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Observation of the main sunspot of AR11692 were carried out with the 1m New Vacuum Solar Telescope (NVST) located on the Fuxian Solar Observatory (FSO) with the Hα on 2013 march 13. The high cadence Hα intensity images show that running waves formed periodically within umbra, and propagate outwards with the shape of partial arcs. The running waves run across the umbra-penumbra boundary and disappear at the outer edge of the penumbra eventually. Confusingly, the period of the running waves within umbra is half of that within penumbra, i.e., the former is a typical 3-min oscillation with a period of 156s and the latter is typical 5-min oscillation with a period of 312s. To investigate the nature of the umbral and penumbral oscillations, we selected 300 sample points from the umbra and 150 sample points from the penumbra, respectively, of the sunspot to analyze the distributions of the extreme values related to the intensity profiles of the sample points. It is found that there are significant difference between the distributions of the extreme values related to umbral and penumbral oscillation intensity profiles. When a possible steady solar radiation, which was replaced by the average of troughs of the umbral oscillation intensity profile, was filtered from the umbral oscillation intensity, the distribution of the trough values is a normal distribution symmetrical about zero, the distribution of the crest values is a skewed distribution. When another possible steady solar radiation, which was replaced by the average of the penumbral oscillation intensity profile, was filtered from the penumbral oscillation intensity, while the distribution of the crest values and trough values were a pair of skewed distribution symmetrical about zero. According to the statistical characteristic of umbral and penumbral oscillations, an interpretation seemingly no having physical meaning was used to explain the statistical results. The possible interpretation for the phenomenon is that the pair of oscillations with different period in the intensity are produced by a same wave with a basis oscillation period of 312s. The basis wave can be expressed as Z ( x , y , t ) = A ( x , y ) c o s ( ω t + θ ) , where ω = ( 2 π ) / 312 is the angular frequency of the wave. In the umbra, the intensity due to the wave is proportional to the square of the displacement of the wave, I U = a 2 Z 2 = a 2 2 A 2 ( x , y ) [ 1 + c o s ( 2 ω t + 2 θ ) ] , while the intensity in the penumbra due to the wave is proportional to the basis wave, I P = a 1 Z = A ( x , y ) c o s ( ω t + θ ) . Correspondingly, the period of the intensity within umbra is T U = 156 s , while the period of the intensity within penumbra is T P = 312 s .

Observations of the main sunspot of AR 11692 were carried out with the 1 m New Vacuum Solar Telescope (NVST) located on the Fuxian Solar Observatory (FSO) in $H\alpha$ on March 13, 2013. The high cadence (up to 12 s) $H\alpha$ intensity images help us to investigate the relationship between the 5-min oscillation and 3-min oscillation. It is found that running waves, periodically formed at the wave sources within umbra, propagate outward with the shape of partial arcs. The running waves run across the umbra–penumbra boundary and eventually disappear at the edge of penumbra. But there are obvious differences when we measure the period of running waves in different regions of a sunspot. The period is about 150 s when the running waves are located in umbra, which is a typical 3-min oscillation, and the period is about 300 s when the running waves are located in the penumbra, which is a typical 5-min oscillation. On the basis of time-slice images, we conclude that the waves form in the umbral region with the 5-min oscillation period, which can cause the brightness periodicity change in the umbra region with the 3-min period (in fact, is half of 5-min oscillation) and 5-min in the penumbra.

Observations of the main sunspot of NOAA 10953 were carried out with the Solar Optical Telescope aboard Hinode with the line Ca II H on 2007 April 30. The high-cadence (up to 10 s) Ca II H intensity images permit us to elaborate on the nature of umbral oscillations. It is found there are 19 wave sources within the sunspot and 82 running umbral waves that periodically appear at the wave sources, and propagate outward while forming ellipses or partial arcs. Due to their own expansion, the running waves become more diffused, and eventually disappear at the edge of the umbra. On the basis of the time-interval distribution between adjacent running waves being centered close to 144 s, we observed an approximate oscillation period of 142 s. By tracing the fronts of the running waves, the propagation speed of the running waves ranges from 30 to 45 km s � 1 . The distribution for all running waves has an average propagation speed of � 37.4 km s � 1 . Furthermore, we have found an interesting phenomenon that

Continuous wavelet transform and cross-wavelet transform have been used to investigate the phase periodicity and synchrony of the monthly mean Wolf ($R_{z}$) and group ($R_{g}$) sunspot numbers during the period of June 1795 to December 1995. The Schwabe cycle is the only one common period in Rg and Rz, but it is not well-defined in case of cycles 5-7 of Rg and in case of cycles 5 and 6 of $R_{z}$. In fact, the Schwabe period is slightly different in $R_{g}$ and $R_{z}$ before cycle 12, but from cycle 12 onwards it is almost the same for the two time series. Asynchrony of the two time series is more obviously seen in cycles 5 and 6 than in the following cycles, and usually more obviously seen around the maximum time of a cycle than during the rest of the cycle. $R_{g}$ is found to fit $R_{z}$ better in both amplitudes and peak epoch during the minimum time time of a solar cycle than during the maximum time of the cycle, which should be caused by their different definition, and around the maximum time of a cycle, $R_{g}$ is usually less than $R_{z}$. Asynchrony of $R_{g}$ and $R_{z}$ should somewhat agree with different sunspot cycle characteristics exhibited by themselves.