KLEIN-GORDON EQUATION AND REFLECTION OF ALFVEN WAVES IN NONUNIFORM MEDIA

A new analytical approach is presented for assessing the reflection of linear Alfven waves in smoothly nonuniform media. The general one-dimensional case in Cartesian coordinates is treated. It is shown that the wave equations, upon transformation into the form of the Klein-Gordon equation, display a local critical frequency (OMEGA(c)) for reflection: At any location in the medium, reflection becomes strong as the wave frequency descends past this characteristic frequency set by the local nonuniformity of the medium. This critical frequency is given by the transformation as an explicit function of the Alfven velocity (V(A)), and its first (V(A)') and second (V(A)'') derivatives, and hence as an explicit spatial function. The transformation thus directly yields, without solution of the wave equations, the location in the medium at which an Alfven wave of any given frequency becomes strongly reflected and has its propagation practically cut off. The local critical frequency is the square root of the larger coefficient of the zeroth-order term in the two transformed wave equations; which equation has the larger coefficient is determined by the local nonuniformity. Hence, the critical frequency can switch from one equation to the other as the nonuniformity changes. Consequently, it is necessary to transform both wave equations to deduce that the critical frequency is OMEGA(c) = square-root (V(A)')2 + \2V(A) V(A)''\/2.