\thesischapter{Oscillations in Multi-Wavelength Optical Observations of the Quiet-Sun Network}
\label{mode_coup}
\hrule height 1mm
\vspace{0.5mm}
\hrule height 0.4mm 
\noindent 
\\{\bf Oscillations in Network Bright Points (NBPs) are studied at a variety of chromospheric heights. In particular, the three-dimensional variation of NBP oscillations is studied using image segmentation and cross-correlation analysis between images taken in the light of \cak, \ha\ core, \mgb, and \mgbw. Wavelet analysis is used to isolate wave-packets in time, and to search for height-dependent time delays which result from upward- or downward- directed travelling waves. In each NBP studied, evidence is found for kink mode waves (1.3~mHz, 1.9~mHz), travelling up through the chromosphere and coupling with sausage mode waves (2.6~mHz, 3.8~mHz). This provides a means for depositing energy in the upper chromosphere. Other upward- and downward- propagating waves in the 1.3--4.6~mHz range are also found. Some oscillations do not correspond to travelling waves and we attribute these to waves generated in neighbouring regions.}
\\
\hrule height 0.4mm
\vspace{0.5mm}
\hrule height 1mm   

\section{Introduction}
\label{mc_intro}
This chapter deals with the search for the oscillatory signatures of propagating waves in the chromosphere by extending the work in the previous chapter into two further dimensions (namely height in atmosphere and time). This study uses the fact that both longitudinal and transverse travelling waves are observable as intensity changes in carefully chosen wavelength bandpasses. A propagating longitudinal wave results in a periodic compression and hence heating of the surrounding plasma. This leads to increased radiation, which is then viewable as a periodic intensity change in the filter bandpass. Transverse waves propagating through a plasma result in a Doppler shift of the line profile of the emitted radiation. When the wave is not travelling along the observers line-of-sight ,this results in a periodic change in intensity in the filtergram as the emitted line profile moves in and out of the chosen bandpass. Detailed MHD simulations predict various types of oscillatory signatures for different wave modes at different heights in the chromosphere. The search for these oscillatory signatures is therefore vital in providing observational evidence for MHD waves.

Simulations of MHD waves are commonly modelled using the thin flux tube approximation, whereby the flux tube radius is much smaller than the characteristic wavelength. This approximation will break down in the mid chromosphere, where the flux tube will expand radially due to decreased density and hence its diameter will exceed the pressure scale height (Spruit 1981a). Further discussion of the validity of the thin flux tube approximation in modelling the network can be found in Hasan {\etal} (2003). 

In the thin flux tube approximation a magnetic flux tube in an isothermal atmosphere supports three types of waves (Spruit \& Roberts 1983), as discussed in Chapter~2. To summarise, the pure \alf\ torsional wave is non-dispersive and is propagating for any frequency, which negates it as a possible heating mechanism. In a flux tube with a strong magnetic field (as occurs in NBPs), the other types are the transverse, kink-mode wave and the longitudinal, sausage-mode wave. The corresponding tube speeds are given in Kalkofen (1997) in terms of the acoustic speed, c$_S$ (which is around 9 km s$^{-1}$ in the low chromosphere). Both wave modes are dispersive and propagate at frequencies above their respective cut-offs; otherwise they are evanescent. The wave will propagate at the appropriate tube speed, followed by a wake oscillating at the corresponding cut-off frequency. The longitudinal cut-off frequency, $\nu_{l}$, and the transverse cut-off frequency, $\nu_{k}$, are defined by Kalkofen (1997) in terms of the acoustic cut-off frequency, $\nu_{ac}$ (typically around 5.5~mHz in the chromospheric internetwork K$_{\it 2V}$ bright points, Liu 1974) and given in Chapter~2. For ease of reference they are reproduced as,
\begin{equation}
{\nu_{l}} = 
{\nu_{ac}}\sqrt{\frac{63+48\beta}{60+50\beta}} \ ,
\end{equation} 
\begin{equation}
{\nu_{k}} = 
{\nu_{ac}}\sqrt{\frac{1}{2\gamma(1+2\beta)}} \  .
\end{equation} 
With a maximum plasma $\beta$ value of 0.5 (Kneer, Hasan, \& Kalkofen 1996), this gives a minimum ${\nu}_{k}$=2.2~mHz and ${\nu}_{\lambda}$=5.7~mHz. However, the local conditions (e.g., density, magnetic field) in the network can be quite different from the internetwork such that although ${\nu}_{\lambda}$ will not vary by more than 2\% from ${\nu}_{ac}$ (Kalkofen 1997), ${\nu}_{k}$ may be much lower than 2.2~mHz. Roberts (1983) also discuss the possibility of non-adibatic conditions in the network, which will drastically alter ${\nu}_{\lambda}$. The exact value of ${\nu}_{ac}$ in the magnetised atmosphere may also differ from the internetwork. 

Transverse waves may be generated at the photosphere by granular buffeting of network flux tubes at a frequency above the transverse cut-off. Muller \& Roudier (1992) and  Muller {\etal} (1994) have discovered that NBPs do possess a rapid, intermittent motion which Choudhuri, Auffret, \& Priest (1993) and Choudhuri, Dikpati, \& Banerjee (1993) modelled as the creation of transverse mode waves at the photospheric level. These waves can then propagate up along the field lines in a NBP at the corresponding tube speed. Their speed amplitude will increase due to density stratification, so that when it becomes comparable to the tube speed, they enter the non-linear range. At this point mode-transformation can occur (Ulmscheider {\etal} 1991), and the waves can couple (and hence transfer power) to longitudinal waves. This mode-coupling occurs preferentially for transverse waves at a frequency, $\nu$, which can transfer most of their power to longitudinal waves at a frequency 2$\nu$, with a remnant of power at the original frequency. The interaction is greatest when the two tube speeds are equal (i.e., $c_K$ = $c_{\lambda}$). From Eqn.~\ref{kinkwave2} and Eqn.~\ref{longwave2} it turns out this corresponds to $\beta = 0.2$, which gives a speed of maximum interaction of around 0.93 c$_S$. The longitudinal waves can then shock and heat the surrounding plasma (Zhugzhda, Bromm, \& Ulmscheider 1995). 

In order to identify such a mechanism in the chromosphere, a number of tests can be carried out.
\newline $\bullet${\it Test 1}: Power at a frequency $\nu$ in the lower chromosphere should also be identifiable at much reduced power in the upper chromosphere.\newline $\bullet${\it Test 2}: Power at frequency 2$\nu$ should appear in the upper chromosphere.\newline $\bullet${\it Test 3}: Waves at frequencies above the transverse cut-off should propagate at a speed close to c$_S$.\newline $\bullet${\it Test 4}: In the upper chromosphere there may be evidence for waves above the longitudinal cut-off frequency although to a lesser extent as they should shock in this region.\newline $\bullet${\it Test 5}: Oscillations should be quasi-periodic, resulting from the nature of the granular buffeting.

