### 4.6 MHD waves in coronal loops

Besides fast-mode MHD waves, which can be subdivided into kink-mode (asymmetric, see Section 4.5)
and sausage-mode (symmetric) types, there are also slow-mode MHD waves, which propagate with acoustic
(sound) speed. The sound speed can be expressed in terms of the electron temperature (assuming
the coronal approximation , although according to some observations),
where is the adiabatic index, the thermal pressure, the mass
density, and the mean molecular weight. However, if a sound wave propagates at
an angle of to the line-of-sight, the observed speed scales with the sine function of the
line-of-sight angle, i.e., . This projection angle can be determined by stereoscopic
triangulation of the loop structure that guides the sound wave propagating in longitudinal
direction.
Such a measurement has been conducted for slow-mode (acoustic) waves observed with STEREO/A and
B on 2008 Jan 10 (Marsh et al., 2009). The propagating waves were observed in a fan-like loop structure
emanating from one magnetic pole of a bipolar active region (Figure 38). A time-slice plot of the EUV flux
along the fans as a function of time is shown in Figure 39, which confirms the quasi-periodic wave
propagation (diagonal pattern), for which a velocity of was measured with
STEREO/A, and with STEREO/B, respectively. The wave period was
found to be min. Stereoscopic triangulation of the fans yielded an inclination angle
of to the local vertical. 3D projections of the triangulated directions are also
shown in Figure 38 (top panels). Correcting the observed wave speed for the corresponding
line-of-sight angle yields a corrected speed of and ,
which represents the first measurement of the true coronal longitudinal slow-mode speed in
3D. Inserting this true sound speed into Eq. (43), a temperature of is
inferred, which is close to the peak sensitivity of the used 171 Å passband. The temperature
structure of the same fan loop was also investigated with EIS/Hinode and a mean temperature of
was obtained, in excellent agreement with the EUVI measurements, which
confirms that the slow-mode phase speed is identical to the sound speed (Marsh and Walsh,
2009).