### 3.6 Wavelength, phase speed, and group velocity

To derive the wavelength () and phase speed () of oscillations, time signals at different locations on the prominence must be acquired. The signature of a propagating wave is a linear variation of the oscillatory phase with distance. Hence, when several neighbouring points are found to oscillate with the same frequency, one can compute the Fourier phase of the signal at each of the points and check whether it varies linearly with distance. If it does, this gives place to a wave propagation interpretation and the wavelength can be calculated. This approach has been followed by Thompson and Schmieder (1991), Molowny-Horas et al. (1997) (about which more details are given in Section 3.6.2), and Terradas et al. (2002) (see Section 3.6.3). On the other hand, Lin (2005) and Lin et al. (2007) (see Section 3.6.4) detected wave propagation along threads by studying Doppler velocity variations at fixed times. They observed a sinusoidal variation of the Doppler shift with distance along the thread, which allowed them to compute the wavelength. Moreover, the phase velocity of the oscillations can be derived from the inclination of the coherent features in the Doppler velocity time-slice diagrams. Other authors have followed less strict methods to calculate these wave parameters.

It must be mentioned that observations of wave propagation in slender waveguides or plane wave propagation in a uniform medium do not provide the actual value of the wavelength (), but its projection on the plane of the sky, which is shorter than . And if a slit or some points along a straight line are used, then the computed wavelength is the projection of on the slit or the line. The observationally measured period and wavelength can in turn be used to calculate the phase speed, but since the observational wavelength is a lower limit to , this observational phase speed is also a lower limit to (Oliver and Ballester, 2002). Hence, even if it is not explicitly mentioned, the values of and quoted here are observationally derived lower bounds to the actual values.

The results presented in this section are grouped in four parts, the first three of them in increasing order of complexity of the data analysis; the fourth one is devoted to thread oscillations. The reported wavelength values cover a range from less than 3000 km (for waves propagating along some threads) to 75,000 km (for waves propagating in a large area of a quiescent prominence). These numbers must be taken into account in the theoretical study of these events.

#### 3.6.1 Simple analyses

Malville and Schindler (1981) observed a loop prominence some 90 min before the onset of a nearby flare and detected periodic changes with a wavelength along the loop of 37,000 km. This value, together with the period of 75 min, results in a phase speed of about 8 km s–1.

Subsequent reports, which we now describe, are based on sheet-like prominences. Thompson and Schmieder (1991) detected periodic variations with periods between 3.5 and 4.5 min in a filament thread. They then computed the Fourier phase of the points along the thread and, after confirming its linearity from a phase versus distance plot, the value  km was derived, from which the phase speed is  km s–1. In other works (e.g., Tsubaki and Takeuchi, 1986; Tsubaki et al., 1987, 1988; Suematsu et al., 1990) the signal in some consecutive locations along the slit has been found to be in phase. Although this seems to indicate that the wavelength of oscillations is much larger than the distance between the first and last of those points, this may not be necessarily true and a proper determination of the wavelength requires computing the Fourier phase corresponding to the oscillatory period.

Blanco et al. (1999) detected 15 – 20 min periodic variations corresponding to a pulse travelling with a speed of 170 km s–1. Such a large phase velocity is hard to reconcile with the typical speeds in a prominence, but it must be taken into account that this result has been obtained using Si iv and O iv lines, which are formed at transition region temperatures. Still, Blanco et al. (1999) mention that the sound speed in the prominence-corona transition region must be considerably faster than 170 km s–1, which leads them to conclude that the detected wave is of fast or Alfvénic character. Assuming a density of 1010 cm–3, a magnetic field of 8 G is derived.

