Eigenvalues of the Dirichlet Laplacian in an interval
of length π (pi)
are given by n2,
where n is any positive integer.
The eigenfrequencies (the square roots of the eigenvalues)
are thus equidistantly distributed on the real axis:
this is the harmonic series which the Western music is based on.
The four lowest frequencies
together with the nodal points of
the corresponding eigenfunctions
are listed below.
1
2
3
4
individual tones
time modulation
Rectangular drum
Eigenvalues of the Dirichlet Laplacian in a square
of sides of length π
are given by n2 + m2,
where n,m are any positive integers.
The eigenfrequencies do not form the harmonic series any more.
The four lowest frequencies
(multiplicities not counted but depicted)
together with the nodal lines of
the corresponding eigenfunctions
are listed below.
1.4
2.2
2.8
3.2
Combining the eigenfunctions corresponding to degenerate eigenvalues,
one can get fancy curves, including a closed nodal line
for the fifth eigenvalue (counting multiplicities).
individual tones
time modulation
Circular drum
Eigenvalues of the Dirichlet Laplacian in a disk
of radius √π
(so that the rectangular and circular drums have the same area)
are given by roots of Bessel functions.
The eigenfrequencies do not form the harmonic series either.
The four lowest frequencies
(multiplicities not counted but depicted)
together with the corresponding nodal lines of
the corresponding eigenfunctions
are listed below.
1.36
2.16
2.90
3.12
individual tones
time modulation
Now the sixth eigenvalue (counting multiplicities)
has an eigenfunction with a closed nodal line.
While higher eigenvalues can obviously have eigenfunctions
with closed nodal lines,
Payne's conjecture from 1967 states that, for any domain,
the second eigenfunction
(the first eigenfunction with a non-trivial nodal-line structure)
cannot have a closed nodal.
This conjecture is still open for simply connected domains.
The property is apparently related to musical qualities of the drum.
Note also that the fundamental frequency of the circular drum
is lower than that of the rectangular drum of the same area.
This (and other analogous comparisons)
led Lord Rayleigh to conjecture the related spectral
optimality of the disk in 1877,
later proved by Faber and Krahn in 1923-4.
