Derive the wave equation for fields E, B from the potentials

[phenomenologic]

I'm studying for my electrodynamics exam and one of the past exam questions is:

From the scalar and vector potentials, derive the homogenous wave equations for E and B fields in vacuum.

I did derive the wave equation for the B field by simply taking the curl of the homogenous wave equation for the vector potential A. But I got stuck deriving the wave equation for the E field. I think it has to come from a tricky combination of the following four equations:
1) Definition of the E field E = -∇Φ - ∂A/dt
2&3) Wave equations for Φ and A
4) Lorenz gauge condition: ∇.A - (1/c^2)(∂^2Φ/∂t^2) = 0

Any help is appreciated!

 

[TSny]

If Φ satisfies the wave equation, can you show that -∇Φ also satisfies the wave equation?

 

[phenomenologic]

Hmm, so I write -∇(ΔΦ-(1/c^2)(∂^2Φ/∂t^2))=Δ(-∇Φ)-(1/c^2)(∂^2(-∇Φ)/∂t^2)=0. And this operation (i.e, taking nabla 'inside' the Laplacian and the partial time derivative) is allowed, right? There are no restrictions on it as far as I know but I'm not 100% sure.

Then I perform the same trick for -∂A/∂t. Similarly, I take the partial derivative inside (again, assuming it's allowed), showing the equation is satisfied for -∂A/∂t. Then add up the new equations, use formula 1) above and done?

 

[TSny]

Yes, that's it. We make the usual assumption in physics that our functions are well-behaved enough to allow reordering of partial derivatives. You can search the web for information on the sufficient conditions to be "well-behaved". For example https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Schwarz.27s_theorem

 

[phenomenologic]

That was what I wasn't 100% sure. Thanks!