Let us now pause to explain in more detail why we called the elementary solutions (9.9) and (9.26) plane waves. The reason is that the displacement q(r, t) has the symmetry of a plane. To see this, fix a time t (take a “snapshot” of the wave) and pick a location r. Examine the wave displacement q (at the fixed time) at all points in a plane that is (i) perpendicular to k, and (2) passes through r. The wave displacement will be the same at each point of this plane.
Plane waves, displacement, chapter ten
Torre, Charles G., "10 Why "Plane" Waves?" (2012). Foundations of Wave Phenomena. Book 13.