Wave equation is $$\nabla ^2 \bf{E}=\mu \epsilon \frac{\partial^2 E}{\partial t^2}\tag{1}$$

How can I prove that, given a solution ##\bf{E}## that satisfies both Maxwell equations and ##(1)##, then the vector ##\bf{E+E'}## where ##\bf{}{}E'## is a vector such that ##\nabla \times \bf{}{}E'=0## (and ##\nabla \cdot E \bf{}##) is a solution of ##(1)##?

Can anyone provide a proof of the fact that a non divergence free (but irrotational) vector can be solution of wave equation without being solution of maxwell equation?