  We characterize waves in small magnetic elements and investigate their
propagation in the lower solar atmosphere from observations at high spatial and
temporal resolution. We use the wavelet transform to analyze oscillations of
both horizontal displacement and intensity in magnetic bright points found in
the 300 nm and the Ca II H 396.8 nm passbands of the filter imager on board the
Sunrise balloon-borne solar observatory. Phase differences between the
oscillations at the two atmospheric layers corresponding to the two passbands
reveal upward propagating waves at high frequencies (up to 30 mHz). Weak
signatures of standing as well as downward propagating waves are also obtained.
Both compressible and incompressible (kink) waves are found in the small-scale
magnetic features. The two types of waves have different, though overlapping,
period distributions. Two independent estimates give a height difference of
approximately 450+-100 km between the two atmospheric layers sampled by the
employed spectral bands. This value, together with the determined short travel
times of the transverse and longitudinal waves provide us with phase speeds of
29+-2 km/s and 31+-2 km/s, respectively. We speculate that these phase speeds
may not reflect the true propagation speeds of the waves. Thus, effects such as
the refraction of fast longitudinal waves may contribute to an overestimate of
the phase speed.
