Unformatted text preview: plane as in the preceding problem.
Suppose that the eigenvalues for the Laplace operator ∆ on D are λ1 , λ2 , . . . ,
once again. Show that the general solution to the wave equation
together with Dirichlet boundary conditions (u vanishes on ∂D) and the initial
(x, y, 0) = 0,
∞ u(x, t) = bn φn (x, y ) cos( −λn t), n=1 where the bn ’s are arbitrary constants. 5.8 Eigenvalues of the disk To calculate the eigenvalues of the disk, it is convenient to utilize polar coordinates r, θ in terms of which the Laplace operator is
r ∂r r ∂f
∂r + 1 ∂2f
r2 ∂θ2 f |∂D = 0. (5.28) Once again, we use separation of variables and look for product solutions of the
f (r, θ) = R(r)Θ(θ), where R(a) = 0, Θ(θ + 2π ) = Θ(θ). Substitution into (5.28) yields
r dr r dR
dr Θ+ R d2 Θ
r2 dθ2 144 We multiply through by r2 ,
dr r r dR
dr Θ+R d2 Θ
= λr2 RΘ,
dθ2 and divide by RΘ to obtain
R dr r dR
dr − λr2 = − 1 d2 Θ
Θ dθ2 Note that in this last equation...
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This document was uploaded on 01/12/2014.
- Winter '14