Klein-Gordon Equation and Reflection of Alfv\'en Waves
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**Session 121 -- Early Stars, X-Ray Binaries, Theory**
*Oral presentation, Thursday, 12, 1995, 2:00pm - 3:30pm*

## [121.06] Klein-Gordon Equation and Reflection of Alfv\'en Waves

*Z. E. Musielak (UAH), R. L. Moore (NASA/MSFC)*
It is of some interest to know the physical conditions that lead to
efficient reflection of Alfv\'en waves in the solar and stellar
atmospheres. The problem seems to be important because these waves
may play some role in non-radiative heating of the solar and stellar
chromosphere and coronae, and may also be responsible for acceleration
of the solar and cool massive stellar winds. A significant effort has
been made by a number of authors to understand the behavior of these
waves in highly inhomogeneous stellar atmospheres. The simplest
treatment of the problem seems to be the so-called Klein-Gordon
equation approach, which allows obtaining local critical frequencies
by transforming the wave equations into their Klein-Gordon forms and
then choosing the largest positive coefficient to be the square of
the local critical frequency. In this paper, we show that the local
critical frequency can be alternatively defined by using the
turning-point property of Euler's equation. Our results are
obtained specifically for Alfv\'en waves propagating in an isothermal
atmosphere with constant gravity and uniform vertical magnetic field.
We demonstrate that Alfv\'en waves in the upper (above the wave source)
part of our model always form a standing wave pattern and that the
waves in the lower (below the wave source) part of the model are always
propagating (but partially reflected) waves. We also show that the
turning point for the upward and downward waves is located at the
height where the condition $\omega = \Omega_A$ is satisfied and that
$\Omega_A = V_A / 2 H$, where $V_A$ is the Alfv\'en velocity and $H$
is the scale height, can be taken as a local critical frequency because
the waves undergo strong reflection in this region of the atmosphere
where $\omega \leq \Omega_A$. By applying our turning-point analysis
to the Alfv\'en wave equations for the velocity and magnetic field
perturbation, we obtain an interesting result: for our particular
model atmosphere the magnetic-field-perturbation wave equation
yields the local critical frequency but the velocity-perturbation
wave equation does not. A physical interpretation of the obtained
results will be given.

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