Notice how the scalar and vector parts have neatly partitioned themselves. This is a wave equation, except that a sign is flipped. Here is the equation for a longitudinal wave like sound. Thus every value of n will satisfy the longitudinal wave equation. One of the signs must be switched by doing some operation twice.
V acts like an imaginary phase factor that rotates the spatial component.
That is a quantized, transverse wave. That's fortunate, because light is a quantized transverse wave. The equations were generated by taking the classical Maxwell equations, and making them simpler.