I'm curious to receive comments from Leigh and others about how to
properly think about a question I've put on a midterm for tomorrow. The
question itself is trivial. You have a child on a swing of some length
and therefore work out its period. We can pump the swing by pushing on
the child with that period, but we can also push every second time they
come back (twice the period half the frequency), every third time etc.
So I've asked them to identify those frequencies.
The thing I've been pondering is that as compared to other "sympathetic
vibration" / "resonance" / ... type examples I can think of, in this
case (or the comparable spring question) the system responds at its own
frequency which is a harmonic of the frequency at which you pump it....
So it *appears* to be multiplication of a frequency. (i.e. if an
optical analog existed it would amount to something such as putting 800
nm in and getting 400 nm out.)
However, is it really accurate to think of this as multiplying the
pumping frequency? It is multiplying the frequency that I relate to -
the frequency at which I apply impulses - however the swing system would
have to be described by multiple frequencies (since it is increasing and
decreasing in the amplitude of its oscillations), and the pumping itself
is periodic impulses which should be fourier decomposed into a set of
frequencies, one of which is the resonant frequency of the swing.