| Suppose a musician
wants a guitar string that
has a certain specific sound.
This specification can be
translated into eigenvalue
locations in the complex plane:
persistent tones correspond
to eigenvalues near the imaginary
axis; highly damped tones
give eigenvalues farther in
the left half of the complex
plane. Can we design a string
that has the desired response?
In this talk, we investigate
the construction of a spatially-varying
damping coefficient that
induces a wave operator
having specified eigenvalues.
This problem is distinguished
from the more familiar inverse
Sturm-Liouville problem
by the fact that the operator
in question in non-self-adjoint.
Still, many of the same
theoretical devices, based
on eigenvalue asymptotics,
prove useful. The constructed
damping functions are even
about the midpoint and should
vary mildly about their
mean. Numerical results,
in the absence and presence
of noise, demonstrate the
efficacy of this technique.
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