Foullon et al. (2004) analyzed the intensity on a set of points along the main axis of a filament in 195 Å images. A time-space plot shows a clear oscillatory pattern at one end of the filament (around position 25 in Figure 9a). The oscillatory phase, displayed in Figure 9b, presents oscillatory fronts that are well correlated along the filament, meaning that the oscillations of neighbouring points along the filament are almost in phase. This is true in particular for positions around 25, although in positions around 5 and 10 the phase presents a linear trend in neighbouring points, which can be interpreted as wave propagation along the filament axis. Lower bounds to the wavelength and phase speed in this area could be determined as explained above. It is remarkable that the most pronounced periodic intensity variations, those around position 25, were detected during 6 days, which suggests that they suffered very little damping or were excited continuously during this time span.

Berger et al. (2008) used high-resolution observations of limb prominences made by SOT on Hinode and detected oscillations that do not affect the whole prominence body. They considered three horizontal time slices at heights separated by 4.7 Mm and detected the presence of coherent oscillations in the three slices (Figure 2). A phase matching of the sinusoidal profiles of these oscillations results in a vertical propagation speed (i.e., phase speed) around 10 km s–1. Again this value comes from a projection on the plane of the sky and is therefore a lower bound of the real value.

#### 3.6.2 An elaborate one-dimensional analysis

Molowny-Horas et al. (1997) took into account the projection effects in their analysis of the Doppler velocity along the spectrograph slit. They detected periodic velocity variations with period of 7.5 min some 7300 km along the slit and found that the Fourier phase of the velocity at this period changes linearly with distance (Figure 10). The value was derived. The corresponding phase speed is .

To obtain the group velocity of this event, Molowny-Horas et al. (1997) performed a wavelet analysis of the same set of data, which revealed the presence of a train of 7.5-min waves in the slit locations (Figure 8). Moreover, the time of occurrence of the train of waves increases linearly along the slit, which agrees with the assumption of a propagating disturbance. The velocity of propagation along the slit can then be computed and the value is obtained. Taking into account that is the projection of the group velocity, , on the slit, one concludes that the above value provides a lower limit for the group velocity, so .

#### 3.6.3 A two-dimensional analysis

This section is devoted to review the work by Terradas et al. (2002), that stands out among all other works in which wave properties have been determined since in this one a fully two-dimensional analysis is carried out. Figure 11 shows a time series of H filtergrams of the prominence studied by Terradas et al. (2002). The corresponding time series of the Doppler signal is presented in Figure 12.

The data used by Molowny-Horas et al. (1999) were re-analysed by Terradas et al. (2002) and clear evidence for propagating and standing waves was uncovered. These authors started from the Doppler velocity, which in many areas of the two-dimensional field of view can be very well fitted by a damped sinusoid (Figures 6 and 7). The subsequent analysis was performed in a rectangle (black box in Figure 13) that includes an area in which the correlation coefficient of the fit is large. The period of the oscillations in this rectangle is quite uniform and with a value around 75 min. First, Terradas et al. (2002) conducted an analysis of the phase along two straight lines inside the rectangle. Along the continuous line in Figure 13, it is found that waves emanate from a point and propagate away from it (Figure 14). It is clear both from the raw and the fitted signals in Figure 14 that the slope of wave propagation to the left is larger than that to the right. To derive the wavelength, Terradas et al. (2002) plotted the Fourier phase associated to the most relevant period in the Fourier spectrum (i.e., the one with 75 min period) along the selected path (right panel of Figure 14). There is an almost linear decrease of the phase between positions 5 and 30, a linear increase between positions 50 and 62 and a region of roughly constant phase in between. The first two patterns correspond to propagation to the left and right along the path, such as was pointed out from the first two panels of Figure 14, while the third pattern is caused by standing wave motions. The slope of a straight line fitted to the Fourier phase in each of the regions with wave propagation gives the wavelength of oscillation (projected on the selected path) which is around 75,000 km and 70,000 km for propagation to the left and right, respectively. The corresponding phase velocities are around 17 km s–1 and 15 km s–1.

