On the understanding of pulsations in the atmosphere of roAp stars: phase diversity and false nodes
Studies based on high-resolution spectroscopic data of rapidly oscillating Ap stars show a surprising diversity of pulsation behavior in the atmospheric layers, pointing, in particular, to the co-existence of running and standing waves. The correct interpretation of these data requires a careful modelling of pulsations in these magnetic stars. In light of this, in this work we present a theoretical analysis of pulsations in roAp stars, taking into account the direct influence of the magnetic field. We derive approximate analytical solutions for the displacement components parallel and perpendicular to the direction of the magnetic field, that are appropriate to the outermost layer. From these, we determine the expression for the theoretical radial velocity for an observer at a general position, and compute the corresponding pulsation amplitude and phase as function of height in the atmosphere. We show that the integral for the radial velocity has contributions from three different types of wave solutions, namely, running waves, evanescent waves, and standing waves of nearly constant amplitude. We then consider a number of case studies to illustrate the origin of the different pulsational behaviour that is found in the observations. Concerning pulsation amplitude, we find that it generally increases with atmospheric height. Pulsation phase, however, shows a diversity of behaviours, including phases that are constant, increasing, or decreasing with atmospheric height. Finally, we show that there are situations in which the pulsation amplitude goes through a zero, accompanied by a phase jumps of , and argue that such behaviour does not correspond to a pulsation node in the outermost layers of the star, but rather to a visual effect, resulting from the observers inability to resolve the stellar surface.
keywords:roAp stars – radial velocity – running and standing waves.
The rapidly oscillating Ap (roAp) stars are found among the coolest subgroup of chemically peculiar Ap stars, which is located in the main sequence part of the classical instability strip. They pulsate with periods typically within the range from 5 to 21 minutes, (e.g. Kurtz et al. (2005)), have oscillations with amplitudes between 0.5 and 5 km s in velocity, and, as in other Ap stars, have strong, large scale magnetic fields, with typical intensities of a few kG, although in some stars the magnetic field strength can be higher than 20 kG,(e.g, Hubrig et al. (2009)). Moreover, they have about two solar masses (Kurtz, 1990), and temperatures that range from about 6400 to 8100 K (Kochukhov (2009)).
The present number of known roAp stars is more than 40. Due to their characteristics, with roAp stars we have the unique opportunity to observe the interaction of acoustic modes with strong large-scale, magnetic fields. The roAp stars have been observed photometrically since their discovery by Kurtz (1982) and until recently the knowledge about their acoustic oscillations was essentially based on high-speed, ground-based, photometric observations. However, in the past few years numerous exciting observational results have been published, both as a result of the acquisition of space-based photometry (e.g. Gruberbauer et al., 2008; Bruntt et al., 2009; Balona et al., 2010) and as a result of the analysis of high-resolution spectroscopic data (e.g. Kurtz et al., 2006a, b; Ryabchikova et al., 2007; Sachkov et al., 2007). Such high-resolution spectroscopic data hold unique information about the structure and dynamics of the peculiar atmospheres of roAp stars, and have revealed a surprising diversity in the pulsation behavior of different lines in the roAp spectra.
The general picture that emerges from the analysis of time-series of high-resolution spectroscopic data of roAp stars is that pulsational variability is seen predominantly in lines of rare-earth ions, especially those of Pr and Nd, which are strong and numerous in the roAp spectra and are formed in the higher layers of the atmosphere. On the other hand, lines of light and iron-peak elements, enhanced in the lower atmospheric layers, often do not show pulsation variability within the observational detection limit for most roAp stars (e.g. Ryabchikova et al., 2002). Thus, generally pulsations are found to be weak or non-detectable in the lower atmosphere, while often reach amplitudes of several km s higher in the atmosphere.
Another interesting feature that can be inferred from spectroscopic observations of roAp stars is the presence of significant shifts in pulsation phase when comparing radial velocities derived from lines of different rare-earth elements, or even from different lines of the same element (e.g. Kochukhov & Ryabchikova, 2001). Likewise, phase shifts are seen when performing depth-in-line analysis of the H and Nd III 6145 Å lines, (e.g. Kurtz et al., 2006a). Along with the phase shifts, depth variations of pulsation amplitude are also inferred, which in the generality of the cases point to an increase of amplitude with atmospheric height. Moreover, in some stars the radial velocity amplitudes measured in lines of the rare-earth ion Pr III and in the core of the H line, are seen to vary on a time scale of a few pulsation cycles (Kurtz et al., 2006b).
