Given:
$$\nabla \times \mathbf H = \frac{4\pi}{c} \mathbf j \ \ \ \ \ \ \ \ \ (1)$$
$$\nabla \times \mathbf E = -\frac{1}{c} \frac{\partial \mathbf H}{\partial t} \ \ \ (2)$$
$$\mathbf E = -\frac{\mathbf v}{c} \times \mathbf H \ \ \ \ \ \ \ \ \ (3)$$
$$\rho \frac{\partial \mathbf v}{\partial t} = \frac{1}{c}(\mathbf j \times \mathbf H) \ \ (4)$$
Derive the following PDE (1D Wave Equation):
$$\frac{\partial ^2 \mathbf H'}{\partial z^2} = \frac{4\pi\rho}{H_0^2}\frac{\partial ^2 \mathbf H'}{\partial t^2}$$
Where $H'$ means a small perturbation of $H$.
We assume that the magnetic field is uniform and equal to:
$$\mathbf B = H_0 \hat z$$
Alright, so let's introduce the following perturbations:
$$\mathbf H = H_0 \hat z + H' \hat x$$
$$\mathbf E = + E' \hat y$$
$$\mathbf j = + j' \hat y$$
$$\mathbf v = + v' \hat x$$
While the density is assumed to remain constant.
All the perturbed quantities are assumed to be very small. Thus, the product of each other is negligible.
But how can we derive the stated wave equation out of the above equations?
EDIT:
Alright, so let's write down all equations.
- Combining $(1)$ and $(4)$ we get:
$$4\pi \rho \frac{\partial {\mathbf v}}{\partial t} = (\nabla\times \mathbf H)\times \mathbf H$$
Inserting perturbation equations into it and working out the RHS we get (let me know if you want me to include the steps):
$$4 \pi \rho \frac{\partial}{\partial t} v' \hat x = H_0 \Big(\frac{\partial H'}{\partial z} - \frac{\partial H_0}{\partial x}\Big) \hat x -\Big(H' \frac{\partial H'}{\partial y} + H_0\frac{\partial H_0}{\partial y}\Big) \hat y -H' \Big(\frac{\partial H'}{\partial z} - \frac{\partial H_0}{\partial x}\Big) \hat z$$
By equating components we get:
$$4 \pi \rho \frac{\partial}{\partial t} v' = H_0 \Big(\frac{\partial H'}{\partial z} - \frac{\partial H_0}{\partial x}\Big) \ \ \ (I)$$
$$H' \frac{\partial H'}{\partial y} = -H_0\frac{\partial H_0}{\partial y} \ \ \ (II)$$
$$\frac{\partial H'}{\partial z} = \frac{\partial H_0}{\partial x} \ \ \ (III)$$
- Inserting perturbation equations into equation $(2)$ and working out the LHS we get:
$$\frac{\partial E'}{\partial x} \hat z - \frac{\partial E'}{\partial z} \hat x = -\frac{\partial}{c\partial t} \Big(H_0 \hat z + H' \hat x \Big) = -\frac{\partial}{c\partial t} H' \hat x$$
By equating components we get:
$$\frac{\partial E'}{\partial x} = -\frac{\partial}{c\partial t} H_0 = 0 \ \ \ (IV)$$
$$\frac{\partial E'}{\partial z} = \frac{\partial}{c\partial t} H' \ \ \ (V)$$
- Inserting perturbation equations into equation $(3)$ and working out the RHS we get:
$$E' \hat y = \frac{1}{c} v' H_0 \hat y$$
By equating components we (of course) get:
$$E' = \frac{1}{c} v' H_0 \ \ \ (VI)$$
OK! So I just have to work out the above six equations to get the desired wave equation...