Derivation of the wave equation

The following is based on the “Derivation of the Wave Equation in Time” here with Faraday’s and Ampere-Maxwell’s laws completed for three dimensions of duration. With electric field e, electric displacement d, magnetic induction b, magnetic intensity h, current density j, length coordinates x, and duration coordinates z, these are as follows:

$\mathbf{\nabla} \times \mathbf{e}= -\frac{\partial b_1}{\partial z_1} -\frac{\partial b_2}{\partial z_2} -\frac{\partial b_3}{\partial z_3} =-\frac{\partial }{\partial \mathbf{z}} \cdot \mathbf{b} =-\frac{\partial \mathbf{b}}{\partial \mathbf{z}} \; \; \; (309a)$

and

$\mathbf{\nabla} \times \mathbf{h}=\mathbf{j}+\frac{\partial d_1}{\partial z_1} +\frac{\partial d_2}{\partial z_2}+\frac{\partial d_3}{\partial z_3}= \mathbf{j}+\frac{\partial }{\partial \mathbf{z}} \cdot \mathbf{d} = \mathbf{j}+\frac{\partial \mathbf{d}}{\partial \mathbf{z}} \; \; \; (310a)$

where the dot is understood as a formal dot product, similar to the del as a formal cross product:

$\mathbf{\nabla} \times \mathbf{e} = \frac{\partial }{\partial \mathbf{x}} \times \mathbf{e}$

Note: to recover the original laws set z₁ = z₂ = z₃ = t.

The three constitutive relations are:

$\mathbf{j}=\sigma \mathbf{e} \; \; \; (311a)$

$\mathbf{d}=\epsilon \mathbf{e}\; \; \; (312a)$

$\mathbf{b}=\mu \mathbf{h}\; \; \; (313a)$

with conductivity σ = 1/ρ, permittivity ε, and permeability μ.

Derivation for the Electric Field

To derive the wave equation for e, we first take the curl of Faraday’s Law shown in equation (309a):

$\mathbf{\nabla} \times (\mathbf{\nabla} \times \mathbf{e})=-\mathbf{\nabla} \times \left (\frac{\partial b_1}{\partial z_1} +\frac{\partial b_2}{\partial z_2} +\frac{\partial b_3}{\partial z_3} \right )=-\nabla \times \frac{\partial \mathbf{b}}{\partial \mathbf{z}} \; \; \; (314a)$

The appropriate constitutive relations can be substituted into equation (314a) to get the following expression in terms of only the fields e and h:

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{e}=-\mathbf{\nabla} \times \left (\frac{\partial (\mu h_1)}{\partial z_1} +\frac{\partial (\mu h_2)}{\partial z_2} +\frac{\partial (\mu h_3)}{\partial z_3} \right )=-\nabla \times \left (\frac{\partial }{\partial \mathbf{z}} (\mu \mathbf{h})\right ) \; \; \; (315a)$

Assuming the physical properties are homogeneous throughout the domain, μ, ϵ, and σ can be moved out front of the derivative terms. This simplifies the above expressions:

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{e}=-\mu \mathbf{\nabla} \times \left (\frac{\partial h_1}{\partial z_1} +\frac{\partial h_2}{\partial z_2} +\frac{\partial h_3}{\partial z_3} \right )=-\mu \nabla \times \frac{\partial \mathbf{h}}{\partial \mathbf{z}} \; \; \; (316a)$

If we further assume that we can take the first and second derivatives of e, we can either take the length space derivatives first or the duration space derivatives first. Equation (316a) can then be written as:

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{e}=-\mu \frac{\partial }{\partial \mathbf{z}} (\mathbf{\nabla} \times \mathbf{h}) \; \; \; (317a)$

This expression is now solely in terms of ∇ × e and ∇ × h. Thus, we can use equation (310a) to generate an equation with only e. We substitute equation (310a) into equation (317a) and simplify using the constitutive relations:

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{e} = -\mu \frac{\partial }{\partial \mathbf{z}} \left (\mathbf{j} +\frac{\partial \mathbf{d}}{\partial \mathbf{z}} \right )$

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{e} = -\mu \frac{\partial }{\partial \mathbf{z}} \left (\sigma \mathbf{e} +\frac{\partial \epsilon \mathbf{e}}{\partial \mathbf{z}} \right )$

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{e}=-\mu \sigma \frac{\partial \mathbf{e}}{\partial \mathbf{z}} -\mu \epsilon \frac{\partial^2 \mathbf{e}}{\partial \mathbf{z}^2}\; \; \; (318a)$

