often carried out in the frequency domain, where the system response for a monochromatic excitation is studied. In general, solutions to a one-dimensional wave problem are the sum of a forward and backward traveling wave. Since a sum of a cosine and a sine of a single frequency with arbitrary phase can be written as a single cosine, it makes sense to do so. With Euler's formula, this cosine can be written as the real part of a complex exponential in order to simplify derivatives. That is: $$E_0\cos(\omega t - kz + \phi) = \mathfrak{R}\{E_0e^{i(\omega t-kz+\phi)}\} =\mathfrak{R}\{E_0e^{i\phi}e^{i(\omega t-kz)}\}.$$ This part $E_0e^{i\phi}$ is generally written as a single complex variable called the complex amplitude. Since the system response is usually linear (isotropic, linear materials) it makes sense to omit the $e^{i\omega t}$ part for brevity. This is called phasor notation. Rarely a $e^{-i\omega t}$ convention is chosen. The electric field of to the one-dimensional wave equation can then be written as:
$$E(z) = Ae^{-ikz} + Be^{ikz},$$ where A is the complex amplitude of the forward travelling wave, and B the complex amplitude of the backward traveling wave (relative to the positive $\hat{z}$ direction). Clearly this form is much easier for calculating derivatives and integrals and is more concise than the time domain form.
The instantaneous electric field, which in frequency domain studies is usually given a different symbol such as $\mathcal{E}$, is then obtained from the phasor of the field as:
$$ \mathcal{E}(z,t) = \mathfrak{R}\{ E(z)e^{i\omega t}\}.$$
