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 \pm 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 \pm 2 km s ^ { -1 } and 31 \pm 2 km s ^ { -1 } , 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 .