What does it mean to decouple Maxwell's equations with the curl operator?

Question

I'm reading Introduction to Electrodynamics 4th Edition by D.J. Griffiths where after listing the Maxwell equations in empty space (i.e. $\rho = 0, \mathbf{J} = 0, \mu = \mu_0, \epsilon = \epsilon_o$) he says on p.g. 393:

[The equations] constitute a set of coupled, first-order, partial differential equations for E and B. They can be decoupled by applying the curl to (iii) and (iv):

where equations (iii) and (iv) are:

$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$

$$\nabla \times \mathbf{B} = - \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

Griffiths then goes on to apply the curl operator to both functions and arrives to the wave equation for electromagnetic waves.

I don't understand why we use the curl operator. It seems arbitrary to me without understanding what Griffiths means by coupled PDEs and decoupling them by using the curl operator. I know that the two equations rely on each other; one field affects the other and that is reflected in the equations, maybe that is what is meant by coupled.

My questions is made of two parts:

1. What does it mean for equations to be coupled, and why must we decouple them?

2. Why does using the curl operator decouple them?

Answer 1

Coupling means simply that both equations involve both fields, so that you can't solve them separately.

Now, both equations involve a relation between the curl of one field and the other field. So if we take the first equation and apply curl, the right hand side will contain $\nabla \times \mathbf{B}$, which we can write in terms of $\mathbf{E}$ and get an equation written only in terms of the electric field: that's what decoupling means.

The curl only complicates a simple idea: if we have a pair of first order equations

$$\begin{align}\dot{f} &= a g \\ \dot{g} &= bf,\end{align}$$

how do we decouple them? Just take derivatives! After a bit of algebra, that will turn them into the two second order equations

$$\begin{align}\ddot{f} &= ab f \\ \ddot{g} &= ab g.\end{align}$$

As a final note, the curl is not the only option: you can also take time derivatives, go through the same steps, and arrive at the same result.