Wave equation in a magnetic field.

Question

$\nabla\times\nabla\times B=\nabla(\nabla B)-\nabla(\nabla B)=-\Delta B$ since $\nabla B=0$

That seems to be odd, since one might expect $=0$ instead of $=-\Delta B$

Answer 1

You have your Vector Identity messed up, it should be:

$\nabla \times \left( \nabla \times \mathbf{B} \right) = \nabla(\nabla \cdot \mathbf{B}) - \nabla^{2}\mathbf{B}$, where $\nabla^{2}=\Delta$ is the Vector Laplacian.

And we know from Maxwell's equations that $\nabla \cdot \mathbf{B} = 0$, and so we are left with $$\nabla \times \left( \nabla \times \mathbf{B} \right) = -\nabla^{2}\mathbf{B}$$

What happens from this point depends on how your vector field $\mathbf{B}$ is behaving.