In this chapter I search for observational evidence of MHD waves by studying intensity changes in NBPs at several wavelengths, corresponding to a range of heights from the photosphere to the upper chromosphere. Section~\ref{mc_data_anal} provides a summary of the observations and data analysis. Both Fast Fourier Transforms (FFTs) and wavelet transforms are used to study NBP oscillations in space and time. The wavelet transforms are also cross-correlated at all frequencies to check for the signature of possible travelling waves. In Section~\ref{mc_resu} I present and discuss the results from each NBP individually and collectively. These results are then further discussed in Section~\ref{mc_dis} in relation to the observational tests mentioned above. 

\section{Data Analysis}
\label{mc_data_anal}

\begin{table}
\begin{center}
\caption[Details of {\dst} dataset]{Details of {\dst} dataset, including an estimation of the height of formation of each bandpass}
\label{wavelengths}
\begin{tabular}{lccccc}
\hline
\hline
Line		&Wavelength 	&FWHM		&Instrument	&Image Scale 		&Formation  \\
		&(\AA)		& (\AA)		&		&(\arcsec\ per pixel)	&Height (km)\\
\hline
\cak		&3933.7		&0.30		&Halle    	&0.33  			&1800-2000$^1$\\
\ha		&6562.8		&0.21$^a$	&{\ubf}$^b$  	&0.18  			&1200-1700$^1$\\
\mgb		&5172.7		&0.13$^a$	&{\ubf}$^b$	&0.18  			&700-750$^2$\\
\mgbw		&5183.2		&0.13$^a$	&{\ubf}$^b$	&0.18  			&200$^3$\\
\hline
\end{tabular}
\end{center}
REFERENCES:\\
(a) Beckers, Dickson, \& Joyce (1975)\\
(b) Universal Birefringent Filter, Bonaccini {\etal} (1989)\\
(1) Vernazza {\etal} (1981)\\
(2) Schmieder (1979)\\
(3) Keller \& Koutchmy (1991)
\end{table}

The observational set-up at the {\dst} has been described in detail in Chapter~3. For this study, time series analysis of the \cak, \ha\ core, \mgb, and \mgbw\ data were used to provide information as a function of height throughout the chromosphere (Table~\ref{wavelengths}). 

\begin{figure}
%\epsscale{0.80}
\begin{center}
\epsfig{figure=3dplot.eps, width=5in}
\caption[Layered plot of the quiet Sun throughout the solar atmosphere]{Example time-averaged data of the 100$\arcsec$ $\times$ 100$\arcsec$ FOV in the corona (Fe~{\sc xv} 284~{\AA}), transition region (C~{\sc iv} 1550~{\AA}) and chromosphere ({\dst} wavelengths). The bottom plot is the corresponding magnetogram. The vertical lines show the four stable NBPs analysed in this paper.}
\label{3d_plot}
\end{center}
\end{figure}

A 150-minute sequence of 100\arcsec\ $\times$ 100\arcsec\ images were obtained in all wavelengths, centred on a quiet Sun field (Solar\_X = -25{\arcsec}, Solar\_Y = -35{\arcsec}) with a 45~s cadence. This provided seven NBPs (Figure~\ref{3d_plot}), four of which (NBPs 1, 2, 5 \& 7) were distinguishable throughout the entire series in all wavelengths, and hence are used here. As discussed in Chapter~3 the {\ubf} cycle (Table~\ref{wavelengths}) also included two Fe~{\sc{i}} (6302.5~{\AA}) images, linearly polarised at $90\,^{\circ}$ to each other. These images were destretched, integrated and combined to create the magnetogram at the bottom of Figure~\ref{3d_plot}. Coalignment of the ground based images with the {\sl TRACE} data led to a calculated spatial size of 0.33\arcsec\ per pixel in the \cak\ data and 0.18\arcsec\ per pixel for the other wavelengths (Table~\ref{wavelengths}). Figure~\ref{3d_plot} shows that the chromospheric NBPs have a one-to-one spatial correlation with the photospheric magnetic field and exist throughout the transition region, and even into the corona.

Analysis of the {\dst} observations was carried out using standard routines within {\ssw}. The data were corrected for CCD readout bias and flat-fielded. Each image was aligned to the first image of the series by means of cross-correlation. {\sl TRACE} images were reduced using the standard calibration and despiking routines provided. Light curves in each wavelength were then created using the contour and contrast method, high-pass filtered and subjected to a Fourier analysis as described in Chapter~4.

\subsection{Heights of Formation}

\begin{figure}
\begin{center}
\epsfig{figure=filter_profiles.eps,width=5in}
\caption[Filter profile plotted with line profiles for {\dst} wavelengths]{Filter profile plotted with line profile for (a) \cak, (b) \ha, (c) \mgb\ and (d) \mgbw. In each case the leftmost y axis is solar intensity, the rightmost y  axis is normalised filter transmission and the x axis is wavelength.}
\label{filter_profiles}
\end{center}
\end{figure}

An estimation of the height of formation of each bandpass is provided in Table~\ref{wavelengths}. It is important to note both the difficulty of estimating these, and their physical meaning. Essentially the temperature and density of the plasma where the optical depth of light emitted in each bandpass is unity ($\tau = 1$, photons escape from the solar atmosphere without any further absorption) is estimated. These parameters are then compared to models of the chromosphere in order to assign a height of formation. However this $\tau = 1$ surface will vary dramatically according to the local plasma parameters. This, combined with the fact that most models are of a magnetic-field free plasma, means that care must be taken not to apply too much physical meaning to any height of formation estimation. 

Figure~\ref{filter_profiles} shows the profile of each filter overplotted on the line profile of the respective wavelength. As the \cak\ profile is narrowband, most of the emission is assumed to be from the K$_3$ core. However there will be some contribution from the self-reversed K$_2$ peaks. The VAL model is used here to estimate a height of 1800--2000~km. The height of formation of the \ha\ core is also estimated from the VAL model. It is well known that \ha\ contains a double-peaked response function, so a much larger range of heights is assumed (1200--1700~km). The \mgb\ core is discussed in Kneer \& von Uexk\"{u}ll (1993) who, with a bandpass of 0.6~\AA, place it at $\sim$350~km. However, the data presented in this paper have a much smaller bandpass, and as such include less of the wings, so are placed at 700--750~km (Schmieder 1979). Keller \& Koutchmy (1991) use data 0.4~\AA\ into the red wing of the Mg~{\sc{i}}~b$_1$ line, and adopt 200~km as the height of formation. Although our data are 0.4~\AA\ in the blue wing, we also adopt this height of formation. 

Compared to the relatively clean images of \cak, \mgb\ and \mgbw, the \ha\ images are notoriously difficult to study due to the presence of extra structure (Chapter~1 of this thesis; Zirin 1966). Hence extra care was taken to correctly identify each NBP. The contour method was particularly useful for these data as it ensured non-NBP brightenings were not included. 