The phase behaviour derived from spectroscopic studies of roAp stars is usually interpreted as resulting from the presence of running waves, or standing waves, in the atmosphere, depending, respectively, on whether, or not, the phase shows a variation with depth. In particular, it was found by Sachkov et al. (2007) and Ryabchikova et al. (2007), that for stars with pulsation frequencies below the acoustic cut-off frequency, the pulsations seem to have standing wave character in the deeper layers and then, in the outer layers, behave like running waves. On the other hand, for stars which have pulsation frequency close to, or higher than, the cut-off frequency, the authors found that the pulsations behave like running waves from the deepest layers.
One important and interesting feature that can be seen in spectroscopic data of roAp stars is the existence of pulsation zeros in the amplitude diagram accompanied by phase jumps in the phase diagram. This was interpreted as the presence of a pulsation node at some depth in the atmosphere. The first alleged node found in a roAp star was detected by Baldry et al. (1998) in the atmosphere of Cir. More recently, in the case of the star 10Aql, Elkin et al. (2008) found that the pulsation phase derived from the lines of Tb III and Dy III differ by . According to the authors this could be an indication of a radial node between the line-forming layers of these elements. Moreover, they found that the line bisectors for strong NdIII line profiles show significant changes of phase, or even phase jumps, as in the case of the line NdIII at 5102.41 Å. For this line, the authors found a phase shift of and at the same depth, an amplitude close to zero, which suggests the existence of a radial node. In addition, Mkrtichian et al. (2003) studied the radial velocity variations in the roAp star 33 Librae. The authors found that the NdIII lines pulsate nearly in anti-phase with those of NdII, a feature that the authors attribute to the presence of a pulsation node in the atmosphere of this star. The pulsational behaviour of the Nd ii and Nd iii lines in 33 Lib has also been discussed by Kurtz et al. (2005). They found that the line depth versus pulsation amplitude supported the hypothesis of a node between the line-forming layers of these two ions. Ryabchikova et al. (2007) found similar results for this star, confirming a phase jump between the NdIII and NdII lines, and the amplitude decreasing towards zero at the same atmospheric height.
The results of the our theoretical analysis will be presented in two separate papers, as detailed in sec. 2. In sec. 3 we discuss the physics underlying our model, along with the assumptions made. In sec. 4 we derive the expression for the theoretical radial velocity and describe a Toy Model that will be useful for the interpretation of the results. The results of the analysis of six case studies are presented in sec. 5, followed by a general discussion in sec. 6.
2 Parameter space
Given an underlying equilibrium stellar model, the general problem that we set out to study depends on a number of input parameters. Some of these are intrinsic to the star, namely: the magnetic field intensity and topology, characterized by the vector ; the unperturbed (i.e., in the absence of a magnetic field) oscillation mode, characterized by its radial order, , angular degree, , and the azimuthal order ; the location of the chemical elements whose spectral lines are used in the derivation of the radial velocity, characterized by an atmospheric depth (or region in depth), as well as by longitudinal and latitudinal limits. Moreover, some parameters depend on the position of the observer, such as: the inclination angle between the magnetic axis and the direction of the observer (at a given phase of rotation); the inclination angle between the latter and the rotation axis, .
In our analysis we will keep some of the above fixed. In particular, we will only consider magnetic fields of a dipolar topology. We will not consider the effect of rotation on the dynamics and, consequently, will assume that the magnetic and pulsation axis are aligned. This is generally a good approximation for roAp stars, except possibly in cases when the manetic field is rather low (below 1 kG) (Bigot & Dziembowski, 2002). Moreover, the unperturbed pulsation modes to be considered are only those axisymmetric about the magnetic axis, thus, characterized by in a spherical harmonic decomposition about that axis of symmetry. Finally, we will consider only a fixed phase of rotation, so that our problem will have no dependence on the inclination angle . We note that the relaxation of most of the above conditions is relatively straightforward, although in some cases will lead to significant additional work.
The results of our study will be presented in two papers. The present paper will deal with the underlying mathematical analysis and the in-depth investigation of a number of test cases – thecase studies discussed in section 5. For that, we will restrict the parameter space further, by considering only modes of degree and the case in which the pulsation and magnetic axis is along the line-of-sight (i.e., ). The latter is justified by the fact that in this case the spherical coordinate system associated with the observer coincides with the one used to solve the pulsation problem in the star. As will become clear in sec. 4.2, that is a necessary condition for the in-depth study that we aim at in the present paper.
