The wave equation behaves nicely in one dimension and in three dimensi — Wave equation

"The wave equation behaves nicely in one dimension and in three dimensions but not in two dimensions. In one dimension, waves on a uniform string propagate without distortion. In three dimensions, waves in a homogeneous isotropic medium propagate in an undistorted way except for a spherical correction factor. However, in two dimensions, wave propagation is complicated and distorted. By its very nature, 2D processing never can account for events originating outside of the plane. As a result, 2D processing is broken up into a large number of approximate partial steps in a sequence of operations. These steps are ingenious, but they can never give a true image."

Wave equation

Wave equation

Wave equation

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The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves or electromagnetic waves. It arises in fields like acoustics, electromagnetism, and fluid dynamics.