Oddball applied math and numerical project ideas
These are ideas for investigation of the natural world that I have not seen
anywhere else. I have not had time to try them!
They might make fun projects for any young or experienced
Fractal dimension of lightning strike
The electrical discharge and subsequent explosion of high-temperature
gas occurs in of order 1 millisecond or less, simultaneously along
the lightning channel. However the thunder heard is spread out over often
many seconds, due to the range of travel times from points along the
channel to the listener (there are also reflections from buildings which
prolong the sound).
That thunder initially sounds like `tearing', especially when it strikes
close by, is no coincidence:
the discharge channel is a fractal, therefore the arrival time distribution
will be fractal too. Tearing of paper or other sheets is
a chaotic process with build-up of elastic energy followed by catastrophic
releases of energy (rips) occuring on all scales (this self-similarity
is also known in earthquakes).
In principle by studying the acoustic thunder signal at a
one could compute the fractal dimension of the strike.
How, and what is it? Does it
depend on weather conditions?
Note that we are projecting a curve in 3D down to 1D. This constrasts
the visual estimation of fractal dimension which involves projecting from
3D to 2D.
By recording at three or more locations, one might be able to reconstruct the
full 3D shape of the strike. How tractable an inverse problem is this?
(It certainly seems harder than optical reconstruction!)
How much of a problem are reflections off the ground or other objects?
PHYSICS AND EFFECTS (ENCYCLOPEDIA OF LIGHTNING)
by V.A. RAKOV and M.A. UMAN
(Cambridge University Press, 2003). Chapter 11 is on acoustics
of lightning, ie thunder (I have not yet read).
Gaussian curvature of lettuce leaf
Estimate the negative radius of curvature of a lettuce leaf, using a method
of your choosing. This
probably changes as a function of distance to the edge. However, since almost
all of the area is within the last couple of cm from the edge, the
radius must be quite tight. How much area would there be if the leaf
continued with this curvature, for another 10 cm? (I suspect this would
be huge--exponentially large).