In a second paper we will present the conclusions of a systematic search of the parameter space, including also variation of mode degree and of the position of the observer. Additionally, in that paper we will compare our results with those derived by simulations of line profiles, as well as with general trends of pulsation phase and amplitude found in the observations.
3 Pulsations in the presence of a magnetic field
The study of pulsations in the presence of a magnetic field has been undertaken by a number of authors in the past, adopting different approximations, or mathematical approaches (Dziembowski & Goode, 1996; Bigot et al., 2000; Cunha & Gough, 2000; Saio & Gautschy, 2004; Saio, 2005; Cunha, 2006).
The magnetic field has a direct influence on the oscillations only in the region where the Lorentz forces are comparable to or larger than the gas pressure gradient, that is to say, in the surface layers of the stars. Therefore, we follow the approach of previous studies, and divide the star into two regions, a region where the magnetic pressure is greater than or comparable to the gas pressure, the surface boundary layer, and a region where the gas pressure is much larger than the magnetic pressure, the interior, defined to be the region between the center of the star and the boundary layer. In the interior the dynamics is not directly affected by the magnetic field and the oscillations there will be modified only as a consequence of the change in the conditions of the surface layers.
The surface boundary layer itself can be divided into two regions, one where the magnetic pressure is comparable to the gas pressure, that we shall call the coupling region, and one where the magnetic pressure is much larger than the gas pressure, which is the magnetically dominated region.
Concerning the oscillations, the studies referred to above have shown that in the coupling region the magnetic and acoustic wave components are coupled into a magnetoacoustic wave. Moreover, the authors have shown that in the interior of the star, where the gas pressure is much larger than the magnetic pressure, the magnetoacustic wave decouples into a fast (acoustic in nature) and a slow (magnetic in nature) component, the latter being an inwardly propagating wave. Likewise, higher in the atmosphere, where the magnetic pressure is much larger than the gas pressure, the magnetocaoustic wave decouples into acoustic and magnetic components, the latter being a standing wave with almost constant amplitude (Sousa & Cunha, 2008).
In Fig. 1, we show a schematic representation of a vertical cut in the star. On the left hand side we show the interior and the surface boundary layer and on the right hand side we show a zoom of the surface boundary layer in which the coupling and magnetically dominated regions are depicted. Moreover, the figure shows also the outer atmospheric region, which we define as a subsection of the magnetically dominated region in which the plasma is assumed to be approximately isothermal. In this study we will concentrate in the outer atmospheric region, and will take as input the velocity field at the base of that region which, in turn, is obtained from the study of the star as a whole, following the approach of Cunha (2006).
3.1 Surface boundary layer
In our calculations we follow Cunha & Gough (2000) and assume a dipolar magnetic field, , of polar magnitude which is force free, and hence does not influence the equilibrium state. Therefore, in this layer the equilibrium state is governed by the following system of equations,
where is the pressure, is the density, is the gravitational field, is the time, and the subscript denotes the equilibrium quantities. The equilibrium gravitational field, , can be written as the gradient of the gravitational potential, ,
where satisfies Poisson’s equation,
where is the gravitational constant.
In the limit of perfect conductivity, adiabatic motions associated with a velocity, , are governed by the following system of magneto-hydrodynamic equations,
where stands for total derivatives, stands for partial derivatives, is the first adiabatic exponent and is the current density.
Assuming small adiabatic perturbations of the equilibrium state, such that,
for any scalar or vector component, , and neglecting the terms of higher order in the perturbations, the two previous systems of equations can be rearranged to give a new system of equations which govern linear adiabatic pulsations in the Cowling approximation, namely,
where is the magnetic permeability and is the vector displacement, defined by the relation =, and where we used the fact that the magnetic field in the equilibrium state is irrotational.
As expected, the magnetic term in equation (10) is perpendicular to the direction of and, thus, within the approximations considered, there is no magnetic component of the restoring force along the direction of the unperturbed magnetic field. To proceed we follow the assumptions adopted in previous studies and consider, at each latitude, a local plane-parallel layer with locally constant and , and neglect and , but allow to change with latitude. In the above, we considered a local coordinate system (x,y,z), with x and y pointing along the latitudinal and longitudinal directions, respectively. Moreover, z is taken to be zero at some fiducial radius (that shall be identified later) located in the outer atmospheric region, and increasing outwardly.
