# Derivation of the Wave Equation in Time

Here, we derive the wave equations in time for the electric and magnetic fields.To accomplish this, we begin with Faraday’s Law and Ampere-Maxwell’s Law:

(309)$\boldsymbol{\nabla} \times \mathbf{e} = -\frac{\partial \mathbf{b}}{\partial t}$
(310)$\boldsymbol{\nabla} \times \mathbf{h} = \mathbf{j} + \frac{\partial \mathbf{d}}{\partial t}$

as well as the three constitutive relations:

(311)$\mathbf{j} = \sigma \mathbf{e}$
(312)$\mathbf{d} = \epsilon \mathbf{e}$
(313)$\mathbf{b} = \mu \mathbf{h}$

## Derivation for the Electric Field

To derive the wave equation for $$\mathbf{e}$$, we first take the curl of Faraday’s Law, shown in equation (309):

(314)$\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf{e}) = - \boldsymbol{\nabla} \times \frac{\partial \mathbf{b}}{\partial t}$

The appropriate constitutive relations can be substituted into Equation (314) to get the following expressions in terms of only the fields $$\mathbf{e}$$ and $$\mathbf{h}$$ instead of fields and fluxes:

(315)$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{e} = - \boldsymbol{\nabla} \times \left ( \frac{\partial}{\partial t} (\mu \mathbf{h}) \right )$

Assuming the physical properties are homogeneous throughout the domain, $$\mu$$, $$\epsilon$$, and $$\sigma$$ can be moved out front of the derivative terms. This simplifies the above expressions:

(316)$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{e} = - \mu \boldsymbol{\nabla} \times \frac{\partial \mathbf{h}}{\partial t}$

If we further assume that we can take the first and second derivatives of $$\mathbf{e}$$, we can either take the spatial derivatives first or the time derivatives. Equation (316) can then also be written as:

(317)$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{e} = - \mu \frac{\partial}{\partial t} \left ( \boldsymbol{\nabla} \times \mathbf{h} \right )$

This expression is now solely in terms of $$\boldsymbol{\nabla} \times \mathbf{e}$$ and $$\boldsymbol{\nabla} \times \mathbf{h}$$. Thus, we can use Equation (310) to generate an equation with only $$\mathbf{e}$$. We substitute in Equation (310) into Equation (317) and simplify using the constitutive relations in Equations (103) and (312):

$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{e} = - \mu \frac{\partial}{\partial t} \left ( \mathbf{j} + \frac{\partial \mathbf{d}}{\partial t} \right )$
$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{e} = - \mu \frac{\partial}{\partial t} \left ( \sigma \mathbf{e} + \frac{\partial (\epsilon \mathbf{e})}{\partial t} \right )$
(318)$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{e} = - \mu \sigma \frac{\partial \mathbf{e}}{\partial t} - \mu \epsilon \frac{\partial^2 \mathbf{e}}{\partial t^2}$

Additionally, we can simplify the first term of this expression by using the vector identity $$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{x} = \boldsymbol{\nabla} \boldsymbol{\nabla} \cdot \mathbf{x} - \boldsymbol{\nabla}^2 \mathbf{x}$$. Recalling that both $$\boldsymbol{\nabla} \cdot \mathbf{e}$$ and $$\boldsymbol{\nabla} \cdot \mathbf{h}$$ are zero in a homogenous space, the vector identity simply becomes $$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{x} = - \boldsymbol{\nabla}^2 \mathbf{x}$$. If we now substitute that into (318), we get the following expression:

(319)$\boldsymbol{\nabla}^2 \mathbf{e} - \mu \epsilon \frac{\partial^2 \mathbf{e}}{\partial t^2} - \mu \sigma \frac{\partial \mathbf{e}}{\partial t} = 0$

This is the wave equation for the electric field in the time domain.

## Derivation for the Magnetic Field

To derive the wave equation for $$\mathbf{h}$$, we repeat the above derivation but start by taking the curl of Ampere’s Law, shown in equation (310):

(320)$\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf{h}) = \boldsymbol{\nabla} \times \mathbf{j} + \boldsymbol{\nabla} \times \frac{\partial \mathbf{d}}{\partial t}$

The constitutive relations can be substituted into Equation (320) to get the following expressions in terms of only $$\mathbf{e}$$ and $$\mathbf{h}$$:

(321)$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{h} = \boldsymbol{\nabla} \times (\sigma \mathbf{e}) + \boldsymbol{\nabla} \times \left ( \frac{\partial}{\partial t} (\epsilon \mathbf{e}) \right )$

We simplify the expression just like we did before for the electric field.

(322)$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{h} = \sigma \boldsymbol{\nabla} \times \mathbf{e} + \epsilon \boldsymbol{\nabla} \times \frac{\partial \mathbf{e}}{\partial t}$

We can assume that we can take the first and second derivatives of $$\mathbf{e}$$ and $$\mathbf{h}$$ and can either take the spatial derivatives or time derivatives first. Equation (322) can then also be written as:

(323)$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{h} = \sigma \boldsymbol{\nabla} \times \mathbf{e} + \epsilon \frac{\partial}{\partial t} \left ( \boldsymbol{\nabla} \times \mathbf{e} \right )$

These expressions are now in terms of $$\boldsymbol{\nabla} \times \mathbf{e}$$ and $$\boldsymbol{\nabla} \times \mathbf{h}$$. Thus, we can use Equation (309) to generate an equation with only $$\mathbf{h}$$. We then again use the vector identity $$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{x} = \boldsymbol{\nabla} \boldsymbol{\nabla} \cdot \mathbf{x} - \boldsymbol{\nabla}^2 \mathbf{x}$$ and the fact that $$\boldsymbol{\nabla} \cdot \mathbf{h}$$ is zero in a homogenous space to simplify the vector identity to $$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{x} = - \boldsymbol{\nabla}^2 \mathbf{x}$$. This is then substituted into the wave equation. The following shows these derivations.

$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{h} = - \sigma \frac{\partial \mathbf{b}}{\partial t} - \epsilon \frac{\partial}{\partial t} \left (\frac{\partial \mathbf{b}}{\partial t} \right )$
$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{h} = - \sigma \frac{\partial (\mu \mathbf{h}) }{\partial t} - \epsilon \frac{\partial}{\partial t} \left (\frac{\partial (\mu \mathbf{h})}{\partial t} \right )$
$\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{h} = - \sigma \mu \frac{\partial \mathbf{h}}{\partial t} - \epsilon \mu \frac{\partial^2 \mathbf{h}}{\partial t^2}$
$- \boldsymbol{\nabla}^2 \mathbf{h} = - \sigma \mu \frac{\partial \mathbf{h}}{\partial t} - \epsilon \mu \frac{\partial^2 \mathbf{h}}{\partial t^2}$
(324)$\boldsymbol{\nabla}^2 \mathbf{h} - \epsilon \mu \frac{\partial^2 \mathbf{h}}{\partial t^2} - \sigma \mu \frac{\partial \mathbf{h}}{\partial t} = 0$

Equation (324) is then the wave equation for the magnetic field in the time domain.