\subsection{Wavelet Analysis}
\label{wave_anal}

\begin{figure}
\begin{center}
\epsfig{figure=fft_mor.eps, width=4in}
\caption[Windowed FFT and wavelet analysis]{A pictorial description of how varying the scale ensure both time - and frequency- localisation are retained in (a) wavelet analysis but lost in (b) a windowed-FFT}
\label{fft_mor}
\end{center}
\end{figure}

Like many time series in solar physics (and astronomy in general), light curves from NBPs are known to exhibit non-stationarity. This means that while the light curve may contain periodic signals, these may vary in both frequency and amplitude with time. As such, a simple Fourier analysis may not detect the time- and frequency--localised oscillations. There are several solutions to this problem. One such solution is to use a {\em windowed FFT}, whereby a portion of the light curve is selected, and this window is slid along the light curve, calculating the FFT at each step. However this approach means the shape of the curve is dependent on the window size. As is shown in Figure~\ref{fft_mor}b, at low frequencies so few oscillations will occur inside the window that frequency--localisation will be lost. At high frequencies there will be so many oscillations inside the window that the time--localisation will be lost. The use of a box-car shape window will also lead to ringing effects.

Wavelet analysis circumvents these difficulties by using a base wave-packet of finite duration, $d$, and specific frequency, $\nu$. A convolution of this wave-packet with a section of light curve will give an indication of the oscillatory power (at frequency $\nu$) in that section (of duration $d$) of the light curve. Sliding this wave-packet along the light curve provides time--localisation. Changing the width of the wave-packet (known as the scale) provides frequency--localisation. The major improvements of wavelet analysis over a windowed FFT are, \newline\indent (i) wavelet analysis will fit the same number of oscillations for each frequency whereas the windowed FFT does not (Figure~\ref{fft_mor}). \newline\indent (ii) a suitable choice of base-wavelet avoids edge effects.    

\begin{figure}
\begin{center}
\epsfig{figure=morlet.ps, width=5in}
\caption[The morlet wavelet]{The real part (blue) of the Morlet wavelet is simply a sine curve (green), with a Gaussian envelope (red)}
\label{morlet}
\end{center}
\end{figure}

In this work a wavelet analysis (Torrence \& Compo 1998) was carried out on all the maximum threshold light curves using a Morlet wavelet of the form
\begin{equation}
\psi(\eta)=\pi^{-1/4}\exp(i\omega_0\eta)\exp\left(\frac{-\eta^{2}}{2}\right) \ ,
\end{equation}
where $\eta = t/s$ is the dimensionless time parameter, $t$ is the time, $s$ the scale of the wavelet (i.e., its duration), $\omega_0 = s\omega$ 
is the dimensionless frequency parameter (i.e., the number of oscillations within the wavelet; $\omega_0$=6 to satisfy the admissibility condition, Farge 1992), and $\pi^{-1/4}$ is a normalization term. The real part of this wavelet is basically a sine curve, with a Gaussian envelope (Figure~\ref{morlet}). This continuous (and hence nonorthogonal), complex function is ideal for capturing smooth, continuous oscillatory behaviour, but is known to be redundant at large scales (and hence low frequencies) where the wavelet spectrum will be highly correlated (another reason for filtering off low frequencies). The wavelet transform, $W_{n}$, at scale, $s$, of a discrete sequence, $x_n$ ($n=0, 1, 2, .... N-1$), is a convolution of $x_n$ with a translated $\psi(\eta)$,

\begin{equation}
W_n(s) = \sum_{n'=0}^{N-1} x_{n'} \psi^{*} \left[ \frac{(n' - n){\delta}t}{s} \right] \ .
\end{equation}
The smallest scale, $s_0$ is normally chosen as 2$\delta$t (i.e., inverse of the Nyquist frequency) with the larger scales then $2^n$ multiples of $s_0$, 
\begin{equation}
\label{Jeqn}
S_j = s_0 \times 2^{j{\delta}j} \ ,
\end{equation}
where ${\delta}j$ is the spacing between scales and $j=0, 1, 2... J-1$. Hence the scales values increase logarithmically from $s_0$ to $s_0 \times 2^{J{\delta}j}$ and $J$ is calculated by rearranging Equation~\ref{Jeqn}, 
\begin{equation}
J = \frac{1}{{\delta}j}log_2\left(\frac{N{\delta}t}{s_0}\right) \ .
\end{equation}

The wavelet transform will have a real and imaginary part, from which the amplitude, power, and phase can be determined as for the FFT. These will all be 2-dimensional maps, plotted against both time and scale (here scale, period, and frequency can be interchanged). Significance levels can also be calculated using a method similar to that for the FFT (e.g., for $sig$ = $0.95$, {\large$\chi^2$}(0.05,2) = 5.99, which is then multiplied by the mean power spectrum to give a power value for the 95\% level).  

Bocchialini \& Baudin (1995) note that although wavelet analysis can be used to obtain phase information, this is very sensitive to frequency. They preferred to determine lags from correlations between their signals, and it is an extension of their method which is used here. Essentially, from the 2D wavelet power density maps, `power curves' at each frequency were created by taking slices of the wavelet power transform. These were then cross correlated across pairs of wavelengths to search for timelags (it should also be noted here that the chosen width of the frequency band should be small enough to negate errors arising from the different weighting of each scale, Torrence \& Compo 1998). 

\subsection{Search for Travelling Waves}
\label{sftw}

For a pair of wavelengths, $\lambda_{1}$ and $\lambda_{2}$, at a frequency $\nu$, the power curves, P$_{\lambda_{1}}(\nu)$ and P$_{\lambda_{2}}(\nu)$ were created. The cross correlation coefficient, C($\Delta t, \nu$), of these power curves as a function of timeshift, $\Delta t$, is given by 
\begin{equation}C(\Delta t,\nu) = \frac{\sum \left(P_{\lambda_{1}}(\nu,t)- \overline{P_{\lambda_{1}}(\nu,t)}\right) . \left(P_{\lambda_{2}}(\nu,t+\Delta t)- \overline{P_{\lambda_{2}}(\nu,t+\Delta t)}\right)}{\sqrt{\sum \left(P_{\lambda_{1}}(\nu,t)- \overline{P_{\lambda_{1}}(\nu,t)}\right)^2 . \sum \left(P_{\lambda_{2}}(\nu,t)- \overline{P_{\lambda_{2}}(\nu,t)}\right)^2}} \ .
\end{equation}
This is used to evaluate the shift in P$_{\lambda_{2}}(\nu)$ which results in maximum correlation with P$_{\lambda_{1}}(\nu)$. This shift, $\Delta t_{max\_corr}$, then corresponds to the timelag of P$_{\lambda_{2}}(\nu)$ onto P$_{\lambda_{1}}(\nu)$ at frequency $\nu$, and C($\Delta t_{max\_corr},\nu$) is an indication of how good the correlation is. This procedure was then repeated for all frequencies, and between all pairs of wavelengths. From this timelag information, and from a knowledge of the approximate heights of formation (Table~\ref{wavelengths}), the speed of possible travelling waves at specific frequencies were calculated (Bocchialini \& Baudin 1995; Baudin, Bocchialini, \& Koutchmy 1996).  