3.1.1 Components of the displacement
Let us first begin with the projections of equation (10) in the directions parallel and perpendicular to the magnetic field. For that we will define and such that , with , and , the unit vectors in the direction parallel and perpendicular to , respectively, and and the unit vectors in the direction of the x and y axes, of the adopted Cartesian coordinate system, respectively. We note that due to the axisymmetry of both the magnetic field and the oscillation modes to be considered, the problem reduces to fourth-order (two second order coupled differential equations), and the only projections that need to be considered in the analysis lie in the (, ) plane.
where is the directional derivative in the direction parallel to the magnetic field which is defined as , and .
Next, we assume that all horizontal derivatives of the displacement, and its components, are much smaller than the corresponding vertical derivatives. For low degree, high radial order modes in the absence of a magnetic field this is clearly a good approximation. In the presence of a magnetic field there are additional dependencies of the eigenfunctions on the latitude, but these occur on a scale of , where is the radius of the star, and thus introduce local horizontal derivatives that are much smaller than the vertical derivatives in the atmosphere. Thus, this approximation remains good in general, in the presence of a magnetic field. Note, however, that this approximation may fail, at particular latitudes, for particular frequencies, when the phase and/or amplitude variations of the displacement with latitude are very sharp (e.g. Cunha & Gough, 2000). However, since the observer sees only an integral of the eigenfunctions over the stellar disk, that failure is likely not to have a strong impact on the results, unless the eigenfunction is trapped within a given latitudinal region, in which case the approach considered here will not be adequate.
So, neglecting the horizontal derivatives of the displacement when compared with the vertical derivatives, we get, after some algebra, the following expression for the projection of equation (10) in the direction along the magnetic field,
Let us now consider the projection of equation (10) in the direction perpendicular to the magnetic field. Taking the scalar product with we get,
where is the directional derivative in the direction perpendicular to the magnetic field, defined as , and and are, respectively, the components of along and perpendicular to the magnetic field direction.
In equation (16), the first term on the right hand side corresponds to the restoring force that would act in the absence of the magnetic field while the second term corresponds to the magnetic response. After developing both terms and combining with equations (11), (12) and (13), equation (16) becomes,
Furthermore, similarly to what we have done for the projection in the direction along the magnetic field, we neglect the horizontal derivatives of the displacement components when compared with the vertical derivatives, and equation (17) becomes,
3.2 Magnetically dominated region
There is a close relation between the location of the coupling region in a roAp star and the strength of the magnetic field in that star. The photosphere of a typical roAp star is located in different parts of the surface boundary layer, depending on the magnetic field intensity. For magnetic fields of the order of 1 kG, it is located in the coupling region, while for magnetic fields of the order of 3 kG it is located in the magnetically dominated region (Cunha, 2007). Therefore, the oscillations in the atmospheres of these stars might look significantly different from star to star, and in different regions within the same star. In our analysis we are particularly interested in the form of the velocity in the outer atmospheric region. This means that in the region of interest, the magnetic pressure is much larger than the gas pressure and, consequently, the magnetoacoustic wave is decoupled into its acoustic and magnetic components.
Thus, we next consider regions where is much smaller than one, and assume the displacement can be expressed as the sum of a slow (essentially acoustic) component, , and a fast (essentially magnetic) component, , i.e.,
such that , where and stand, respectively, for any of the components of the vectors and . If the two components are indeed decoupled, equations (19) - (20) must be satisfied separately by and . Moreover, we will consider solutions of the form,
where is the oscillation frequency, so that . Finally, we note that, as discussed in the introduction, spectral lines of the rare earth elements form very high in the atmosphere where the it becomes close to isothermal. Since our ultimate aim is to compare the velocity field derived from this analysis with that deduced observationally from the lines of the rare earth elements, we will make an additional approximation and consider only those outermost layers where we may expect the isothermal approximation to be appropriate. Also, we take the first adiabatic exponent, , to be constant in that region, and equal to 5/3.
In these layers we can define the pressure, , and the density, , as (e.g. Gough, 1993)
where and are the pressure and density, respectively, at the base of the isothermal region, and is the scale height defined by . With these expressions we simultaneously define as being the position of the base of the isothermal region.
3.2.1 Slow Component
In the limit when the fast and slow components are decoupled, the approximate form of the space-dependent part of the slow component, can be obtained from the system of equations,
These are derived from the system of equations (19) - (20), after neglecting small terms (as discussed in Appendix A). Moreover, the latter implies that , as expected for a wave that is essentially acoustic in nature, in a magnetically dominated region.
