Quasi-periodic Oscillations of Solar Active Regions in Connection with Their Flare Activity – NoRH Observations

The sunspot-associated sources at the frequency of 17 GHz give information on plasma parameters in the regions of magnetic field about B=2000 G at the level of the chromosphere-corona transition region. The observations of short period (from one to ten minutes) oscillations in sunspots reflect propagation of magnetohydrodynamic (MHD) waves in the magnetic flux tubes of the sunspots. We investigate the oscillation parameters in active regions in connection with their flare activity. We confirm the existence of a link between the oscillation spectrum and flare activity. We find differences in the oscillations between pre-flare and post-flare phases. In particular, we demonstrate a case of powerful three-minute oscillations that start just before the burst. This event is similar to the cases of the precursors investigated by Sych et al. (Astron. Astrophys. 505, 791, 2009). We also found well-defined eight-minute oscillations of microwave emission from sunspot. We interpret our observations in terms of a relationship between MHD waves propagating from sunspots and flare processes.     

The Pixelised Wavelet Filtering Method to Study Waves and Oscillations in Time Sequences of Solar Atmospheric Images

Pixelised wavelet filtering (PWF) for the determination of the spatial, temporal, and phase structure of oscillation sources in temporal sequences of 2D images, based upon the continuous wavelet transform, has been designed and tested. The PWF method allows us to obtain information about the presence of propagating and nonpropagating waves in the data and localise them precisely in time and in space. The method is tested on the data sets obtained in microwaves with the Nobeyama Radioheliograph and in the EUV with TRACE. The method reveals fine spatial structuring of the sources of 3-, 5-, and 15-minute periodicities in the microwave and EUV emission generated in sunspot atmospheres. In addition, the PWF method provides us with unique information about the temporal variability of the power, amplitude, and phase narrowband maps of the observed oscillations and waves. The applicability of the method to the analysis of coronal wave phenomena is discussed.     

On the Nature and Genesis of EUV Waves: A Synthesis of Observations from SOHO, STEREO, SDO, and Hinode (Invited Review)

A major, albeit serendipitous, discovery of the SOlar and Heliospheric Observatory mission was the observation by the Extreme Ultraviolet Telescope (EIT) of large-scale extreme ultraviolet (EUV) intensity fronts propagating over a significant fraction of the Sun??s surface. These so-called EIT or EUV waves are associated with eruptive phenomena and have been studied intensely. However, their wave nature has been challenged by non-wave (or pseudo-wave) interpretations and the subject remains under debate. A string of recent solar missions has provided a wealth of detailed EUV observations of these waves bringing us closer to resolving the question of their nature. With this review, we gather the current state-of-the-art knowledge in the field and synthesize it into a picture of an EUV wave driven by the lateral expansion of the CME. This picture can account for both wave and pseudo-wave interpretations of the observations, thus resolving the controversy over the nature of EUV waves to a large degree but not completely. We close with a discussion on several remaining open questions in the field of EUV waves research.     

Coronal Shock Waves,EUV Waves,and Their Relation to CMEs. III. Shock-Associated CME/EUV Wave in an Event with a Two-Component EUV Transient

On 17 January 2010, STEREO-B observed in extreme ultraviolet (EUV) and white light a large-scale dome-shaped expanding coronal transient with perfectly connected off-limb and on-disk signatures. Veronig et al. (Astrophys. J. Lett. 716, L57, 2010) concluded that the dome was formed by a weak shock wave. We have revealed two EUV components, one of which corresponded to this transient. All of its properties found from EUV, white light, and a metric type II burst match expectations for a freely expanding coronal shock wave, including correspondence with the fast-mode speed distribution, while the transient sweeping over the solar surface had a speed typical of EUV waves. The shock wave was presumably excited by an abrupt filament eruption. Both a weak shock approximation and a power-law fit match kinematics of the transient near the Sun. Moreover, the power-law fit matches the expansion of the CME leading edge up to 24 solar radii. The second, quasi-stationary EUV component near the dimming was presumably associated with a stretched CME structure; no indications of opening magnetic fields have been detected far from the eruption region.     