\section{Results}
\label{mc_resu}

\begin{figure}
\begin{center}
\epsfig{figure=cak_wave.eps,width=5in}
\caption[Example wavelet power transform for \cak\ data]{Example wavelet power transform for \cak\ data. (a) Filtered light curve from the \cak\ data for NBP2. (b) Wavelet power transform. The lighter an area, the stronger the oscillatory power. (c) The global wavelet spectrum.}
\label{wave_cak}
\end{center}
\end{figure}

Example wavelet power transforms for NBP2 are shown in Figures~\ref{wave_cak} and~\ref{wave_mgb2}, where the wavelet diagrams are divided into three boxes. The top panel (a) shows the filtered light curve, while the main panel (b) shows the wavelet power transform, with frequency as the vertical axis and time along the horizontal axis. This time axis is the same as in panel (a), so that direct comparisons between the occurrence of oscillatory power and the parent time series can be made. The lighter an area, the stronger the oscillatory power. The contours refer to the $\chi^{2}$-derived 95\% confidence level, while the cross-hatched area is known as the Cone-Of-Influence (COI; Torrence \& Compo 1998). Inside this hatched area, edge effects can be significant, so that any power appearing inside the hatched area has to be treated with caution. The right-hand panel (c) shows the global wavelet spectrum. This is the sum of power over all time as a function of frequency, analogous to the FFT. However it should be noted that at large wavelet scales (low frequency), peaks will be sharper and hence have a larger amplitude. So it is best to think of the global significance level as a good, but biased, estimate of the true Fourier power. The wavelet transforms clearly show the quasi-periodic nature of the oscillations in all wavelengths (agreeing with {\it Test 5}) where power at any frequency occurs in the form of wave-packets lasting for a few oscillations before disappearing. Its also clear that \cak\ exhibits power around 3--4~mHz, whereas \mgb\ contains more power at lower frequencies (1.5--2~mHz)

\begin{figure}
\begin{center}
\epsfig{figure=mgb_wave.eps,width=5in}
\caption[Example wavelet power transform for \mgb\ data]{Example wavelet power transform for \mgb\ data. (a) Filtered light curve from the \mgb\ data for NBP2. (b) Wavelet power transform. The lighter an area, the stronger the oscillatory power. (c) The global wavelet spectrum.}
\label{wave_mgb2}
\end{center}
\end{figure}

Tables~\ref{NBP7} -- \ref{NBP1} contain a summary of the behaviour of each NBP. The top part of each table refers to peaks in the FFT power spectrum (and is hereafter referred to as an {\em FFT table}). It contains a summary of the behaviour of the NBP in each wavelength as a function of thresholding intensity (as in Chapter~4). The `99' and `95' refer to the fact that the power at this frequency reached at least the 99\% and 95\% confidence levels, respectively. Wavelet diagrams were then used to look at the duration of wave-packets at each peak in the FFT. For a frequency to be considered, it must last for at least one full oscillation outside the COI at greater than 95\% confidence. Any frequencies which showed more than two complete oscillation periods are presented in bold type.  Together with the FFT confidence level, this gives an indication of the strength and significance of any oscillations. As in Chapter~4 the frequency resolution of 0.11~mHz is used as the error in the FFT values, and hence all values are grouped in $\pm$ 0.11~mHz blocks, with the first row in the table corresponding to the lower limit, and second row to the upper limit (e.g. the first data column contains any peaks at 1.00, 1.11, and 1.22~mHz) 

\begin{figure}
\begin{center}
\epsfig{figure=corr_curve.eps,width=5in}
\caption[Peak correlation against frequency and timelag]{The three curves resulting from the correlation of the light curves of NBP2 in \cak\ and \mgb. (a) Peak correlation against frequency. (b) Peak correlation against timelag. (c) Timelag against frequency. The x-axis of (a) and (c) are the same. The y-axis of (a) and (b) are the same. Hence the timelag corresponding to the peaks in (a) can be determined by drawing horizontal and vertical lines across to (b) and down to (c) respectively.}
\label{corr_curve}
\end{center}
\end{figure}

The bottom part of each table contains a summary of the possible travelling waves for each wavelength pair. Across every pair of wavelengths and for each frequency,  the maximum correlation coefficient and corresponding timelag were recorded. Figure~\ref{corr_curve} shows the result of this procedure, and is now used to explain further steps. In Figure~\ref{corr_curve}(a) there are two peaks above 1~mHz in the frequency-correlation curve: 1.3~mHz (correlation = 0.78) and 1.9~mHz (correlation = 0.68). By drawing horizontal lines from these peaks across into (b), the timelag-correlation curve, the corresponding timelags can be obtained. This can also be achieved by drawing vertical lines from these peaks, down into (c), the frequency-timelag curve. Maxima in the frequency-correlation curve were retained only if the correlation was above 0.6, and the corresponding timelag was less than 450~s (10 $\times$ 45~s cadence). By individual inspection of the power curves, it is clear that correlations of greater than 10 time units could not be attributed to individual wave packets. Any correlations below 1~mHz were also disregarded, as the high-pass filter (Figure~\ref{cak_digifilt}) used dictates that there will be no wave-packets remaining below this frequency. Wavelet analysis and Fourier transforms were both featureless above 5~mHz, so this is taken as the upper limit to search for correlations. The range of values given in correlation and timelag in Tables~\ref{NBP7} -- \ref{NBP1} is determined by the maximum and minimum correlation and timelag in the frequency range of the chosen frequency $\pm$ 0.1~mHz. The final column denotes the speed of the wave.  It is noted that a data cadence of 45~s, combined with the spread in height estimates (Table~\ref{wavelengths}) leads to a large range of speed estimates. A `$<$' symbol signifies a maximum speed, whereas a `+' symbol signifies the speed may be faster than the quoted value. It should also be noted here that a negative timelag refers to a downward speed, whereas a zero timelag could suggest either evanescent waves or upward- or downward- propagating waves, within the available time resolution. 

