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Invariance under Translations

To find the eigenfunctions of the Laplacian that are invariant under horizontal and vertical translations, we seek eigenfunctions f(x,y) with f(x+K,y) = f(x,y) and f(x,y+L) = f(x,y) for numbers K and L (the lengths of the two translations).

Of course, one way to solve a problem in PDEs is to guess the solution correctly. It's very simple, by means of another application of the technique of separation of variables , to find that the totality of translation invariant eigenfunctions is the set consisting of

where m and n are integers. For a given pair (n,m), we have a four-dimensional eigenspace with eigenvalue -(n² + m²). For convenience we adopt shorthands:

The values of K and L may be adjusted to give the rectangular cell any desired dimensions.

Invariance under Half-Turns Lattice Waves -- Translational Invariance Example in a Rectangular Lattice Constructing Wallpaper Functions