# On the electromagnetic wave equation

Solving the Maxwell equations leads us to, $$\nabla^{2} \vec{E}-\frac{1}{c^{2}}\frac{\partial^2 \vec{E}}{\partial t^2}=0$$ And $$\nabla^{2} \vec{B}-\frac{1}{c^{2}}\frac{\partial^2 \vec{B}}{\partial t^2}=0$$ I am aware that, $$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{v^{2}}\frac{\partial^2 \psi}{\partial t^2}$$ Represents a (one-dimensional) wave equation. Now,

Questions:-

$$(1)$$- I was told that the first two equations are for 3-dimensional wave equation. I wonder how to represent it geometrically. I mean how does a three dimensional wave looks like?

$$(2)$$ How do we know that electric and magnetic components are perpendicular to each other?

$$\nabla \cdot \vec B = 0$$ and $$\nabla \cdot \vec E = 0$$ Since this is the case, using the curl equations it implies that they must be perpendicular (for instance if the value of $$\vec B$$ in direction $$x$$ exists it means that the Electric field has $$y$$ and$$z$$ components only and so on. Thus $$f(E_z,E_x)\cdot E_y +f(E_z,E_y)\cdot E_x+f(E_y,E_x)\cdot E_z$$ and using $$\nabla\cdot E = 0$$ one can see that the fields are perpendicular.