Some general comments can be made, before concentrating on each NBP individually. In the lower atmosphere (e.g., {\mgbw}), the oscillations are concentrated in the bright central portion of each NBP. However, in the upper chromospheric (e.g., {\cak}), there is a greater tendency for some oscillatory power to exist in the outer regions (i.e., further from the centre of the NBP). This is best demonstrated by the moving threshold FFTs (Chapter~4, Figure~\ref{cak_thr}), where the sudden appearance of power at a particular threshold indicates that it is confined in the central portion only. It can be explained in terms of the magnetic canopy effect, whereby the magnetic field lines diverge in the upper chromosphere. The FFT tables also show a tendency for higher frequency power ($\sim$3~mHz) to exist in the upper chromosphere, with lower frequencies ($\sim$2~mHz) more powerful in the lower chromosphere. This is in general agreement with {\it Test 2}. The following four subsections contain a qualitative description of the wave behaviour found in each NBP, especially pertaining to the the five tests stated in Section~\ref{mc_intro}

\begin{landscape}
\begin{table}
\begin{center}
\caption[Summary of frequencies and correlations for NBP7]{Summary of frequencies (mHz) and correlations for NBP7. See Section~\ref{res_NBP7} for details.}
\label{NBP7}
\begin{tabular}{lrlllllllllll}
\hline
\hline
			&From:	&1.00	&1.33	&1.67	&2.00	&2.33	&2.67	&3.00	&3.33	&3.67	&4.00	&4.33  	\\
$\lambda$		&To:	&1.22	&1.56	&1.89	&2.22	&2.56	&2.89	&3.22	&3.56	&3.89	&4.22	&4.55	\\
\hline
\multicolumn{2}{l}{\cak} 	&\bf{99}&\bf{95}&\bf{99}&\bf{99}&       &       &       &       &       &       &95,95	\\
\multicolumn{2}{l}{\ha}	       	&\bf{99,99}&\bf{99,95}& &95,95  &       &       &       &\bf{99}&       &       &	\\
\multicolumn{2}{l}{\mgb}       	&\bf{99}&\bf{99,95}&\bf{99},\textmd{95}& &       &95     &       &\bf{95}&\bf{95}&       &\bf{99}\\
\multicolumn{2}{l}{\mgbw}      	&95,99  &95,95,95&99    &       &95     &       &99     &       &       &       &	\\ \\ \\ \\
\end{tabular}

\begin{tabular}{lllllll} 
\hline
\hline
		&		&Correlation	&Frequency    &Timelag   	      &Speed       \\
$\lambda_{1}$	&$\lambda_{2}$	&Coefficient	&  (mHz)      &(Units of 45~s)  &(km s$^{-1}$)\\
\hline
\cak		&\mgb		&0.66 -- 0.72	&1.5 --1.7    &8--11	      &2--4 \\
\cak		&\mgbw		&0.81 -- 0.91	&1.3 -- 1.5   &0 -- 3	      &12--40+	      \\
\ha		&\mgb		&0.69 -- 0.81	&1.8 -- 2.0   &0 -- -3        &3--22+      \\
		&		&0.75 -- 0.79	&2.8 -- 3.0   &-1 -- -3       &3--22      \\
		&		&0.83 -- 0.84	&3.55 -- 3.75 &4 -- 5	      &2--6	      \\
\ha		&\mgbw		&0.72 -- 0.81	&1.8 -- 2.0   &1 -- 2	      &11--33	      \\
\mgb		&\mgbw		&0.96 -- 0.98	&1.8 -- 2.0   &1 -- 5	      &2--12      \\
		&		&0.69 -- 0.70	&3.3 -- 3.5   &2	      &5--6   \\
\end{tabular}
\end{center}
\end{table}
\end{landscape}

\begin{landscape}
\begin{table}
\begin{center}
\caption[Summary of frequencies and correlations for NBP2]{Summary of frequencies (mHz) and correlations for NBP2. See Section~\ref{res_NBP2} for details.}
\label{NBP2}
\begin{tabular}{lrlllllllllll}
\hline
\hline
			&From:	&1.00	&1.33	&1.67	&2.00	&2.33	&2.67	&3.00	&3.33	&3.67	&4.00	&4.33  	\\
$\lambda$		&To:	&1.22	&1.56	&1.89	&2.22	&2.56	&2.89	&3.22	&3.56	&3.89	&4.22	&4.55	\\
\hline
\multicolumn{2}{l}{\cak}	&       &       &       &       &       &\bf{99,95}&99,95&99,95 &\bf{99}&\bf{95}&\bf{95}\\
\multicolumn{2}{l}{\ha}		&\bf{99}&\bf{99}&       &\bf{99}&       &       &       &       &\bf{95}&       &\bf{95}\\
\multicolumn{2}{l}{\mgb}	&       &\bf{99,99}&\bf{99,95}& &       &       &       &\bf{99,99}&    &       &\bf{95}\\
\multicolumn{2}{l}{\mgbw}	&95,99  &95,95  &95,95  &95     &       &       &       &       &       &       &       \\ \\ \\ \\
\end{tabular}
\begin{tabular}{llllll} 
\hline
\hline
		&			&Correlation	&Frequency	   	&Timelag		&Speed	   \\
$\lambda_{1}$	&$\lambda_{2}$		&Coefficient	&  (mHz)	   	&(Units of 45~s)  	&(km s$^{-1}$)\\
\hline
\cak		&\ha			&0.72 -- 0.76   &2.4 -- 2.6		&0 -- 4	   		&0--17+     \\
\cak		&\mgb			&0.66 -- 0.78	&1.25 -- 1.45   	&2 -- 4	   		&6--14     \\
		&			&0.68 -- 0.68	&1.75 -- 1.95	    	&3 -- 8		    	&3--10     \\
\cak		&\mgbw			&0.67 -- 0.78	&1.2 -- 1.4	    	&1 -- 3		    	&12--40      \\
		&			&0.70 -- 0.73	&1.7 -- 1.9	   	&7 -- 11	   	&3--6	   \\
\ha		&\mgb			&0.65 -- 0.67	&2.97 -- 3.17	    	&1 -- 2		    	&5--22      \\
		&			&0.63 -- 0.65	&3.85 -- 4.05	  	&3	  		&3--7 	  \\
\ha		&\mgbw			&0.35 -- 0.57	&1.15 -- 1.35	  	&0 -- -13	  	&2--33+ 	  \\
		&			&0.61 -- 0.62	&3.2 -- 3.4	  	&5		  	&4--7 	  \\
		&			&0.62 -- 0.63	&4.7 -- 4.9		&-2			&11--16     \\
\mgb		&\mgbw			&0.83 -- 0.93	&1.3 -- 1.5	  	&0 -- 1	  		&11+ 	  \\
		&			&0.95 -- 0.97	&1.6 -- 1.8	    	&4 -- 6			&2--3        \\
		&			&0.94 -- 0.94	&4.2 -- 4.4      	&-2		    	&5--6	\\
\end{tabular}
\end{center}
\end{table}
\end{landscape}

\begin{landscape}
\begin{table}
\begin{center}
\caption[Summary of frequencies and correlations for NBP5]{Summary of frequencies (mHz) and correlations for NBP5. See Section~\ref{res_NBP5} for details.}
\label{NBP5}