Present and Future Observing Trends in Atmospheric Magnetoseismology

With modern imaging and spectral instruments observing in the visible, EUV, X-ray, and radio wavelengths, the detection of oscillations in the solar outer atmosphere has become a routine event. These oscillations are considered to be the signatures of a wave phenomenon and are generally interpreted in terms of magnetohydrodynamic (MHD) waves. With multiwavelength observations from ground- and space-based instruments, it has been possible to detect waves in a number of different wavelengths simultaneously and, consequently, to study their propagation properties. Observed MHD waves propagating from the lower solar atmosphere into the higher regions of the magnetized corona have the potential to provide excellent insight into the physical processes at work at the coupling point between these different regions of the Sun. High-resolution wave observations combined with forward MHD modeling can give an unprecedented insight into the connectivity of the magnetized solar atmosphere, which further provides us with a realistic chance to reconstruct the structure of the magnetic field in the solar atmosphere. This type of solar exploration has been termed atmospheric magnetoseismology. In this review we will summarize some new trends in the observational study of waves and oscillations, discussing their origin and their propagation through the atmosphere. In particular, we will focus on waves and oscillations in open magnetic structures (e.g., solar plumes) and closed magnetic structures (e.g., loops and prominences), where there have been a number of observational highlights in the past few years. Furthermore, we will address observations of waves in filament fibrils allied with a better characterization of their propagating and damping properties, the detection of prominence oscillations in UV lines, and the renewed interest in large-amplitude, quickly attenuated, prominence oscillations, caused by flare or explosive phenomena.     

Advances in Observing Various Coronal EUV Waves in the SDO Era and Their Seismological Applications (Invited Review)

Global extreme-ultraviolet (EUV) waves are spectacular traveling disturbances in the solar corona associated with energetic eruptions such as coronal mass ejections (CMEs) and flares. Over the past 15 years, observations from three generations of space-borne EUV telescopes have shaped our understanding of this phenomenon and at the same time led to controversy about its physical nature. Since its launch in 2010, the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) has observed more than 210 global EUV waves in exquisite detail, thanks to its high spatio–temporal resolution and full-disk, wide-temperature coverage. A combination of statistical analysis of this large sample, more than 30 detailed case studies, and data-driven MHD modeling, has been leading their physical interpretations to a convergence, favoring a bimodal composition of an outer, fast-mode magnetosonic wave component and an inner, non-wave CME component. Adding to this multifaceted picture, AIA has also discovered new EUV wave and wave-like phenomena associated with various eruptions, including quasi-periodic fast propagating (QFP) wave trains, magnetic Kelvin–Helmholtz instabilities (KHI) in the corona and associated nonlinear waves, and a variety of mini-EUV waves. Seismological applications using such waves are now being actively pursued, especially for the global corona. We review such advances in EUV wave research focusing on recent SDO/AIA observations, their seismological applications, related data-analysis techniques, and numerical and analytical models.     

Oscillatory Response of the 3D Solar Atmosphere to the Leakage of Photospheric Motion

The direct propagation of acoustic waves, driven harmonically at the solar photosphere, into the three-dimensional solar atmosphere is examined numerically in the framework of ideal magnetohydrodynamics. It is of particular interest to study the leakage of 5-minute global solar acoustic oscillations into the upper, gravitationally stratified and magnetised atmosphere, where the modelled solar atmosphere possesses realistic temperature and density stratification. This work aims to complement and bring further into the 3D domain our previous efforts (by Erdélyi et al., 2007, Astron. Astrophys. 467, 1299) on the leakage of photospheric motions and running magnetic-field-aligned waves excited by these global oscillations. The constructed model atmosphere, most suitable perhaps for quiet Sun regions, is a VAL IIIC derivative in which a uniform magnetic field is embedded. The response of the atmosphere to a range of periodic velocity drivers is numerically investigated in the hydrodynamic and magnetohydrodynamic approximations. Among others the following results are discussed in detail: i) High-frequency waves are shown to propagate from the lower atmosphere across the transition region, experiencing relatively low reflection, and transmitting most of their energy into the corona; ii) the thin transition region becomes a wave guide for horizontally propagating surface waves for a wide range of driver periods, and particularly at those periods that support chromospheric standing waves; iii) the magnetic field acts as a waveguide for both high- and low-frequency waves originating from the photosphere and propagating through the transition region into the solar corona. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.     

