  Recently, complex horizontal patterns of umbral oscillations have been
reported, but their physical nature and origin are still not fully understood.
Here we show that the two-dimensional patterns of umbral oscillations of slow
waves are inherited from the subphotospheric fast-body modes. Using a simple
analytic model, we successfully reproduced the temporal evolution of
oscillation patterns with a finite number of fast-body modes. In this model,
the radial apparent propagation of the pattern is associated with the
appropriate combination of the amplitudes in radial modes. We also find that
the oscillation patterns are dependent on the oscillation period. This result
indicates that there is a cutoff radial mode, which is a unique characteristic
of the model of fast-body modes. In principle, both internal and external
sources can excite these fast-body modes and produce horizontal patterns of
umbral oscillations.
