To examine running horses it may be best to jump on a horse. Likewise, to examine moving waves, it may be better to move along with them. An alternative to the moving coordinate system is to define retarded coordinates where .The classical example of retarded coordinates is solar time. Time seems to stand still on an airplane that moves westward at the speed of the sun. The migration process resembles the simulation of wave propagation in either a moving coordinate frame or a retarded coordinate frame.
Retarded coordinates are much more popular than moving coordinates.
In a moving coordinate system the velocity could depend on all three variables, thus unnecessarily increasing the complexity of the calculations.