VW - CVM 1.1

# 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

cos(n 2px/K) cos(m 2py/L),   cos(n 2px/K) sin(m 2py/L),
sin(n 2px/K) cos(m 2py/L),   sin(n 2px/K) sin(m 2py/L),

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:

X = 2px/K   and   Y = 2py/L.

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

Communications in Visual Mathematics, vol 1, no 1, August 1998.