Jan Panteltje wrote:
'It is written' that this cannot (has not?) been done [1].

Before I delve deeper in the math, let us look at it from
a slightly different perspective, a 'mechanical' analogy
with waves.

Just for arguments sake imagine a surface with hollow extrusions.
A ball is sitting in each 'dip'.


--- ----------
\ O /
--------

When we excitate the surface with a sine wave, the ball gets excited
by the maxima and minima of the wave (it starts jumping up and down a bit).


ABOVE A SPECIFIC LEVEL the ball will jump over the edge and roll away, or
even jump further up.
You can envision this as shaking the container in the frequency of the wave.

When the frequency doubles, then 2 x more maxima (excitations) will occur,
and 2 x more balls will leave their hole.


Non sequitur.