Fig. 5.3 The first four eigenfunctions for pressure fluctuations in a clarinet
dB
Hz
Fig. 5.4 Clarinet frequency decomposition
The corresponding oscillation frequencies are given by ωn = cπ
(n−12).
In contrast to the string, the model predicts that the clarinet’s spectrum will contain only odd multiples of the fundamental frequencyω1. Figure5.4shows the frequency decomposition for a clarinet sound sample. The prediction holds true for the first few modes, but the simple model appears to break down at higher frequencies. ♦
The polar form of the Laplacian is computed by writing = ∂2
∂x12 + ∂2
∂x22
and then converting the partials with respect tox1andx2 intor andθ derivatives using the chain rule. The result is
= 1 r
∂
∂r
r ∂
∂r + 1 r2
∂2
∂θ2. (5.14)
Note that there are no mixed partials involving bothrandθ, and that the coefficients do not depend onθ. This allows separation ofrandθ, provided the domain is defined by specifying ranges ofrandθ.
To solve the radial eigenvalue equation, we will useBessel functions, named for the astronomer Friedrich Bessel. Bessel’s equation is the ODE:
z2f(z)+z f(z)+(z2−k2)f(z)=0, (5.15) withk∈Cin general. For our applicationkwill be an integer. The standard pair of linearly independent solutions is given by the Bessel functionsJk(z)andYk(z).
The Bessel J-functions, a few of which are pictured in Fig.5.5, satisfy
J−k(z)=(−1)kJk(z), (5.16) for allk∈Z. Bessel represented these solutions as integrals:
Jk(z):= 1 π
π
0
cos
zsinθ−kθ dθ.
One can also writeJkas a power seriesk∈N0,
Fig. 5.5 The first four
Bessel J-functions J0
J1
J2 J3
Jk(z)=z 2
k∞ l=0
1 l!(k+l)!
−z2 4
l
. (5.17)
Together with (5.16), this shows that Jk(z) ∼ckz|k| as z → 0 for anyk ∈ Z. In contrast, the Bessel Y-function satisfiesYk(z)∼ckz−|k|asz→0.
A change of sign in (5.15) gives the equation
z2f(z)+z f(z)+(z2+k2)f(z)=0. (5.18) Its standard solutions are themodified Bessel functions Ik(z)andKk(z). Asz→0 these satisfy the asymptotics Ik(z)∼ckz|k|, as illustrated in Fig.5.6, andKk(z)∼ ckz−|k|.
Lemma 5.4 Supposeφ∈C2(R2)is a solution of
−φ=λφ,
that factors as a product h(r)w(θ). Then, up to a multiplicative constant,φhas the form
φλ,k(r,θ):=hk(r)ei kθ, (5.19) for some k ∈Z, with
hk(r):=
⎧⎪
⎨
⎪⎩
r|k|, λ=0, Jk(√
λr), λ>0, Ik(√
−λr), λ<0.
Proof Under the assumptionφ=hw, the Helmholtz equation reduces by (5.14) to w
r
∂
∂r
r∂h
∂r + h r2
∂2w
∂θ2 +λhw=0.
Fig. 5.6 The first four
modified Bessel I-functions I0
I1
I2
I3
With some rearranging, we can separate therandθvariables, 1
h
r ∂
∂r
2
h+λ2r2 = −1 w
∂2w
∂θ2, (5.20)
providedhandware nonzero.
As in Lemma5.1, we conclude that both sides must be equal to some constantκ. Theθequation is
−∂2w
∂θ2 =κw. (5.21)
The functionw(θ)is assumed to be 2π-periodic. By the arguments used in Theo- rem5.2, a nontrivial solution is possible only ifκ=k2wherekis an integer. A full set of 2π-periodic solutions of (5.21) is given by
wk(θ):=ei kθ, k∈Z.
Before examining the radial equation, let us note that the assumption thatφis C2imposes a boundary condition atr =0. To see this, first note that the function r =
x12+x22is continuous at(0,0)but not differentiable. Forr >0,
∂r
∂xj = xj
r ,
which does not have a limit asr →0. On the other hand, the functions r e±iθ=x1±i x2
areC∞. Similarly, fork∈Zwe have r|k|ei kθ =
(x1+i x2)k, k∈N0,
(x1−i x2)−k, −k∈N. (5.22) These functions are polynomial and henceC∞. We will see below that the solutions of the radial equation corresponding toκ=k2satisfyh(r)∼ar±kasr → 0, for some constanta. The differentiability ofφat the origin will require the asymptotic condition
hk(r)∼ar|k| (5.23)
asr →0.