An EUV Jet and Hα Filament Eruption Associated with Flux Cancelation in a Decaying Active Region

Using data from the Transition Region and Coronal Explorer (TRACE), Solar and Heliospheric Observatory (SOHO), Ramaty High Energy Solar Spectroscopic Imager (RHESSI), and Hida Observatory (HO), we present a detailed study of an EUV jet and the associated Hα filament eruption in a major flare in the active region NOAA 10044 on 29 July 2002. In the Hα line wings, a small filament was found to erupt out from the magnetic neutral line of the active region during the flare. Two bright EUV loops were observed rising and expanding with the filament eruption, and both hot and cool EUV plasma ejections were observed to form the EUV jet. The two thermal components spatially separated from each other and lasted for about 25 minutes. In the white-light corona data, a narrow coronal mass ejection (CME) was found to respond to this EUV jet. We cannot find obvious emerging flux in the photosphere accounting for the filament eruption and the EUV jet. However, significant sunspot decay and magnetic-flux cancelation owing to collision of opposite flux before the events were noticed. Based on the hard X-ray data from RHESSI, which showed evidence of magnetic reconnection along the main magnetic neutral line, we think that all of the observed dynamical phenomena, including the EUV jet, filament eruption, flare, and CME, should have a close relation to the flux cancelation in the low atmosphere.     

Oscillations in EUV emission lines during a loop brightening

Oscillations in the emission in the ultraviolet lines of Cii, Oiv, and Mg x, detected by the Harvard College Observatory EUV spectroheliometer on Skylab are observed on August 7, 1973, during a loop brightening. The intensity of the EUV lines varies with a period of 141 s during the time of enhanced intensity of the coronal loop, lasting 10 min. The periodic oscillation is not only localized in the loop region but extends over a larger area of the active region, maintaining the same phase. We suggest that the intensity fluctuation of the EUV lines is caused by small-amplitude waves, propagating in the plasma confined in the magnetic loop and that size of the loop might be important in determining its perferential heating in the active region.On leave from the University of Torino, Italy.     

Arbitrary amplitude ion-acoustic solitary waves in superthermal electron-positron-ion magnetoplasma

Arbitrary amplitude ion-acoustic solitary waves propagating in a magnetized plasma composed of positive ions, superthermal electrons and positrons are investigated. For this purpose, the ions are represented by the hydrodynamical fluid equations while the non-Maxwellian electrons and positrons densities are assumed to follow kappa (κ) distribution. The basic equations are reduced to a pseudoenergy-balance equation. Existence conditions for large amplitude solitary waves are presented. The analytical and numerical analysis of the latter show that the ion-acoustic solitary wave can propagate only in the subsonic region in our plasma system and it is significantly influenced by the plasma parameters. The present analysis could be helpful for understanding the nonlinear ion-acoustic solitary waves propagating in interstellar medium and pulsar wind, which contain an excess of superthermal particles.     

Ion-Sound Model of Microwave Spikes with Fast Shocks in the Reconnection Region

A new model for solar spike bursts is considered based on the interaction of Langmuir waves with ion-sound waves: l+st. Such a mechanism can operate in shock fronts, propagating from a magnetic reconnection region. New observations of microwave millisecond spikes are discussed. They have been observed in two events: 4 November 1997 between 05:52–06:10 UT and 28 November 1997 between 05:00–05:10 UT using the multichannel spectrograph in the range 2.6–3.8 GHz of Beijing AO. Yohkoh/SXT images in the AR and SOHO EIT images testify to a reconstruction of bright loops after the escape of a CME. A fast shock front might be manifested as a very bright line in T _e SXT maps (up to 20 MK) above dense structures in emission measure (EM) maps. Moreover one can see at the moment of spike emission (for the 28 November 1997 event) an additional maximum at the loop top on the HXR map in the AR as principal evidence of fast shock propagation. The model gives the ordinary mode of spike emission. Sometimes we observed a different polarization of microwave spikes that might be connected with the depolarization of the emission in the transverse magnetic field and rather in the vanishing magnetic field in the middle of the QT region. Duration and frequency band of isolated spikes are connected with parameters of fast particle beams and shock front. Millisecond microwave spikes are probably a unique manifestation of flare fast shocks in the radio emission.     

