1. Design of experiments and data analysis
Determine experimentally the order of accuracy of the
finite difference formulas given below
(i.e. how does the error scale with
).
Also, determine the optimum
for each formula. Based on your
limited data, can you suggest a relation between machine precision
eps and ?
You must explain your approach to solving this problem.
where u(x,y,t) is the elevation of the membrane which depends on position x,y and time t. The parameter c2 depends on material properties. The wave equation has solutions that are periodic in time and do not change shape, called ``eigenmodes''.
A circular drum has axisymmetric eigenmodes of
the form 
where
and
, with
the ``eigenfrequency" (also called ``characteristic'' frequency).
If the drum is attached around its rim, then we must have
where R is the radius of the drum.
Hint: the function
is
the Bessel function of the first kind of order 0.
Asymptotic analysis for large z tells us that
Bessel functions are very important in Science and Engineering. They appear for diffusion and wave problems in cylindrical geometry.