How does the Lorenz Gauge condition lead to four wave equations?

Question

The 1972 book by L. Eyges's, The Classical Electromagnetic Field, on p. 184, in $\S$11.7, Integral Forms of The Potential, the statement

"We now turn to the problem of finding $\mathbf{A}$ and $\mathbf{\Phi}$ in terms of $\mathbf{J}$ and $\rho$. For this purpose, the Lorenz gauge is the more convenient one. In this gauge we have four equations in (11.33)."

appears. Equation 11.33 is stated on p. 182 as

$$ \nabla^2 \mathbf{A}- \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = - \frac{4 \pi \mathbf{J}}{c}, \\ \nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = - 4 \pi \rho $$

Why does the author claim that this is four equations when only two are clearly written?

Answer 1

The Equation,

$$ \nabla^2 \mathbf{A}- \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = - \frac{4 \pi \mathbf{J}}{c} $$ ,

is actually three seperate equations in three dimensions. In Cartesian coordinates this Equation expands to,

$$\nabla^2 A_x- \frac{1}{c^2} \frac{\partial^2 A_x}{\partial t^2} = - \frac{4 \pi J_x}{c}, \\ \nabla^2 A_y- \frac{1}{c^2} \frac{\partial^2 A_y}{\partial t^2} = - \frac{4 \pi J_y}{c}, \\ \nabla^2 A_z- \frac{1}{c^2} \frac{\partial^2 A_z}{\partial t^2} = - \frac{4 \pi J_z}{c}. $$

Therefore, given these three equations, and the equation for $\phi$, there are four total equations.