Deriving the wave equation using small perturbations

[JD_PM]

Homework Statement

Given:

$$\nabla \times \mathbf H = \frac{4\pi}{c} \mathbf j$$



$$\nabla \times \mathbf E = -\frac{1}{c} \frac{\partial \mathbf H}{\partial t}$$



$$\mathbf j = \sigma(\mathbf E + \frac{\mathbf v}{c} \times \mathbf H)$$



$$\rho \frac{\partial \mathbf v}{\partial t} = \frac{1}{c}(\mathbf j \times \mathbf H) - \nabla p$$


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 $H$

Relevant Equations

Please see Homework statement

Note that the wave equation we want to derive was introduced by Alfven in his 1942 paper (please see bottom link to check it out), but he did not include details on how to derive it. That's what we want to do next.

Alright, writing the above equations we assumed that:

$$\mu = 1 \ \ \ ; \ \ \ \mathbf H = \mathbf B$$

Let me now make new assumptions:

1) $$\sigma \rightarrow \infty$$

2) The pressure contribution is negligible.

Then our set of equations reduces to:

$$\nabla \times \mathbf H = \frac{4\pi}{c} \mathbf j$$
$$\nabla \times \mathbf E = -\frac{1}{c} \frac{\partial \mathbf H}{\partial t}$$
$$\mathbf E = -\frac{\mathbf v}{c} \times \mathbf H$$
$$\rho \frac{\partial \mathbf v}{\partial t} = \frac{1}{c}(\mathbf j \times \mathbf H)$$

So at this point, my approach was to apply the curl of the curl method (please see bottom link for more details) to get the wave equation. But after more reading, one sees this is not the right method. We instead have to introduce perturbations

Let's assume that initially we have a uniform system where there is no velocity, no current and no electric field.

We assume that the magnetic field is uniform and equal to:

$$\mathbf B = H_0 \hat z$$

Note that all these assumptions were made by Alfven

We now introduce a perturbation of the magnetic field, the electric field, the current and the velocity:

$$\mathbf H = H_0 \hat z + H' \hat x$$

$$\mathbf E = + E' \hat y$$

$$\mathbf j = + j' \hat y$$

$$\mathbf v = + v' \hat x$$

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 once here how can I get $\frac{\partial ^2 \mathbf H'}{\partial z^2} = \frac{4\pi\rho}{H_0^2}\frac{\partial ^2 \mathbf H'}{\partial t^2}$ ?

Note: Any help is appreciated. I've been stuck in this derivation since November. I asked this question on MSE but got little attention. You also can find the paper on that post:

https://math.stackexchange.com/ques...tion-out-of-nabla-times-vec-h-frac4-pic-vec-j

 

[JD_PM]

Bit by bit we're moving forward! :)

For anyone interested: https://math.stackexchange.com/ques...-equation?noredirect=1#comment7158100_3480896

EDIT: Finished :)