We continue our study of waves and oscillations observed in sunspots using an improved method for enhancing the waves, giving the opportunity to identify them and determine their properties in far Hα wings. We found that the running penumbral waves are observable at least up to the formation height of the Hα ± 0.5 Å line, but not in the H Hα ± 0.75 Å or the FeI ± 0.12 Å. We found a time lag between the waves in the blue and the red wing of the Hα line corresponding to a phase shift of 180°, that indicates a pure Doppler shift of the line. There is a lag in the propagation of the waves seen at Hα center and at Hα wings. Also there is a lag in the variation of the umbral oscillations as they are observed from lower to higher atmospheric layers. The correlation between umbral oscillations at various atmospheric heights and running penumbral waves strongly indicates that the latter are excited by photospheric umbral oscillations and not the chromospheric ones. We found a new category of photospheric waves that originate at approximately 0.7 of the distance between the umbra and the penumbra boundary and propagate beyond the outer penumbra boundary with a velocity of the order of 3-4 km s^(-1). Further, we found 3 min penumbral oscillations apparent in the inner penumbra at lower chromospheric layers (far Hα wings)
