Editing Plane wave (section)

==Properties==

A plane wave can be studied by ignoring the directions perpendicular to the direction vector <math>\vec n</math>; that is, by considering the function <math>G(z,t) = F(z \vec n, t)</math> as a wave in a one-dimensional medium. 

Any [[local operator]], [[linear operator|linear]] or not, applied to a plane wave yields a plane wave.  Any linear combination of plane waves with the same normal vector <math>\vec n</math> is also a plane wave.

For a scalar plane wave in two or three dimensions, the [[gradient]] of the field is always collinear with the direction <math>\vec n</math>; specifically, <math>\nabla F(\vec x,t) = \vec n\partial_1 G(\vec x \cdot \vec n, t)</math>, where <math>\partial_1 G</math> is the partial derivative of <math>G</math> with respect to the first argument. 

The [[divergence]] of a vector-valued plane wave depends only on the projection of the vector <math>G(d,t)</math> in the direction <math>\vec n</math>. Specifically,
<math display="block"> \nabla \cdot \vec F(\vec x, t) \;=\; \vec n \cdot \partial_1 G(\vec x \cdot \vec n, t)</math>
In particular, a transverse planar wave satisfies <math>\nabla \cdot \vec F = 0</math> for all <math>\vec x</math> and <math>t</math>.<!-- More: curl, laplacian...-->