Forwk(θ)=ei kθ, the radial component of (5.20) is
r ∂
∂r
2
hk+(λr2−k2)hk=0. (5.24)
The caseλ=0 is relatively straightforward to analyze. In this case (5.24) is homoge- neous in thervariable (meaning invariant under scaling). Such equations are solved by monomials of the formhk(r)=rαwithα∈R. If we substitute this guess into (5.24) withλ=0, the equation reduces to
α2−k2=0,
with solutionsα = ±k. Since a second order ODE has exactly two independent solutions, the functionsr±kgive a full set of solutions fork=0. Fork=0 the two possibilities are 1 and lnr. By the condition (5.23), the solutions lnrandr−|k|must be ruled out. The only possible solutions forλ=0 are thus
hk(r)=r|k|. Note that the resulting solutions,
φ0,k(r,θ):=r|k|ei kθ, are precisely the polynomials (5.22).
Forλ > 0 (5.24) can be reduced to the Bessel form (5.15) by the change of variables z = √
λr. The possible solutions Yk(√
λr)are ruled out because they diverge atr = 0. On the other hand, the power series (5.17) shows that the func- tion hk(r) = Jk(√
λr)satisfies the condition (5.23). Thus forλ >0 the possible eigenfunction withk∈Zis
φλ,k(r,θ):= Jk(√ λr)ei kθ.
We should check that this function is at leastC2at the origin. In fact, it follows from the power series expansion (5.17) thatφλ,kisC∞onR2.
Similar considerations apply for λ < 0, except that this time the substitution z = √
−λr reduces (5.24) to (5.18). The condition (5.23) is satisfied only for the solutionIk(√
−λr).
Example 5.5 The linear model for the vibration of a drumhead is the wave equation (4.30). For a circular drum we can take the spatial domain to be the unit diskD:=
{r <1} ⊂R2. Lemma5.1reduces the problem of determining the frequencies of the drum to the Helmholtz equation,
−φ=λφ, φ|∂D=0. (5.25)
The possible product solutions are given by Lemma 5.4, subject to the boundary conditionhk(1)=0. This rules outλ≤0, because in that casehk(r)has no zeros forr >0.
Forλ>0, we havehk(r)=Jk(√
λr), and the boundary condition takes the form Jk(√
λ)=0.
Table 5.1 Zeros of the Bessel functionJk. For eachk, the spacing between zeros approachesπas
m→ ∞
k jk,1 jk,2 jk,3 jk,4
0 2.405 5.520 8.654 11.792
1 3.832 7.016 10.174 13.324
2 5.136 8.417 11.620 14.796
3 6.380 9.761 13.015 16.223
4 7.588 11.065 14.373 17.616
(This is analogous to the condition sin(√
λ)=0 from the one-dimensional string problem.) Although Jkis not a periodic function, it does have an infinite sequence of positive zeros with roughly evenly spacing. It is customary to write these zeros in increasing order as
0< jk,1< jk,2< . . . . By the symmetry (5.16),
j−k,m= jk,m. Table5.1lists some of these zeros.
Restricting√
λto the set of Bessel zeros gives the set of eigenvalues λk,m= jk2,m,
indexed byk∈Z,m∈N. The corresponding eigenfunctions are
φk,m(r,θ):=Jk(jk,mr)ei kθ. (5.26) The first set of these are illustrated in Fig.5.7.
The collection of functions (5.26) yields a complete list of eigenfunctions and eigenvalues forD, although that is not something we can prove here. ♦ The eigenvalues calculated in Example5.5correspond to vibrational frequencies
ωk,m:=cjk,m,
for k ∈ Zandm ∈ N. The propagation speed cdepends on physical properties such as tension and density. The relative size of the frequencies helps to explain the lack of definite pitch in the sound of a drum. The ratios of overtones above the fundamental ω0,1 are shown in Table5.2. In contrast to the vibrating string case, where the corresponding ratios were integers 1,2,3, . . ., or the clarinet model of Example5.3with ratios 1,3,5, . . ., the frequencies of the drum are closely spaced with no evident pattern.
m= 1 m= 2 m= 3
k= 2 k= 1 k= 0
Fig. 5.7 Contour plots of the spatial component of the eigenfunctions ofD
Table 5.2 Frequency ratios
for a circular drumhead k m ωk,m/ω0,1
0 1 1
1 1 1.593
2 1 2.136
0 2 2.295
3 1 2.653
1 2 2.917
4 1 3.155