For a given domain, we consider the eigenfunctions
um(x) of the Laplace operator with Dirichlet or Neumann
boundary condition. If the domain is the shape
of a drum, the eigenfunctions describe its vibration
modes, while the associated eigenvalues give the
squared frequencies (as well as typical length
scales of oscillations). The square of an eigenfunction
can be interpreted as the spatial distribution over
the drum of the potential energy stored in this vibration
mode. Dirichlet boundary condition describes a drum with
a fixed boundary (vibrations vanish here), while Neumann
boundary condition represents a membrane with freely
moving boundary (in this case, the normal derivative
of the eigenfunctions vanishes at the boundary).
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