Consider the standard relativity configuration. Let a spherical light wave be emitted from the coincident coordinate origins at t=0 and t′=0. For the rest frame, the spherical wave front is given by
x² + y² + z² = c²t².
For the frame moving at velocity v parallel to the x-axis, two-way light must be considered. In Newtonian mechanics, the light wave front going out is given by
(x′ + vt′)² + y′² + z′² = (c + v)²t′²,
which expanded equals
x′² + 2x′t′ + vt′² + y′² + z′² = (c² + 2cv + v²)t′².
In Newtonian mechanics, the returning light wave front is given by
(x′ − vt′)² + y′² + z′² = (c − v)²t′²,
which expanded equals
x′² − 2x′t′ + v²t′² + y′² + z′² = (c² − 2cv + v²)t′².
Add these two expanded equations and divide by two for their average:
x′² + v²t′² + y′² + z′² = (c² + v²)t′²,
which reduces to
x′² + y′² + z′² = c²t′².
This demonstrates the invariance of a wave front in Newtonian mechanics.