\begin{tabular}{lrlllllllllll}
\hline
\hline
			&From:	&1.00	&1.33	&1.67	&2.00	&2.33	&2.67	&3.00	&3.33	&3.67	&4.00	&4.33  		\\
$\lambda$		&To:	&1.22	&1.56	&1.89	&2.22	&2.56	&2.89	&3.22	&3.56	&3.89	&4.22	&4.55		\\
\hline
\multicolumn{2}{l}{\cak}        &99,99,95&	  &	  &\bf{99}&	  &95	  &	  &\bf{99}&\bf{95}&	  &\bf{95,95}	  \\
\multicolumn{2}{l}{\ha}	       	&\bf{99,95}&\bf{99},\textmd{95}&   &\bf{99}&	  &99	  &	  &	  &\bf{95,95}&\bf{95,99}&	  \\
\multicolumn{2}{l}{\mgb}        &\bf{99,99}&    &	  &	  &	  &\bf{99}&\bf{95}&\bf{99}&	  &	  &\bf{99}	  \\
\multicolumn{2}{l}{\mgbw} 	&\bf{99}&99,95  &	  &\bf{99}&	  &\bf{95}&\bf{99}&\bf{95}&	  &	  &		  \\ \\ \\ \\
\end{tabular}
\begin{tabular}{llllll} 
\hline
\hline
		&		&Correlation	&Frequency	 &Timelag		 &Speed	 \\
$\lambda_{1}$	&$\lambda_{2}$	&Coefficient	&  (mHz)	 &(Units of 45~s)  &(km s$^{-1}$)\\
\hline
\cak		&\ha		&0.82 -- 0.84   &3.65 -- 3.85	 &-8 -- -9	 &$<$2  	 \\
		&		&0.6 -- 0.76	&3.3 -- 3.5	 &0 -- 1	 &2+	 \\
\cak		&\mgb		&0.64 -- 0.76	&0.9 -- 1.1	 &-1 -- -8	 &3--29	 \\
		&		&0.56 -- 0.77	&1.05 -- 1.25	 &0 -- -1	 &23+  	 \\
\cak		&\mgbw		&0.58 -- 0.60	&4.55 -- 4.75	 &-6 -- -7	 &5--7  	 \\
\ha		&\mgb		&0.55 -- 0.67	&1.2 -- 1.4	 &0 -- -5	 &2-22+	 \\
\ha		&\mgbw		&0.67 -- 0.68	&4.15 -- 4.35	 &2		 &11-17  	 \\
\mgb		&\mgbw		&0.80 -- 0.86	&1.48 -- 1.68	 &0 -- 2	 &6--12+	 \\
		&		&0.62 -- 0.63	&4.0 -- 4.2	 &-3		 &4   \\
\end{tabular}
\end{center}
\end{table}
\end{landscape}

\begin{landscape}
\begin{table}
\begin{center}
\caption[Summary of frequencies and correlations for NBP1]{Summary of frequencies (mHz) and correlations for NBP1. See Section~\ref{res_NBP1} for details.}
\label{NBP1}
\begin{tabular}{lrllllllllll}
\hline
\hline
			&From:	&1.00	&1.33	&1.67	&2.00	&2.33	&2.67	&3.00	&3.33	&3.67	&4.00  \\
$\lambda$		&To:	&1.22	&1.56	&1.89	&2.22	&2.56	&2.89	&3.22	&3.56	&3.89	&4.22	\\
\hline
\multicolumn{2}{l}{\cak}         	&       &       &       &       &       &95     &\bf{99}&\bf{99}&       &\bf{95}\\
\multicolumn{2}{l}{\ha	}             	&\bf{99,95}&\bf{99}&    &95     &       &99     &       &       &\bf{99}&\bf{99,95}\\
\multicolumn{2}{l}{\mgb }         	&99,99  &       &       &       &       &99     &95     &95     &       &       \\
\multicolumn{2}{l}{\mgbw}  		&99     &\bf{99,95}&    &99     &       &95     &99     &95     &       &       \\ \\ \\ \\
\end{tabular}
\begin{tabular}{llllll} 
\hline
\hline
		&	       &Correlation   &Frequency	&Timelag	       &Speed       \\
$\lambda_{1}$	&$\lambda_{2}$ &Coefficient   &  (mHz)  	&(Units of 45~s) &(km s$^{-1}$)\\
\hline
\cak		&\ha	       &0.66 -- 0.79  &1.2 -- 1.4	&-3 -- -12	      &$<$6\\
		&	       &0.61 -- 0.72  &2.3 -- 2.5	&4 -- 6 	      &$<$4\\
\cak		&\mgb	       &0.68 -- 0.75  &1.9 -- 2.1	&0		      &\\
\cak		&\mgbw	       &0.57 -- 0.62  &1.73 -- 1.93	&3 -- 7 	      &5--13\\
\ha		&\mgb	       &0.77 -- 0.84  &1.2 -- 1.4	&-3 -- -7	      &1--7 \\
\ha		&\mgbw	       &0.69	      &4.1 -- 4.3	&3		      &7--11\\
\mgb		&\mgbw	       &0.69 -- 0.79  &1.5 -- 1.7	&0 -- 3 	      &4--12+\\
\end{tabular}
\end{center}
\end{table}
\end{landscape}
\newpage

\subsection{NBP7 --Table~\ref{NBP7}}
\label{res_NBP7} 
Correlations across \ha, \mgb\ and \mgbw\ show a possible travelling wave at $\sim$1.9~mHz at a speed around c$_S$. The FFTs also reveal a strong peak around this frequency in the lower chromospheric lines, with a weaker peak in \ha, and the latter also show the appearance of a strong peak around 3.6~mHz. There is also a correlation around 3.6~mHz (\ha\ to \mgb) and at 3.4~mHz (\mgb\ to \mgbw). This is consistent with {\it Tests 1--4}, where a transverse wave at 1.9~mHz travels up through the atmosphere and couples to a longitudinal wave at around twice the original frequency when it reaches the height of \ha\ (core) formation temperature. This longitudinal wave then continues to travel upward and eventually shocks, explaining the lack of power at $\sim$3.5~mHz in \cak. The strong peaks at $\sim$2.0~mHz in \cak\ are unexpected, but do not appear to be correlated with oscillations at the same frequency in the lower chromosphere.

In the 1.33--1.56~mHz range the oscillations are initially weak but increase in power in \mgb\ and \ha, before decreasing again in \cak. However, there is no power at twice this frequency in \cak, and the only correlation is between \cak\ and \mgbw. There is a downward- moving wave between \ha\ and \mgb\ (2.9~mHz) and a slow upward- moving wave between \cak\ and \mgb\ (1.6~mHz). Below 1.3~mHz there is significant power in all wavelengths, but no indication of any correlation. Several frequencies seem to be localised in height, specifically the peaks around 2.5~mHz in the two lower lines, and the peaks at 4.4~mHz in \mgb\ and \cak. These may be driven by events spatially removed from the NBP at each height, e.g. granular overshoot (Lites {\etal} 1993).
  