Another interesting feature of this data set can be discerned by considering the dashed path in Figure 13. A representation of the Doppler velocity versus position and time (Figure 15) shows that, at least for the first half of the observational time, positive and negative velocities seem to alternate in phase separated by a region, around position 25, with nearly zero amplitude. This pattern suggests that rather than a propagating feature, the signal in this area behaves like a standing wave with two regions completely out of phase. The Fourier phase (right panel of Figure 15) is practically constant in a small region around position 10 and in a larger region for positions greater than 30, which indicates that there is no signal propagation in these locations. The phase difference between positions 10 and 50 is close to , which, together with the fact that between these points the amplitude takes low values, is in close agreement with the standing wave picture and so a tentative identification of nodes and antinodes is possible. The estimated distance between the two antinodes visible in the left panel of Figure 15 is around 22,000 km. This implies that the (projected) wavelength of the standing wave is about 44,000 km and the corresponding phase speed is 10 km s–1. These values are about half those obtained for propagation along the other selected path and are a consequence of the anisotropic propagation of the perturbation.

In addition to the identification of standing and propagating wave features in the prominence, Terradas et al. (2002) went on to perform an investigation of the two-dimensional distribution of the wavelength and phase speed. They started by plotting the Fourier phase for the most relevant period in the Fourier spectrum at each point (Figure 16), which shows that a deep global minimum is found around the central position of the plot. This particular phase structure is an indication that motions have their origin at the position of the minimum and propagate, although in an anisotropic way, from this point. Terradas et al. (2002) gave a much more clear interpretation of the two-dimensional phase by plotting the wavevector field (Figure 17), computed as the gradient of the Fourier phase. The arrows in this figure indicate the direction of wave propagation, their length being proportional to the modulus of the wavenumber, . The projection of the phase velocity on the plane of the sky (computed from ) is also displayed in Figure 17. The analysis of the wavevector field shown in this figure clearly indicates that motions seem to be generated in a narrow strip close to positions x = 35 – 50 and y = 20 – 30 and spread out from this region. It is remarkable that the direction of the propagating waves from the source region is essentially parallel to or towards the prominence edge, revealing the anisotropic character of the observed wave propagation. The values of the phase velocity in Figure 17 are also quite different for both directions, being greater for the direction parallel to the edge, with , than for the direction perpendicular to the edge, with . This is an indication of the possible existence of some wave guiding phenomenon, which shows a preferential direction of propagation. Note the good agreement between the values of the phase velocity in the directions parallel and perpendicular to the edge and those derived from the analysis of the two selected paths based on Figures 14 and 15.

Yi et al. (1991) and Yi and Engvold (1991) used two-dimensional spectral scans and investigated the presence of periodic variations of the Doppler shift and central intensity of the He i 10,830 Å line in two filaments. Yi et al. (1991) performed a first examination of the data and found oscillations with well-defined periods along particular threads in each prominence. For this reason, Yi and Engvold (1991) plotted the Doppler velocity versus position for different times in a given thread, so that a periodic spatial structure would directly yield a measure of the wavelength. Instead of this pattern, an almost linear variation of the velocity along the thread was found and consequently a value of much larger than the length of the threads, some 20,000 km in the two cases considered, was reported. Given that the periods are between 9 and 22 min, the corresponding phase speed is . This result suggests that the thread is oscillating in the fundamental kink mode (whose wavelength is of the order of the length of the supporting magnetic tube, that is, around 100,000 – 200,000 km; see Section 4.4.1), rather than being disturbed by a travelling wave. Let us mention that, in general, this analysis may be misleading since the velocity signal does not generally consist of the detected periodic component only, but it is made of this component mixed with other velocity variations. If the periodic component is weak, then the method used by Yi and Engvold (1991) may fail because the signature of the propagating wave is masked by the rest of the signal.

In the analysis of the Doppler velocity in two threads (denoted as T1 and T2) belonging to the same filament, Lin (2005) found a clear oscillatory pattern in time-slice diagrams along the two thin structures. She determined the following wave properties for thread T1:  = 60 km s–1,  = 22, 12, 15 arcsec, and in the range 2.5 – 5 min (the 4.4 min period being particularly pronounced). For thread T2, the wave properties are:  km s–1,  = 38, 23, 18 arcsec, and in the range 2.5 – 5 min (the 5-min period being particularly pronounced).

