How does a general electromagnetic wave equation looks like?

Question

I know that electromagnetic wave consists of electric field and magnetic field, say:

$ \overrightarrow{E}\left(x,t\right)=E_{0}\cos\left(kx-\omega t\right) $

$ \overrightarrow{B}\left(z,t\right)=B_{o}\cos\left(kz-wt\right) $

But if I want to represent the wave consists of both fields as one equation, should I just add them ? like :

$ \psi\left(x,z,t\right)=E_{0}\cos\left(kx-\omega t\right)+B_{0}\cos\left(kz-\omega t\right) $

I'll be glad to see a correct example of how a general electromagnetic wave looks like. Also, is there a relation between the magnetic field amplitude and the electric field magnitude?

Thanks in advance.

Answer 1

The answer is, I don't know but I know that why you should not do.

First of all wave equation on OP's question are wrong (they don't have any direction). So I'm assuming the following component of electric and magnetic field vector.

$$\mathbf{E}=E_x(z,t)\hat{i}$$ $$\mathbf{B}=B_y(z,t)\hat{j}$$

We need to combine these vector component, the first thing that comes to mind is to make a vector-like $$\Psi(z,t)=E_x(z,t)\hat{i}+B_y(z,t)\hat{j}$$ So that's look a good combination but this good and as long as the electric field component and magnetic component are apart from each other but if we have electric field vector with two components (Like circular ,elliptical wave), the idea of combining these vector will be useless and painful.

Most of the time we don't care about magnetic components because if we know the electric field it's trivial to find magnetic field with following relation.

$$\mathbf{B}=\frac{1}{c}\mathbf{\hat{k}}\times \mathbf{E}$$

One more thing is to note that when you say general wave equation (for EM wave) means that you are talking about $$\nabla^2\mathbf{E}=\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}$$ $$\nabla^2\mathbf{B}=\mu_0\epsilon_0\frac{\partial^2\mathbf{B}}{\partial t^2}$$