I'm reading "Antennas and Wave Propagation" by Harish and Sachidananda. They introduce the phasor form of Maxwell's equations:
$$\nabla\times E=-j\omega\mu H$$ $$\nabla\times H = j\omega\epsilon E + J$$ $$\nabla\cdot D = \rho$$ $$\nabla\cdot B = 0$$
First of all they observe that by taking the curl of the first equation and rewriting the right hand side with the expression for $\nabla\times H$, we can show that $E$ satisfies a wave equation: $\nabla^2E+aE=bJ$, for some constants $a$ and $b$. They also note that in the same way we can show that $H$ satisfies the wave equation.
Then they introduce the notion of vector and scalar potential, and propose to find $V$, the scalar potential of $E$, and $A$, the vector potential of $H$. The motivation for doing this seems to be that finding $A$ and $V$ is easier because they both satisfy... the wave equation.
Why solve the wave equation to get the potentials, and then work out $E$ and $H$ from there? Why not just solve the wave equations for $E$ and $H$ directly?