\subsection{NBP2 --Table~\ref{NBP2}}
\label{res_NBP2} 

Correlations of \cak\ to \mgb\ and \mgbw, and \mgb\ to \mgbw, reveal a possible travelling wave at $\sim$1.4~mHz (also possibly \ha\ to \mgbw). The FFTs also show power at this frequency, up to the \ha\ level. There is some power at the \cak\ level, but it appears inside the COI and hence is not included here. However, \cak\ does contain power around 2.8~mHz, and there is some correlation around 2.6~mHz between \cak\ and \ha. This is all consistent with {\it Tests 1 to 4}, excepting the fact that there is no correlation from \ha\ to \mgb. However, this lack of correlation shows the benefit of multi-wavelength analysis. Although the \ha\ to \mgb\ correlation is missing, we can safely rely on correlations across the other wavelengths. It is worth noting that a transverse frequency of 1.4~mHz, and longitudinal frequency of 2.8~mHz, are low compared to the suggestions of Kalkofen (1997). 

The \cak\ to \mgb\ and \mgbw\ correlations, along with that of \mgb\ to \mgbw, also show a possible $\sim$1.8~mHz wave. There is some power at this frequency in the lower lines and a weak peak in \ha\ at $\sim$3.8~mHz, with a possible travelling wave at $\sim$3.9mHz (\ha\ to \mgb). However, there is no peak in the higher lying lines at the original frequency. Again, there is some power below 1.3~mHz, with no apparent correlation, and several frequencies seem to be localised in height (e.g., $\sim$3.4~mHz in \mgb).

There are several other correlations evident between the lower lines. However in many cases, when the frequencies of these correlation peaks are compared to the FFT tables, there is no associated power in the FFT, or wave-packets in the wavelet power transforms. This demonstrates the advantage of using wavelet analysis to obtain temporal information and allows us to dismiss the 3.07~mHz (\ha\ to \mgb), 3.3~mHz and 4.8~mHz (\ha\ to \mgbw) correlations. The 4.3~mHz correlation (\mgb\ to \mgbw) does correspond to a possible downward- moving wave-packet. Waves at around this frequency have been noted before (Bocchialini \& Baudin 1995). 

\subsection{NBP5 --Table~\ref{NBP5}}
\label{res_NBP5} 

In the 1.2--1.6~mHz range, there appear to be two different correlation patterns. A $\sim$1.3~mHz correlation (\cak\ and \ha\ to \mgb)corresponds to a decrease in power (and hence significance) of this frequency in the upper lines. There is also an appearance of signal at $\sim$2.7~mHz in \ha\ and \cak.  Although this $\sim$2.7~mHz frequency also occurs in the lower lines, it seems to be uncorrelated to the upper lines. However there is a lack of correlation between \ha\ and \cak\ at $\sim$2.7~mHz ({\it Test 4}) (although this could be due to the longitudinal wave shocking) and the waves may be downward- moving. Also, similar to NBP2, it is of a very low frequency compared to Kalkofen (1997). A $\sim$1.5~mHz correlation between \mgb\ and \mgbw\ corresponds to a disappearance of power at this frequency from \mgbw\ to \mgb, as well as the existence of a $\sim$3.1~mHz oscillation in \mgb. However, the lack of a remnant peak ({\it Test 1}) in the upper line makes this a less likely candidate for mode coupling. It would also have coupled too low in the atmosphere to be a viable candidate for chromospheric heating.

The correlation at 4.25~mHz (\ha\ to \mgbw) corresponds to an appearance at this frequency in \ha, and could be the indication of a longitudinal wave appearing at this height. There is also power at 2.1~mHz in the lower lines. However, as there in no correlation at 2.1~mHz, and the power at this frequency does not decrease with height, this is not a good candidate for mode-coupling. There is a further upward- moving wave at 3.4~mHz (\cak\ to \ha).  

There appear to be three downward- moving waves at 1.0~mHz (\cak\ to \mgb), 3.75~mHz (\cak\ to \ha) and 4.65~mHz (\cak\ to \mgbw), corresponding to wave-packets in each wavelength. Bocchialini \& Baudin (1995) also found a downward- moving wave at 3.5~mHz using a similar method and invoked the magneto-gravity waves suggested by Deubner \& Fleck (1990). The 4.1~mHz correlation (\mgb\ to \mgbw) does not correspond to any wave-packets, and is thus dismissed as not being real. Again, there seem to be a few oscillations which are localised in height, mainly in the range 2.67--3.56~mHz in \mgb\ and \mgbw. For the frequencies around 3.33~mHz this could be due to some photospheric leakage. However, as this does not occur in all NBPs, it seems unlikely to be the case.

\subsection{NBP1 --Table~\ref{NBP1}}
\label{res_NBP1} 

The correlations between \cak\ and both \mgb\ and \mgbw\ show a possible travelling wave at $\sim$1.9~mHz (\mgb\ to \mgbw\ also shows a correlation at a slightly lower frequency). This corresponds with an oscillatory signal at $\sim$1.9~mHz in \mgbw\ and reduced power in \ha. There is also a peak at $\sim$4.0~mHz in \ha, which is reduced on reaching the \cak\ level. \ha\ to \mgbw\ shows a travelling wave at $\sim$4.1~mHz ({\it Test 4}). This all indicates a transverse wave of $\sim$1.9~mHz coupling to a longitudinal wave at $\sim$4.1~mHz, leaving behind a small remnant at the original frequency. This longitudinal wave then shocks, thereby explaining the lack of power in lines produced higher up in the chromosphere.

Again, there is also some weak evidence for downward- moving waves. A $\sim$1.3~mHz correlation is found (\cak\ to \ha\ and \ha\ to \mgb), although among these three wavelengths an oscillatory signal at this frequency is found in \ha\ only. Hence the correlation is not necessarily between wave packets. Similarly the $\sim$2.4~mHz correlation (\cak\ to \ha) does not correspond to any wave-packet in either wavelength, and hence is dismissed. As in the other cases, a few frequencies exist which do not correspond to any waves. Below 1.3~mHz there is power up to the \ha\ level. As in NBP5 there are a lot of oscillatory power in the 2.67--3.56~mHz range which does not correspond to any travelling waves.  

\begin{table}
\begin{center}
\caption[Summary of propagating waves]{Summary of propagating waves in all four NBPs. In each case the two wavelengths, wave frequency, and note as to possible mode-coupling transverse (MCT), mode-coupling longitudinal (MCL), upward (U) or downward (D) wave is given. The order of the waves for each NBP reflect the discussion in Sections~\ref{res_NBP7} to \ref{res_NBP1}.}
\label{prop_wave}
\begin{tabular}{llll} 
\hline
\hline
		&		 &Frequency    &Notes\\
$\lambda_{1}$	&$\lambda_{2}$	 &  (mHz)      &\\
\hline
\bf{NBP7}\\
\hline
\ha		&\mgb		 &1.8 -- 2.0   	&MCT\\
\ha		&\mgbw		 &1.8 -- 2.0   	&MCT\\
\mgb		&\mgbw		 &1.8 -- 2.0   	&MCT\\
\mgb		&\mgbw		 &3.3 -- 3.5   	&MCL\\
\ha		&\mgb		 &3.55 -- 3.75 	&MCL\\\\

