1. Electromagnetic Waves in Vacuum
Maxwell’s Equations in Free Space
wave equation in 3D

2. General solution
The one dimensional wave equation is unusual for a partial differential equation in that a very simple general solution may be found by defining new variables:

3. Jean le Rond d'Alembert (1717–1783)
In other words, solutions of the 1D wave equation are sums of a right traveling function F and a left traveling function G. "Traveling" means that the shape of these individual arbitrary functions with respect to x stays constant, however the functions are translated left and right with time at the speed c.

4. For an initial value problem, the arbitrary functions F and G can be determined to satisfy initial conditions:
d'Alembert's formula

5. Example

6. The sinusoidal wave
wave number
phase shift
wavelength
angular frequency
frequency
sinusoidal wave traveling to the right
sinusoidal wave traveling to the left