Appendix B: Determination of the boundary conditions in the case of planar approximation

In this section, we present the calculation of the endpoints conditions of the associated phase system, applied to equations of motion for adiabatic oscillations when the wavelength is much less than the solar radius, the local effects of spherical geometry on the dynamic can be ignored (Deubner & Gough 1984, see also Sect. 2.1.1). To simplify the determination of the endpoints of the phase equations, from the boundary conditions (see Sect. 2.2), we will consider the phase equations, on the approximation proposed in Sect. 3.4 (Eqs. (51) and (52)). Furthermore, we will consider only the dominant terms for $\omega_{+}^2$ and $\omega_{-}^2$in the planar approximation, given by Eq. (26), which are proportional to $S_{\rm l}^2$ and N², respectively. This example is particularly interesting because the boundary conditions corresponding to that case remain the same for the other approximations by the fact that the dominant terms which determine the local behaviour of the phase are presented in this case. The functions, $f_{\rm p}$ and $f_{\rm g}$ are given approximately by

$$f_{\rm p}\approx\frac{\omega}{c}\left(1-\frac{L^2}{\omega^2}\frac{c^2}{r^2} \right)$$ (B.1)

and

$$f_{\rm g}\approx\frac{\omega}{c}\left(1-\frac{N^2}{\omega^2}\right)\cdot$$ (B.2)

The boundary condition, of the $\Psi$ function, can be determined from the boundary conditions of $\delta p$, by the Eq. (6) and the respective boundary conditions for $\delta p$. The regularity condition in the centre for $\Psi$ is

 

$$\frac{{\rm d}\ln{\Psi}}{{\rm d}r} \rightarrow \frac{l}{r}$$ (B.3)

as $r\rightarrow 0$. The boundary condition on the surface, for $\Psi$ is

 

$$\Psi= 0$$ (B.4)

at r=R. Using the phase transformation (Eq. (A.5)),

$$\cot{\theta_i} ={\rm sgn}({\cal M}_i)\vert{\cal M}_i\vert^{2}\lef... ...r}\ln{\vert{\cal M}_i\vert} + \frac{\rm d}{{\rm d}r}\ln{\vert\Psi\vert} \right]$$ (B.5)

for each one of the descriptions i=1,2. We observe that for the centre, from the series expansions of equilibrium quantities we obtain (as $r\rightarrow 0$),

$$\frac{\rm d}{{\rm d}r}\ln{\vert f_{\rm p}\vert^{1/2}} \rightarrow r^{-1}$$ (B.6)

and

$$\frac{\rm d}{{\rm d}r}\ln{\vert f_{\rm g}\vert^{1/2}}\rightarrow 0.$$ (B.7)

Under these considerations, it is possible to determine the initial phase of the associated phase system. The endpoints for the centre are ( $r\rightarrow 0$). In the case of the acoustic description, we obtain

$$\cot{\theta_1}\approx {\rm sgn}(f_{\rm g}) \frac{c_{\rm o}}{\omega}\frac{1}{1-\frac{N^2}{\omega^2}} \; \frac{l}{r} \rightarrow +\infty$$ (B.8)

or

$$\theta_1= 0 \;\; ({\rm mod}\;\; \pi).$$ (B.9)

In the case of the gravity description, we obtain

$$\cot{\theta_2} \approx {\rm sgn}(f_{\rm p})\frac{c_{\rm o}}{\omega} \frac{r^2}{r^2-\frac{L^2}{\omega^2}c_{\rm o}^2} \;\frac{l}{r} \rightarrow 0$$ (B.10)

or

$$\theta_2= \frac{\pi}{2} \;\; ({\rm mod}\;\; \pi).$$ (B.11)

The endpoints for the surface, can also be calculated for both descriptions. It is also possible to determine the amplitude boundary conditions, but they are not interesting to our problem.

At last, it is important to observe that the initial boundary conditions can be understood in terms of the physical nature of the acoustic and gravity waves.