Elastic waves can propagate over large distances in structures like rails and pipes, with a generic name referred to as waveguides. After some distance of propagation in a waveguide of uniform cross section, such guided waves can be described as a sum of just a few discrete propagating modes, each with its own shape and phase speed. The equation governing these modes can be obtained as a spatial Fourier transform of the linearized time-harmonic equation in the waveguide axial z direction or by inserting the assumption that the mode is harmonic in space,

and eliminating all out-of-plane z dependence.

Similar to the full time-harmonic equation, the transformed equation can be solved at a given frequency with a nonzero excitation for most axial wave numbers kz. But at certain discrete values the equation breaks down. These values are the propagation constants or wave numbers of the propagating or evanescent waveguide modes. The eigenvalue solver can solve for these propagation constants together with the corresponding mode shapes.