Additionally, we can simplify the first term of this expression by using the vector identity

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{r}= \mathbf{\nabla}\mathbf{\nabla}\cdot \mathbf{r}-\mathbf{\nabla}^2 \mathbf{r}.$

Recalling that both ∇ · e and ∇ · h are zero in a homogenous space, the vector identity simply becomes

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{r}= -\mathbf{\nabla}^2 \mathbf{r}.$

If we now substitute that into (318a), we get the following expression:

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

This is the wave equation for the electric field in length-duration space. If the charge is zero, then σ = 0, and the wave equation is

$\mathbf{\nabla}^2 \mathbf{e} -\frac{1}{c^2} \frac{\partial^2 \mathbf{e}}{\partial \mathbf{z}^2} =0 \; \; \; (319a)$

Derivation for the Magnetic Field

To derive the wave equation for h, we repeat the above derivati on but start by taking the curl of Ampere’s Law, shown in equation (310a):

$\mathbf{\nabla} \times (\mathbf{\nabla} \times \mathbf{h}) = \mathbf{\nabla} \times \mathbf{j} +\mathbf{\nabla} \times \frac{\partial \mathbf{d}}{\partial \mathbf{z}} \; \; \; (320a)$

The constitutive relations can be substituted into equation (320a) to get the following expressions in terms of only e and h:

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{h} = \mathbf{\nabla} \times (\mathbf{\sigma \mathbf{e}}) +\mathbf{\nabla} \times \frac{\partial }{\partial \mathbf{z}} (\epsilon \mathbf{e})\; \; \; (321a)$

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

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{h} = \sigma \mathbf{\nabla} \times \mathbf{\mathbf{e}} + \epsilon \mathbf{\nabla} \times \frac{\partial \mathbf{e}}{\partial \mathbf{z}} \; \; \; (322a)$

We can assume that we can take the first and second derivatives of e and h and can either take the spatial derivatives or duration derivatives first. Equation (322a) can then also be written as:

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{h} = \sigma \mathbf{\nabla} \times \mathbf{\mathbf{e}} + \epsilon \frac{\partial }{\partial \mathbf{z}} (\mathbf{\nabla} \times \mathbf{e}) \; \; \; (323a)$

These expressions are now in terms of ∇ × e and ∇ × h. Thus, we can use equation (309a) to generate an equation with only h. We then again use the vector identity

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{r}= \mathbf{\nabla}\mathbf{\nabla}\cdot \mathbf{r}-\mathbf{\nabla}^2 \mathbf{r}.$

and the fact that ∇ · h is zero in a homogenous space to simplify the vector identity to

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{r}= -\mathbf{\nabla}^2 \mathbf{r}.$

This is then substituted into the wave equation. The following shows these derivations.

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{h} = \sigma \frac{\partial \mathbf{b}}{\partial \mathbf{z}} +\epsilon \frac{\partial }{\partial \mathbf{z}} \left (\frac{\partial \mathbf{b}}{\partial \mathbf{z}} \right )$

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{h} = \sigma \frac{\partial (\mu \mathbf{h})}{\partial \mathbf{z}} +\epsilon \frac{\partial }{\partial \mathbf{z}} \left (\frac{\partial (\mu \mathbf{h})}{\partial \mathbf{z}} \right )$

$\mathbf{\nabla} \times \mathbf{\nabla} \times \mathbf{h} = \sigma \mu \frac{\partial \mathbf{h}}{\partial \mathbf{z}} +\epsilon \mu \frac{\partial^2 \mathbf{h}}{\partial \mathbf{z}^2}$

$-\mathbf{\nabla}^2 \mathbf{h} = \sigma \mu \frac{\partial \mathbf{h}}{\partial \mathbf{z}} +\epsilon \mu \frac{\partial^2 \mathbf{h}}{\partial \mathbf{z}^2}$

$-\mathbf{\nabla}^2 \mathbf{h} -\epsilon \mu \frac{\partial^2 \mathbf{h}}{\partial \mathbf{z}^2} -\sigma \mu \frac{\partial \mathbf{h}}{\partial \mathbf{z}}=0\; \; \; (324a)$

Equation (324a) is then the wave equation for the magnetic field in length-duration space. If the current zero, then σ = 0, and the wave equation is

$-\mathbf{\nabla}^2 \mathbf{h} -\frac{1}{c^2} \frac{\partial^2 \mathbf{h}}{\partial \mathbf{z}^2} =0\; \; \; (325a)$

Related