v0,1
v0,2
v0,3
v1,1
v1,2
v1,3
v2,1
v2,2
v2,3
v3,1
v3,2
v3,3
Figure 11.5.
Normal Modes for a Disk.
Summarizing our progress, the eigenvalues
2
λm,n = ζm,n
,
n = 1, 2, 3, . . . ,
m = 0, 1, 2, . . . ,
(11.112)
of the Bessel boundary value problem (11.54–55) are the squares of the roots of the Bessel
function of order m. The corresponding eigenfunctions are
wm,n (r) = Jm (ζm,n r) ,
n = 1, 2, 3, . . . ,
m = 0, 1, 2, . . . ,
(11.113)
defined for 0 ≤ r ≤ 1. Combining (11.113) with the formula (11.53) for the angular components, we conclude that the separable solutions (11.51) to the polar Helmholtz boundary
8/4/10
532
c 2010
Peter J. Olver
value problem (11.49) are
v0,n (r, θ) = J0 (ζ0,n r),
vm,n (r, θ) = Jm (ζm,n r) cos m θ,
vbm,n (r, θ) = Jm (ζm,n r) sin m θ,
n = 1, 2, 3, . . . ,
where
m = 0, 1, 2, . . . .
(11.114)
These solutions define the so-called normal modes for the unit disk, and Figure 11.5 plots
the first few of them. The eigenvalues λ0,n are simple, and contribute radially symmetric
eigenfunctions, whereas the eigenvalues λm,n for m > 0 are double, and produce two linearly independent separable eigenfunctions, with trigonometric dependence on the angular
variable.
Recalling the original ansatz (11.48), we have at last produced the basic separable
eigensolutions
2
u0,n (t, r) = e− ζ0,n t J0 (ζ0,n r),
n = 1, 2, 3, . . . ,
m = 1, 2, . . . .
2
um,n (t, r, θ) = e− ζm,n t Jm (ζm,n r) cos m θ,
2
u
bm,n (t, r, θ) = e− ζm,n t Jm (ζm,n r) sin m θ,
(11.115)
to the homogeneous Dirichlet boundary value problem for the heat equation on the unit
disk. The general solution is obtained by linear superposition, in the form of an infinite
series
∞
∞
X
1 X
a0,n u0,n (t, r) +
am,n um,n (t, r, θ) + bm,n u
bm,n (t, r, θ) , (11.116)
u(t, r, θ) =
2 n=1
m,n = 1
where the initial factor of 21 is included, as with ordinary Fourier series, for later convenience. As usual, the coefficients am,n , bm,n are determined by the initial condition,
so
u(0, r, θ) =
∞
∞
X
1 X
a0,n v0,n (r) +
am,n vm,n (r, θ) + bm,n vbm,n (r, θ) = f (r, θ).
2 n=1
m,n = 1
(11.117)
Thus, we must expand the initial data into a Fourier–Bessel series in the eigenfunctions.
As before, it is possible to prove, [35], that the separable eigenfunctions are complete —
there are no other eigenfunctions — and hence every (reasonable) function defined on the
unit disk can be written as a convergent series in the Bessel eigenfunctions.
Theorem 9.32 gurantees that the eigenfunctions are orthogonal† with respect to the
standard L2 inner product
ZZ
Z 1Z π
u(r, θ) v(r, θ) r dθ dr
hu;vi =
u(x, y) v(x, y) dx dy =
0
D
−π
†
For the two eigenfunctions corresponding to one of the double eigenvalues, orthogonality
must be verified by hand.
8/4/10
533
c 2010
Peter J. Olver