In the present case, of an isothermal atmosphere, equation (24) admits solutions of the type,
and and are complex, depth independent, amplitudes.
From equation (27) we can derive the acoustic critical frequency, , defined as the frequency at which . We find,
where is the angle between the direction of the local magnetic field and the local vertical coordinate. If we consider the case of the magnetic pole (), we find that the latter equation is similar to the cutoff frequency for acoustic waves in the absence of the magnetic field. Therefore, hereafter we shall use acoustic cutoff frequency for the acoustic critical frequency at the magnetic pole, and retain the name acoustic critical frequency for the critical frequency at a general latitude.
Going back to the solution expressed by equation (26), we find that when the oscillation frequency is larger than the acoustic critical frequency, must be set to zero, as otherwise energy would be sent in from outside the star. In that case, we find, from equations (25) and (26),
If we consider the case of an oscillation frequency smaller than the acoustic critical frequency, we find that must be zero, so that the energy does not increase with . In that case, equations (25) and (26) imply that,
In both cases, we have set the integration constants that result from the integration of equation (25) to zero, as non-zero constants would correspond to solutions that do not obey the condition set for the slow wave, namely that .
From equation (28) we can see that depends on colatitude, because it depends on the inclination of the magnetic field. So, even when the oscillation frequency is lower than the acoustic cutoff frequency, there will be a colatitude at which, due to the inclination of the magnetic field, the acoustic wave will start to propagate. For this colatitude, above which the acoustic component starts to propagate as a running wave, we define a critical angle, .
3.2.2 Fast Component
Similarly, in the limit when the fast and slow components are decoupled, the approximate form of the space-dependent part of the fast component, can be obtained from the system of equations,
and is the Alfvén velocity.
Equations (31) and (32) are derived from the system of equations (19)-(20), by neglecting small terms (see Appendix A for details). An additional conclusion of that analysis is that , as expected for a wave that is essentially magnetic in nature, in a magnetically dominated region. Moreover, we note that is always positive, which means that unlike the case of the slow component, in this case there is no magnetic critical frequency.
In the present case, of an isothermal atmosphere, equation (32) admits solutions of the type,
where and are depth independent amplitudes. Since the function diverges when its argument tends to zero, diverges as and therefore must be zero to obtain a physically meaningful result. Moreover, when is sufficiently small, equation (31) can be further approximated by,
Thus, we find that the components of the fast displacement are, within the approximations considered,
Since the density tends to zero, as one considers higher layers in the atmosphere, tends to one there. Consequently, the solution for the fast component of the displacement perpendicular to the direction of the magnetic field, will tend to a constant. Moreover, both and tend to zero, as one considers higher layers in the atmosphere. Thus, the solution for the fast component of the displacement parallel to the direction of the magnetic field will tend to zero, as it should, since the pure magnetic wave will have a displacement perpendicular to .
3.3 Dimensionless Equations
Prior to computing the expected radial velocity, we convert the quantities derived to this point that will enter such computation to corresponding dimensionless quantities. To that end we define:
, , , ,
, , ,.
, , where is the radius of the star and =, is the mass of the star and is the gravitational constant. We note that and defined above are not to be confused with the dimensional quantities named by the same symbol and used in equations (6)-(9). With these definitions the dimensionless scale height becomes . Taking these new variables and equations (27) and (33), we find the dimensionless and as,
and we define also the dimensionless acoustic critical frequency, , as
where is the dimensionless sound speed defined as .
With these new variables we derive expressions for the dimensionless slow and fast wave solutions and combine them to obtain the expressions for the dimensionless parallel, , and perpendicular, , components of the displacement. For oscillations with frequencies larger than , we get
and for oscillations with frequencies smaller than , we get
where is a complex, depth independent amplitude which depends on latitude, defined in terms of the previously adopted amplitudes by or , depending on the colatitude. Similarly, is a complex depth independent amplitude which depends on colatitude, , and is the dimensionless time.
The complex amplitudes and can be written in terms of their modulus, and , and phases, and . Before we proceed to calculate the disk integrated line-of-sight projected velocity, we still need to determine these amplitudes and phases. Their relative values are not arbitrary. Instead, they are imposed by the magnetoacoustic coupling that takes place below, in the region where the magnetic and gas pressure are comparable. Thus, in order to find their values we fit our expressions for the displacement to the numerical solutions that come out of the MAPPA code (Cunha, 2006). From the fit, we extract the amplitudes and phases and, simultaneously, confirm that our analytical solutions for the displacement represent well the corresponding numerical solutions, in the limit when both is much smaller than one, and the atmosphere becomes isothermal.
