The Wave Equation

This is what I want to write since a long time ago: The wave equation, derived from Maxwell’s equations in free space. Finally! I can write now 🙂

To start with, we need to realize that an electromagnetic field is described by two vector fields, both are functions of position and time:

  1. The electric field, \vec{\varepsilon}(\vec{r},t)
  2. The magnetic field, \vec{H}(\vec{r},t)

The famous Maxwell’s equations in free space are defined by

Maxwells equations

the \nabla \times and \nabla \cdot are the curl and divergence. the constants of \epsilon_0 and $latex \mu_0$ represents the electric permitivity and the magnetic permeability in the free space.

Basically, we can express the wave equation based on the described Maxwell’s equation above. So, here we go

calculation1

Next, we need to use the vector identitiy (curl of the curl) of \nabla \times \left( \nabla \times \vec{\varepsilon} \right)=\nabla \left( \nabla \cdot \vec{\varepsilon} \right)-\nabla^2 \vec{\varepsilon}. From (3), we arrive to the equation of \nabla \times \left( \nabla \times \vec{\varepsilon} \right)=\nabla \left( 0 \right)-\nabla^2 \vec{\varepsilon} \longrightarrow \nabla \times \left( \nabla \times \vec{\varepsilon} \right)=-\nabla^2 \vec{\varepsilon}. Therefore, we will have

calculation2

By applying the speed of light equation in free space c_0=\frac{1}{\sqrt{\varepsilon_0 \mu_0}}, we arrive to the wave equation in the free space according to the Maxwell’s equation

wave equation electric field

If we start with (1) i.e \nabla \times \left( \nabla \times \vec{H} = \epsilon_0 \frac{\partial \vec{\varepsilon}}{\partial t} \right) and follow the same step, the final result for the wave equation will be

wave equation magnetic field

When we speak about scalar wavefunction, the electric (\vec{\varepsilon}(\vec{r},t)) and magnetic field (\vec{H}(\vec{r},t)) can be represented in the scalar wavefunction (u(\vec(r),t)), and we will have the wave equation as it is explained below,

wave equation general

 

Great! Now I am satisfied enough 😀

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