electromagnetic-dynamics: add waves

docs/physics/electromagnetism/electromagnetic-dynamics.md

@@ -221,6 +221,47 @@ then it follows from Newton's second law that:
 
     Should be rewritten/reconsidered in Lagrangian/Hamiltonian formalism.
 
+### Waves
+
+We may decouple the posed equations in *Axiom 1*, obtaining second order equations describing the electric and magnetic field.
+
+> *Theorem 5*: In regions of space where $\rho = 0$ and $\mathbf{J} = \mathbf{0}$, the electric $\mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t)$ and magnetic field $\mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t)$ adhere to
+>
+> $$
+>   \begin{align*}
+>       \nabla^2 \mathbf{E} &= \frac{1}{c^2} \partial_t^2 \mathbf{E},\\
+>       \nabla^2 \mathbf{B} &= \frac{1}{c^2} \partial_t^2 \mathbf{B},
+>   \end{align*}
+> $$
+>
+> with a wave speed $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.
+
+??? note "Proof:"
+    
+    For the electric field $\mathbf{E}$ in a region for which $\rho=0$ and $\mathbf{J}=\mathbf{0}$ we have
+
+    $$
+        \begin{align*}
+            \nabla^2 \mathbf{E} &= - \nabla \times (\nabla \times \mathbf{E}),\\
+            &= \nabla \times \partial_t \mathbf{B},\\
+            &= \partial_t (\nabla \times \mathbf{B}),\\
+            &= \mu_0 \epsilon_0 \partial^2_t \mathbf{E}.
+        \end{align*}
+    $$
+    
+    For the magnetic field $\mathbf{B}$ in a region for which $\rho=0$ and $\mathbf{J}=\mathbf{0}$ we have
+
+    $$
+        \begin{align*}
+            \nabla^2 \mathbf{B} &= - \nabla \times (\nabla \times \mathbf{B}),\\
+            &= -\nabla \times (\mu_0 \epsilon_0 \partial_t \mathbf{E}),\\
+            &= -\mu_0 \epsilon_0 \partial_t (\nabla \times \mathbf{E}),\\
+            &= \mu_0 \epsilon_0 \partial_t^2 \mathbf{B}.
+        \end{align*}
+    $$
+
+The second order equations posed in *Theorem 5* satisfy exactly the three-dimensional wave equation. Thus, vacuum supports the propagation of electromagnetic waves traveling at a constant speed $c$. In my opinion one of the most mesmerizing results of the theory of electromagnetism. 
+
 ## Electromagnetic dynamics in matter
 
 Let $\mathbf{J}_p = \partial_t \mathbf{P}$ denote the polarisation volume current density such that
@@ -236,7 +277,7 @@ $$
 
 complying to the conservation of charge.
 
-> *Theorem 5*: The electric field $\mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t)$, electric displacement $\mathbf{D}: (\mathbf{r},t) \mapsto \mathbf{D}(\mathbf{r},t)$, magnetic field $\mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t)$ and auxiliary field $\mathbf{H}: (\mathbf{r},t) \mapsto \mathbf{H}(\mathbf{r},t)$ in a medium are described by
+> *Theorem 6*: The electric field $\mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t)$, electric displacement $\mathbf{D}: (\mathbf{r},t) \mapsto \mathbf{D}(\mathbf{r},t)$, magnetic field $\mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t)$ and auxiliary field $\mathbf{H}: (\mathbf{r},t) \mapsto \mathbf{H}(\mathbf{r},t)$ in a medium are described by
 >
 > $$
 >   \begin{align*}
@@ -272,3 +313,58 @@ complying to the conservation of charge.
     or $\nabla \times \mathbf{H} = \mathbf{J}_f + \partial_t \mathbf{D}$.
 
 Which require constitutive relations for closure.
+
+### Waves
+
+The posed equations in *Theorem 6* may de decoupled as well, obtaining, as expected, three-dimensional wave equations describing the electric and magnetic field.
+
+> *Theorem 7*: In a linear, homogeneous, dielectric medium where $\rho_f=0$ and $\mathbf{J}_f=\mathbf{0}$, the electric displacement $\mathbf{D}: (\mathbf{r},t) \mapsto \mathbf{D}(\mathbf{r},t)$ and the auxiliary field $\mathbf{H}: (\mathbf{r},t) \mapsto \mathbf{H}(\mathbf{r},t)$ adhere to
+>
+> $$
+>   \begin{align*}
+>       \nabla^2 \mathbf{D} &= \frac{1}{v^2} \partial_t^2 \mathbf{D},\\
+>       \nabla^2 \mathbf{H} &= \frac{1}{v^2} \partial_t^2 \mathbf{H},
+>   \end{align*}
+> $$
+>
+> with a wave speed $v = \frac{1}{\sqrt{\mu \epsilon}}$. 
+
+??? note "Proof:"
+
+    For the electric displacement in a linear, homogeneous, dielectric medium where $\rho_f=0$ and $\mathbf{J}_f=\mathbf{0}$ we have
+
+    $$
+        \begin{align*}
+            \nabla^2 \mathbf{D} &= \epsilon \nabla^2 \mathbf{E},\\
+            &= - \epsilon \nabla \times (\nabla \times \mathbf{E}),\\
+            &= \epsilon \nabla \times \partial_t \mathbf{B},\\
+            &= \epsilon \partial_t (\nabla \times \mathbf{B}),\\
+            &= \mu \epsilon \partial_t (\nabla \times \mathbf{H},\\
+            &= \mu \epsilon \partial_t^2 \mathbf{D}.
+        \end{align*}
+    $$
+
+    For the auxiliary field in a linear, homogeneous, dielectric medium where $\rho_f=0$ and $\mathbf{J}_f=\mathbf{0}$ we have
+
+    $$
+        \begin{align*}
+            \nabla^2 \mathbf{H} &= \nabla^2 \mathbf{H},\\
+            &= - \nabla \times (\nabla \times \mathbf{H}),\\
+            &= - \nabla \times \partial_t \mathbf{D},\\
+            &= - \partial_t (\nabla \times \mathbf{D}),\\
+            &= - \epsilon \partial_t (\nabla \times \mathbf{E}),\\
+            &= \epsilon \partial_t^2 \mathbf{B},\\
+            &= \mu \epsilon \partial_t^2 \mathbf{H}.
+        \end{align*}
+    $$
+
+Equivalently we may write *Theorem 7* in terms of the electric and magnetic fields in the linear, homogeneous, dielectric medium where $\rho_f = 0$ and $\mathbf{J}_f = \mathbf{0}$
+
+$$
+    \begin{align*}
+        \nabla^2 \mathbf{E} = \frac{1}{v^2} \partial_t^2 \mathbf{E},\\
+        \nabla^2 \mathbf{B} = \frac{1}{v^2} \partial_t^2 \mathbf{B},
+    \end{align*}
+$$
+
+since $\mathbf{D} = \epsilon \mathbf{E}$ and $\mathbf{B} = \mu \mathbf{H}$.