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A plane wave (or plane parallel wave) is a theoretical wave such that the motions/changes occurring are the same across an entire mathematical plane, and the wave is traveling perpendicular to the plane. Contrast this to a sound wave coming from an object a few feet in front of you: each wave peak forms an expanding sphere in all directions from the object, i.e., it is curved. But a sound wave reaching your vicinity from a long distance is nearly a plane wave, showing very little such curvature.
A plane wave is mathematically simpler than waves diverging from something, e.g., as expanding spheres, but a sub-region consisting of a small fraction of such an expanding sphere approximates a plane, and as such a wave progresses, using a plane wave as an approximation of the wave becomes more and more useful. Thus, a wave from a distant source (e.g., light waves from stars) is often modeled as a plane wave to generalize the model and simplify the math. This is analogous to the "plane-parallel atmosphere" simplified model of atmospheres that ignore its curved shape.
A plane wave (if longitudinal) also approximates a wave within a cylindrical musical wind instrument such as a flute or organ pipe, one complicating factor being the effect of friction between the medium (air) and the surrounding inner pipe surface.