The following is based on section 3.3.2 of *Electricity and Magnetism for Mathematicians* by Thomas A. Garrity (Cambridge UP, 2015). See also blog post *Relative Motion and Waves* by Conrad Schiff.

The classical wave equation is consistent with the Galilean transformation. Reflected electromagnetic waves are also consistent with classical physics using the dual Galilean transformation, in which linear location is the independent variable. The dual Galilean transformation for motion in one dimension is: *x′* = *x*; *t′* = *t* − *wx* = *t* − *x*/*v*, where *w* = 1/*v* is the relative pace of the moving observer, which is equivalent to their inverse speed.

Suppose we again have two observers, *A* and *B*. Let observer *B* be moving at a constant pace *w* with respect to *A*, with *A*’s and *B*’s coordinate systems exactly matching up at location *x* = 0. Think of observer *A* as at rest, with coordinates *x′* for location and *t′* for time, and of observer *B* as moving to the right at pace *w*, with location coordinate *x* and time coordinate *t*. If the two coordinate systems line up at location *x* = *x′* = 0, then the dual Galilean transformations are

*x′* = *x* and *t′* = *t* + *wx*,

or equivalently,

*x* = *x′* and *t* = *t′* − *wx*.

Suppose in the reference frame for *B* we have a wave *y*(*x, t*) satisfying the wave equation