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We examine a specific category of eigenfunctions of the lattice Laplacian on$\{p,q\}$-tessellations of the Poincar\'e disk that bear resemblance to planewaves in the continuum case. Our investigation reveals that the latticeeigenmodes deviate from the continuum solutions by a factor that depends solelyon the local inclination of the vertex in relation to the wave's propagationdirection. This allows us to compute certain eigenfunctions by numerical andanalytical methods. For various special cases we find explicit exacteigenfunctions and their eigenvalues on the infinite lattice.

Eigenmodes of the Laplacian on Hyperbolic Lattices