Generally, since in the region of interest, is much smaller than one, we may expect that the second term on the right hand side of equations (40)-(43) can be neglected. That assumption, however, can break down if the relative value of the amplitudes and is such as to compensate for the small value of . We checked whether that was the case and found that the only situation in which the two terms become of the same order of magnitude is in equation (40), for colatitudes very close to 90. In those cases, we have included the second term in our computations. In all other cases the second terms on the right hand side of equations (40)-(43) were neglected.
4 Theoretical Radial Velocity
In the observations what is seen is an integral over the visible stellar disk. Therefore, we shall integrate the line-of-sight projection of the velocity over the visible stellar disk taking into account the limb darkening effect, to simulate what observers would see if they had the means to measure the radial velocity as function of atmospheric depth.
In Fig.2 we show a schematic representation of the star, where the observer is considered to be at a general position. In the figure is the angle between the magnetic field axis and the direction of the observer, which we shall assume to be fixed, meaning that we are considering the observer’s view at a particular rotation phase. Moreover, is the local magnetic field vector and is the angle between the magnetic field axis and the direction of the local magnetic field. In our analysis we consider two spherical coordinate systems: The spherical coordinate system, (r, , ), aligned with the magnetic field axis, and the spherical coordinate system aligned with the direction of the observer, (r, ,). The coordinates in the two systems are related through standard expressions given, e.g., by Karttunen et al. (1996).
Taking the real part of equations (40) and (42), after neglecting the second terms on the right hand side, and differentiating with respect to time, we can write a general expression for the component of the velocity parallel to the direction of the magnetic field in the form,
where and are factors defined in the following way: when the oscillation frequency is larger than the acoustic critical frequency, and ; when the oscillation frequency is smaller than the acoustic critical frequency, and . Similarly, from equations (41) and (43), after neglecting the second terms on the right hand side, we find that the component of the velocity perpendicular to the direction of the magnetic field is given by,
Assuming a linear limb darkening law, with as the limb darkening coefficient, the expression for the velocity component parallel to the line of sight averaged over part of the visible stellar disk for a general position of the observer, , thus becomes,
where , , and are, respectively, the limit in longitude and latitude, in the observer’s spherical coordinate system, of the region of the visible stellar disk over which the integration is to be made. For an integration over the whole visible stellar disk varies within the interval and varies within the interval . Moreover, is a normalization factor defined as,
and and are the line-of-sight projections of the unit vectors and , respectively.
4.1 Fitting to Acos(t+)
In spectroscopic observations the radial velocity derived from the time series analysis is usually fitted to a function of the form Acos(t+). From this fit, the authors derive the amplitude and the phase of the oscillations (e.g. Kurtz et al., 2005, 2006b; Ryabchikova et al., 2007; Sachkov et al., 2007). Since our ultimate goal is to compare the results of our work with the observations, in our study we fit the average line-of-sight velocity, , to a function similar to the one above, and derive the amplitudes and phases of the oscillations as function of height in the atmosphere. From the fit we find the expression for the fitted phase, , namely,
and the expression for the fitted amplitude, A, namely,
where and are given by,
4.2 A toy Model
In sec. 5 we will use expressions (45) to (48) to study the behavior of the phase and amplitude of in a number of case studies. Here we try to anticipate those results through an analytical analysis of the sum of wave-like functions. This simple analysis will help support the interpretation of the results obtained in the case studies.
The motivation for the form of the wave-like functions that we shall consider comes from equation (44). In the case of an observer aligned with the magnetic field axis, this equation can be written as the sum of three integrals, namely,
where is the critical angle above which the acoustic component will start to propagate as a running wave. Thus, contains the information of the evanescent acoustic solution only, contains the information of the running acoustic solution only, and contains the information of the standing magnetic solution only. In the above we have assumed . When , is zero and the lower colatitude limit in is taken to be . Like wise, when , is zero and the upper colatitude limit in is taken to be .
4.2.1 Visual superposition of waves of different nature
In this toy study we define three simple waves, with velocities , and , inspired, respectively, by , , and . The velocities are assumed to have the following form, with the subscripts and indicating the real and imaginary parts of the wavenumber, respectively,
with , and amplitudes and wavenumbers characterized as follows: