I am reading this paper: it solves a vibrational elliptical membrane problem.
In equation 13 and 14, they assumed a solution without the second kind modified Mathieu function:
their explanation is:
Second radial $Ne_m$ and $No_m$ functions are excluded because the wave solutions and their derivatives must to be continuous at $\xi=0$. See figure:
I don't quite understand this interpretation, why continuous at $\xi=0$ need to exclude the second kind, how can I see it from the graph? Please give me a clear and simple explanation...
If it is a circular membrane, the radial solution will be the Bessel function of first kind, and the second kind is excluded because it diverges at origin. This is very intuitive and understandable. The analogy to the elliptical case can't holds, because from the graph, it is finite value.
It is somewhat technical, and due to elliptic cylindrical coordinates system, in which occurs a singularity in the interfocal line (between the two focus of the ellipse) for the gradient along radial as well as angular direction. If the domain includes this interfocal-line then solution needs to be of the form of a product of a radial first-kind and angular first kind, other solutions will exhibit a discontinuity in derivatives when crossing the interfocal line. I name this condition, a regular gradient condition anywhere in the space of coordinate system. This kind of condition will appear whenever the gradient show a singularity in the space of coordinates
Henri Lévêque