The previous study by Lin (2005) is followed by a much more profound one in which the two-dimensional motions and Doppler shifts of 328 features (or absorbing “blobs”) of different threads are examined (see also Lin et al., 2003). Forty nine of these features are observed to flow along the filament axis with speeds of 5 – 20 km s–1 while oscillating in the line-of-sight at the same time with periods of 4 – 20 min (see Figure 18). To simplify the examination of oscillations, Lin (2005) computed average Doppler signals along each thread and found that groups of adjacent threads oscillate in phase with the same period. This has two consequences: first, since the periodicity is outstanding in the averaged signal for each thread, the wavelength of oscillations is larger than the length of the thread. Again the interpretation of this result is that the threads oscillate in their fundamental kink mode. Second, in this data set threads have a tendency to vibrate collectively, in groups, rather than independently.

Horizontally flowing threads that undergo simultaneous transverse oscillations have not only been detected by Lin et al. (2003) and Lin (2005), but also by Okamoto et al. (2007) using SOT on Hinode. A Ca ii H line movie shows continuous horizontal thread motions along an active region prominence (cf. Figure 19). This movie also shows that the threads suffer apparently synchronous vertical oscillatory motions. An example of this phenomenon is shown in Figure 20. Six threads displaying the same behaviour were studied and periods in the range 135 – 250 s were measured. The thread flow velocities range from 15 to 46 km s–1 and the vertical oscillation amplitudes range from 408 to 1771 km. These values are, of course, minimum estimates. A particularly interesting feature of these oscillations is that points along each thread oscillate transversally with the same phase. To reach this conclusion, a given thread is selected and several cuts along its length are considered. A representation of the signal as a function of time reveals that oscillations are synchronous along the entire length of the thread (Figure 21). Once more this points to the kink mode as the responsible for the oscillations, as first pointed out by Van Doorsselaere et al. (2008a).

H observations conducted with the Swedish 1-m Solar Telescope by Lin et al. (2007) allowed to detect waves propagating in some selected threads. Figure 22 serves to illustrate the data analysis procedure for one thread. Here the line intensity shows no coherent behaviour (Figure 22a), while the line-of-sight velocity presents some inclined features caused by waves propagating along the thread; two such features are labelled 1 and 2 in Figure 22b. Figure 22c is another way of presenting Figure 22b and is useful to illustrate more clearly the wavy character of the line-of-sight velocities along an individual thread. Two shorter time sequencies of Doppler velocity are extracted from Figure 22c and shown in Figures 22d and e. It is clear that oscillations are of small amplitude since the Doppler shift has an amplitude of 1 – 2 km s–1. The power spectra of two of the curves in Figure 22c (shown in Figures 22f and g) yield wavelengths of the oscillatory pattern of, respectively, 3.8 arcsec and 4.7 arcsec. The phase velocity of the oscillations can be derived from the inclination of the features appearing in the Doppler time-slice diagrams of Figure 22b. The phase velocities thus obtained correspond to, respectively, 8.8 and 10.2 km s–1. Lin et al. (2007) found similar evidence of travelling waves in eight different threads. The mean phase velocity and period (obviously affected by the projection effect) are 12 km s–1 and 4.3 min. Periods between 3 and 9 min were found; longer period oscillations could not be detected in the data set used in this work because of its limited duration (18 min).

To test the coherence of oscillations over a larger area, covering several threads, Lin et al. (2007) averaged the line-of-sight velocity in a 3.4 arcsec × 10 arcsec rectangle containing closely packed threads. The averaged Doppler signal (left panel of their Figure 4) displays a very clear oscillation. In addition, the power spectrum of this signal has a significant power peak at 3.6 min. Thus, the conclusion is that neighboring threads tended to oscillate coherently in this rectangular area, possibly because they were separated by very short distances. This signal averaging could be analogous to acquiring data with poor seeing, such as in Terradas et al. (2002) (see Section 3.6.3).