Multiwavelength Study of a Solar Eruption from AR NOAA 11112: II. Large-Scale Coronal Wave and Loop Oscillation

We analyze multiwavelength observations of an M2.9/1N flare that occurred in AR NOAA 11112 on 16 October 2010. AIA 211 Å EUV images reveal the presence of a faster coronal wave (decelerating from ≈?1390 to ≈?830 km?s^?1) propagating ahead of a slower wave (decelerating from ≈?416 to ≈?166 km?s^?1) towards the western limb. The dynamic radio spectrum from Sagamore Hill radio telescope shows the presence of a metric type II radio burst, which reveals the presence of a coronal shock wave (speed ≈?800 km?s^?1). The speed of the faster coronal wave, derived from AIA 211 Å images, is found to be comparable to the coronal shock speed. AIA 171 Å high-cadence observations showed that a coronal loop, which was located at a distance of ≈?0.32R _⊙ to the west of the flaring region, started to oscillate by the end of the impulsive phase of the flare. The results indicate that the faster coronal wave may be the first driver of the transversal oscillations of coronal loop. As the slower wave passed through the coronal loop, the oscillations became even stronger. There was a plasmoid eruption observed in EUV and a white-light CME was recorded, having velocity of ≈?340?–?350 km?s^?1. STEREO 195 Å images show an EIT wave, propagating in the same direction as the lower-speed coronal wave observed in AIA, but decelerating from ≈?320 to ≈?254 km?s^?1. These observations reveal the co-existence of both waves (i.e. coronal Moreton and EIT waves), and the type II radio burst seems to be associated with the coronal Moreton wave.     

The Magnetic Structure and Generation of EUV Flare Ribbons

The `ribbons' of two-ribbon flares show complicated patterns reflecting the linkages of coronal magnetic field lines through the lower solar atmosphere. We describe the morphology of the EUV ribbons of the July 14, 2000 flare, as seen in SOHO, TRACE, and Yohkoh data, from this point of view. A successful co-alignment of the TRACE, SOHO/MDI and Yohkoh/HXT data has allowed us to locate the EUV ribbon positions on the underlying field to within ∼ 2′′, and thus to investigate the relationship between the ribbons and the field, and also the sites of electron precipitation. We have also made a determination of the longitudinal magnetic flux involved in the flare reconnection event, an important parameter in flare energetic considerations. There are several respects in which the observations differ from what would be expected in the commonly-adopted models for flares. Firstly, the flare ribbons differ in fine structure from the (line-of-sight) magnetic field patterns underlying them, apparently propagating through regions of very weak and probably mixed polarity. Secondly, the ribbons split or bifurcate. Thirdly, the amount of line-of-sight flux passed over by the ribbons in the negative and positive fields is not equal. Fourthly, the strongest hard X-ray sources are observed to originate in stronger field regions. Based on a comparison between HXT and EUV time-profiles we suggest that emission in the EUV ribbons is caused by electron bombardment of the lower atmosphere, supporting the hypothesis that flare ribbons map out the chromospheric footpoints of magnetic field lines newly linked by reconnection. We describe the interpretation of our observations within the standard model, and the implications for the distribution of magnetic fields in this active region.     

The Relation Between Large-Scale Coronal Propagating Fronts and Type II Radio Bursts

Large-scale, wave-like disturbances in extreme-ultraviolet (EUV) and type II radio bursts are often associated with coronal mass ejections (CMEs). Both phenomena may signify shock waves driven by CMEs. Taking EUV full-disk images at an unprecedented cadence, the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory has observed the so-called EIT waves or large-scale coronal propagating fronts (LCPFs) from their early evolution, which coincides with the period when most metric type II bursts occur. This article discusses the relation of LCPFs as captured by AIA with metric type II bursts. We show examples of type II bursts without a clear LCPF and fast LCPFs without a type II burst. Part of the disconnect between the two phenomena may be due to the difficulty in identifying them objectively. Furthermore, it is possible that the individual LCPFs and type II bursts may reflect different physical processes and external factors. In particular, the type II bursts that start at low frequencies and high altitudes tend to accompany an extended arc-shaped feature, which probably represents the 3D structure of the CME and the shock wave around it, and not just its near-surface track, which has usually been identified with EIT waves. This feature expands and propagates toward and beyond the limb. These events may be characterized by stretching of field lines in the radial direction and may be distinct from other LCPFs, which may be explained in terms of sudden lateral expansion of the coronal volume. Neither LCPFs nor type II bursts by themselves serve as necessary conditions for coronal shock waves, but these phenomena may provide useful information on the early evolution of the shock waves in 3D when both are clearly identified in eruptive events.     

