In this paper, wave propagation in sunspot umbrae is analysed. The stratification in a typical umbra is approximated by a model atmosphere, extending vertically from a depth of a few thousand kilometres below the photosphere, to the transition region. No boundaries are imposed in the horizontal direction. It is assumed that this atmosphere is embedded in a uniform vertical field. Using a Rayleigh-Ritz variational technique, the normal mode frequencies of the umbra are calculated for different values of the horizontal wave number k. A theoretical diagnostic diagram is generated. An interesting feature of the solutions is the existence of ‘avoided crossings’, similar to those found in the study of global oscillations. The nature of the wave modes is examined by decomposing the eigenvectors into longitudinal and transverse components. In general, it is found that the character of a mode changes with height. For large k, modes with periods in the 2-3 min range, correspond to low order modes in the present calculation. Above the photosphere, they resemble slow waves.