\cak		&\mgbw		 &1.3 -- 1.5   	&U\\
\ha		&\mgb		 &2.8 -- 3.0   	&D\\
\cak		&\mgb		 &1.5 --1.7    	&U\\

\hline
\bf{NBP2}\\
\hline
\cak		&\mgb		&1.25 -- 1.45 	&MCT\\
\cak		&\mgbw		&1.2 -- 1.4	&MCT\\
\mgb		&\mgbw		&1.3 -- 1.5	&MCT\\
\cak		&\ha		&2.4 -- 2.6	&MCL\\\\

\cak		&\mgb		&1.75 -- 1.95	&U\\
\cak		&\mgbw		&1.7 -- 1.9	&U\\
\mgb		&\mgbw		&1.6 -- 1.8	&U\\
\ha		&\mgb		&3.85 -- 4.05	&U\\\\

\mgb		&\mgbw		&4.2 -- 4.4	&D\\   
\hline
\bf{NBP5}\\
\hline
\cak		&\mgb		&1.05 -- 1.25	&MCT\\
\ha		&\mgb		&1.2 -- 1.4	&MCT\\
\mgb		&\mgbw		&1.48 -- 1.68	&MCT\\\\

\ha		&\mgbw		&4.15 -- 4.35	&U\\
\cak		&\ha		&3.3 -- 3.5	&U\\\\

\cak		&\mgb		&0.9 -- 1.1	&D\\
\cak		&\ha		&3.65 -- 3.85	&D\\
\cak		&\mgbw		&4.55 -- 4.75	&D\\
\hline
\bf{NBP1}\\
\hline
\cak		&\mgb	       &1.9 -- 2.1	&MCT\\
\cak		&\mgbw	       &1.73 -- 1.93	&MCT\\
\mgb		&\mgbw	       &1.5 -- 1.7	&MCT\\
\ha		&\mgbw	       &4.1 -- 4.3	&MCL\\\\

\cak		&\ha	       &1.2 -- 1.4	&D\\
\ha		&\mgb	       &1.2 -- 1.4	&D\\

\end{tabular}
\end{center}
\end{table}

\section{Discussion}
\label{mc_dis}

A summary of all the propagating waves in each NBP is provided in Table~\ref{prop_wave}. The ordering of the waves in this table, and notes as to whether each wave may be a possible mode-coupling candidate, or simply an upward- or downward- moving wave reflects the discussion in Sections~\ref{res_NBP7} to \ref{res_NBP1}. Correlations which do not correspond to wave packets are omitted.

In each NBP, there is at least one frequency which satisfies the five tests described in Section~\ref{mc_intro}. However, a few points must be addressed concerning this. Firstly, Kalkofen (1997) suggests that transverse waves should not normally be viewable at disk centre. If they are viewable it will be due to a flux tube which, being slightly slanted, will make a small angle with the line-of-sight. In this case the resulting light curve will have two maxima for each complete oscillation of the wave and hence the frequency detected will be twice that of the wave. However, this will only be true for completely symmetrical profiles. If the line profile is asymmetric, or if the filter is in the wing of the line, the Doppler shift from the transverse wave will result in a light curve which will still contain a signal at the original transverse wave frequency. 

Secondly, the main area of disagreement with theory is in the values of the cut-off frequencies determined from Kalkofen (1997). As stated in Section~\ref{mc_intro} it should be noted that the cut-offs calculated have not been derived specifically for the network. In the network several parameters may vary, due to the highly magnetised structure and local depression, or non-adiabatic effects, which may alter ${\nu}_{k}$. A complete understanding of the structure within the network is necessary, which may introduce other factors into the equations given by Kalkofen (1997) for the cut-off frequencies. It is also apparent that the thin flux tube approximation will not be appropriate in the high chromosphere. Further modelling using thick flux tubes will be necessary. The results here suggest transverse cut-offs of $\sim$1.3~mHz, and $\sim$1.9~mHz, with longitudinal cut-offs at around twice these frequencies. Hasan \& Kalkofen (1999) suggest cut-off periods of 534~s and 227~s for the transverse and longitudinal mode respectively, which agree with the higher frequency (1.9~mHz) cut-off suggested here. 

Thirdly, results such as those presented here have never before been so clearly demonstrated. There are several reasons for this. It is important to note that our method uses 2-dimensional images, making it easier to spatially isolate the NBP throughout the entire time series (as first pointed out by Cauzzi {\etal} 2000). The contour method also ensures only pixels inside the NBP are included, ignoring any other bright pixels in the FOV. In addition, some oscillatory components can be missed if each NBP is not followed completely from the outer reaches to the bright centre. In Chapter~4 I demonstrated how spatially averaging over many NBPs can lead to overlooking power at several frequencies. The contrast light curve and digital high-pass filtering techniques used here also remove all very low frequency power which can dominate the power spectrum (Figures~\ref{cak_digifilt} and~\ref{cak_filtered}). It is of course important to remove this low-frequency component without unduly affecting the higher frequencies. It is also of note that the time series used here (150 minutes) is longer than most previous studies. Most importantly, the use of wavelet analysis to create the power curves gives temporal information associated with any wave-packets. A Fourier phase analysis, which will average over time, may result in any correlation across wave-packets being `washed out' by the longer non-periodic component. This may explain the lack of correlation found in previous work (e.g., Lites {\etal} 1993). It is also important to note the multi-wavelength nature of the data used in this study. Although a few papers have previously performed correlations across wavelet power diagrams (Bocchialini \& Baudin 1995; Baudin, Bocchialini, \& Koutchmy 1996), they have used only two wavelengths in a limited frequency range. Using four wavelengths across all observable frequencies gives a much higher chance of detecting correlated oscillations, and makes the findings more significant. 

Finally, the results reported here suggest longitudinal waves shock in the mid- to high- chromosphere, hence heating the surrounding plasma, and the oscillations subsequently disappear. However simulations by Carlsson \& Stein (1992) of K$_{2V}$ grains suggest that these shocks may be coherent which will lead to further oscillations at 5.5~mHz. This lack of agreement suggests that either the magnetic network differs from the internetwork bright points, or the coherence area must be well less than the areas integrated over in this paper.

In this chapter I have used a novel combination of wavelet analysis and cross-correlation to study propagating waves in the chromosphere. For each NBP studied I have found the possibility of transverse wave propagation in the lower chromosphere. There is also evidence of these transverse waves coupling to longitudinal waves in the upper chromosphere, which can then shock. Thus mode-coupling provides a means of energy transport to heat the upper chromosphere. In the next chapter I will present another novel wavelet analysis technique to study oscillatory signals in {\sl TRACE} ultraviolet datasets.
