On the electromagnetic wave equation

Question

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?

Answer 1

For the first question; Consider the 1 dimensional light, that is to be a wave going in one direction. Two get a sense of a two dimensional wave imagine the waves on the surface of a water, if you have seen one it is a circular wave going in every direction (in everu direction would be a circle pattern on a 2d surface) for a three dimensional just generalize the analogy and think of a spherical shape that is getting bigger and bigger with time a good analogy is the mechanical waves in the air (sound).

For the second question I would wonder since we took these wave equations in Vacuum. Where there's no charge the fields has to satisfy:

$$ \nabla \cdot \vec B = 0 $$ 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.