I m and K m denote the modified Bessel function of the first and second kind respectively of order m . For real frequencies takes complex values in the cut-off regions and can therefore not be equal to which is real. The cut-off regions are now given by and . The fast and slow modes are found between and respectively. We can see in Fig.
Vertical lines are asyptotes.
In the marked cut-off regions is complex. The cut-off frequencies are , and . The internal characteristic frequencies are , and . Also notice that functions become zero for and respectively mode can be present here because outside the regions of the fast and slow body modes is positive whereas is always negative. Under coronal conditions they indeed found no surface waves.