Multilevel Analysis of Oscillation Motions in Active Regions of the Sun

The nature of the three-minute and five-minute oscillations observed in sunspots is considered to be an effect of propagation of magnetohydrodynamic (MHD) waves from the photosphere to the solar corona. However, the real modes of these waves and the nature of the filters that result in rather narrow frequency bands of these modes are still far from being generally accepted, in spite of a large amount of observational material obtained in a wide range of wave bands. The significance of this field of research is based on the hope that local seismology can be used to find the structure of the solar atmosphere in magnetic tubes of sunspots. We expect that substantial progress can be achieved by simultaneous observations of the sunspot oscillations in different layers of the solar atmosphere in order to gain information on propagating waves. In this study we used a new method that combines the results of an oscillation study made in optical and radio observations. The optical spectral measurements in photospheric and chromospheric lines of the line-of-sight velocity were carried out at the Sayan Solar Observatory. The radio maps of the Sun were obtained with the Nobeyama Radioheliograph at 1.76 cm. Radio sources associated with the sunspots were analyzed to study the oscillation processes in the chromosphere – corona transition region in the layer with magnetic field B=2000 G. A high level of instability of the oscillations in the optical and radio data was found. We used a wavelet analysis for the spectra. The best similarities of the spectra of oscillations obtained by the two methods were detected in the three-minute oscillations inside the sunspot umbra for the dates when the active regions were situated near the center of the solar disk. A comparison of the wavelet spectra for optical and radio observations showed a time delay of about 50 seconds of the radio results with respect to the optical ones. This implies an MHD wave traveling upward inside the umbral magnetic tube of the sunspot. For the five-minute oscillations the similarity in spectral details could be found only for optical oscillations at the chromospheric level in the umbral region or very close to it. The time delays seem to be similar. Besides three-minute and five-minute ones, oscillations with longer periods (8 and 15 minutes) were detected in optical and radio records. Their nature still requires further observational and theoretical study for even a preliminary discussion.     

Coronal Shock Waves,EUV Waves,and Their Relation to CMEs. II. Modeling MHD Shock Wave Propagation Along the Solar Surface,Using Nonlinear Geometrical Acoustics

We model the propagation of a coronal shock wave, using nonlinear geometrical acoustics. The method is based on the Wentzel–Kramers–Brillouin (WKB) approach and takes into account the main properties of nonlinear waves: i) dependence of the wave front velocity on the wave amplitude, ii) nonlinear dissipation of the wave energy, and iii) progressive increase in the duration of solitary shock waves. We address the method in detail and present results of the modeling of the propagation of shock-associated extreme-ultraviolet (EUV) waves as well as Moreton waves along the solar surface in the simplest solar corona model. The calculations reveal deceleration and lengthening of the waves. In contrast, waves considered in the linear approximation keep their length unchanged and slightly accelerate.     

Two dimensional electromagnetic shock structures in dense electron-positron-ion magnetoplasmas

Linear and nonlinear analysis of low frequency magnetoacoustic waves propagating at an angle θ with the ambient magnetic field are investigated in dense electron-positron-ion (e-p-i) plasmas using the quantum magnetohydrodynamic (QMHD) model. In this regard, a quantum Kadomtsev-Petviashvili-Burgers (KPB) equation is derived in the small amplitude limit. The stability of KPB equation is also presented. The variation of the nonlinear fast and slow magnetoacoustic shock waves with the positron concentration, kinematic viscosity, obliqueness parameter θ, and the magnetic field, are also investigated. It is observed that the aforementioned plasma parameters significantly modify the propagation characteristics of two dimensional nonlinear magnetoacoustic shock waves in dissipative quantum magnetoplasmas. The relevance of the present investigation with regard to dense astrophysical environments is also pointed out.     

