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.