A New Model for Propagating Parts of EIT Waves: A Current Shell in a CME

EIT waves are observed in EUV as bright fronts. Some of these bright fronts propagate across the solar disk. EIT waves are all associated with a flare and a CME and are commonly interpreted as fast-mode magnetosonic waves. Propagating EIT waves could also be the direct signature of the gradual opening of magnetic field lines during a CME. We quantitatively addressed this alternative interpretation. Using two independent 3D MHD codes, we performed nondimensional numerical simulations of a slowly rotating magnetic bipole, which progressively result in the formation of a twisted magnetic flux tube and its fast expansion, as during a CME. We analyse the origins, the development, and the observability in EUV of the narrow electric currents sheets that appear in the simulations. Both codes give similar results, which we confront with two well-known SOHO/EIT observations of propagating EIT waves (7 April and 12 May 1997), by scaling the vertical magnetic field components of the simulated bipole to the line of sight magnetic field observed by SOHO/MDI and the sign of helicity to the orientation of the soft X-ray sigmoids observed by Yohkoh/SXT. A large-scale and narrow current shell appears around the twisted flux tube in the dynamic phase of its expansion. This current shell is formed by the return currents of the system, which separate the twisted flux tube from the surrounding fields. It intensifies as the flux tube accelerates and it is co-spatial with weak plasma compression. The current density integrated over the altitude has the shape of an ellipse, which expands and rotates when viewed from above, reproducing the generic properties of propagating EIT waves. The timing, orientation, and location of bright and faint patches observed in the two EIT waves are remarkably well reproduced. We conjecture that propagating EIT waves are the observational signature of Joule heating in electric current shells, which separate expanding flux tubes from their surrounding fields during CMEs or plasma compression inside this current shell. We also conjecture that the bright edges of halo CMEs show the plasma compression in these current shells.     

Statistical Analysis of Large-Scale EUV Waves Observed by STEREO/EUVI

We statistically analyzed the kinematical evolution and wave pulse characteristics of 60 strong large-scale EUV wave events that occurred during January 2007 to February 2011 with the STEREO twin spacecraft. For the start velocity, the arithmetic mean is 312±115 km?s^?1 (within a range of 100?–?630 km?s^?1). For the mean (linear) velocity, the arithmetic mean is 254±76 km?s^?1 (within a range of 130?–?470 km?s^?1). 52 % of all waves under study show a distinct deceleration during their propagation (a≤?50 m?s^?2), the other 48 % are consistent with a constant speed within the uncertainties (?50≤a≤50 m?s^?2). The start velocity and the acceleration are strongly anticorrelated with c≈?0.8, i.e. initially faster events undergo stronger deceleration than slower events. The (smooth) transition between constant propagation for slow events and deceleration in faster events occurs at an EUV wave start-velocity of v≈230 km?s^?1, which corresponds well to the fast-mode speed in the quiet corona. These findings provide strong evidence that the EUV waves under study are indeed large-amplitude fast-mode MHD waves. This interpretation is also supported by the correlations obtained between the peak velocity and the peak amplitude, impulsiveness, and build-up time of the disturbance. We obtained the following association rates of EUV wave events with other solar phenomena: 95 % are associated with a coronal mass ejection (CME), 74 % to a solar flare, 15 % to interplanetary type II bursts, and 22 % to coronal type II bursts. These findings are consistent with the interpretation that the associated CMEs are the driving agents of the EUV waves.     

Solar TErrestrial Relations Observatory-A (STEREO-A) and PRoject for On-Board Autonomy 2 (PROBA2) Quadrature Observations of Reflections of Three EUV Waves from a Coronal Hole

We investigate the interaction of three consecutive large-scale coronal waves with a polar coronal hole, simultaneously observed on-disk by the Solar TErrestrial Relations Observatory (STEREO)-A spacecraft and on the limb by the PRoject for On-Board Autonomy 2 (PROBA2) spacecraft on 27 January 2011. All three extreme ultraviolet (EUV) waves originate from the same active region, NOAA 11149, positioned at N30E15 in the STEREO-A field of view and on the limb in PROBA2. For the three primary EUV waves, we derive starting velocities in the range of ≈?310 km?s^?1 for the weakest up to ≈?500 km?s^?1 for the strongest event. Each large-scale wave is reflected at the border of the extended coronal hole at the southern polar region. The average velocities of the reflected waves are found to be smaller than the mean velocities of their associated direct waves. However, the kinematical study also reveals that in each case the ending velocity of the primary wave matches the initial velocity of the reflected wave. In all three events, the primary and reflected waves obey the Huygens–Fresnel principle, as the incident angle with ≈?10° to the normal is of the same magnitude as the angle of reflection. The correlation between the speed and the strength of the primary EUV waves, the homologous appearance of both the primary and the reflected waves, and in particular the EUV wave reflections themselves suggest that the observed EUV transients are indeed nonlinear large-amplitude MHD waves.