5.7 Vibrations of a Circular Membrane 327
There are two other relations like this one involving functions from two dif-
ferent series.
We already know that the functions within the first series are orthogonal to
each other: ∫
π
−π
∫a0J 0 (λ 0 nr)J 0 (λ 0 qr)rdrdθ= 0 , n=q.Within the second series we must show that, ifm=porn=q,then
0 =∫π−π∫a0Jm(λmnr)cos(mθ)Jp(λpqr)cos(pθ)rdrdθ. (28)(Recall thatrdrdθ=dAin polar coordinates.) Integrating with respect toθ
first,weseethattheintegralmustbezeroifm=p, by the orthogonality of
cos(mθ)and cos(pθ).Ifm=p, the preceding integral becomes
π∫a0Jm(λmnr)Jm(λmqr)rdrafter the integration with respect toθ.Finally,ifn=q,thisintegraliszero;the
demonstration follows the same lines as the usual Sturm–Liouville proof. (See
Section 2.7.) Thus the functions within the second series are shown orthogonal
to each other. For the functions of the last series, the proof of orthogonality is
similar.
Equipped now with an orthogonality relation, we can determine formulas
for thea’s andb’s. For instance,
a 0 n=∫π
−π∫a
0 f(r,θ)J^0 (λ^0 nr)rdrdθ
2 π∫a
0 J
02 (λ^0 nr)rdr. (29)TheA’s andB’s are calculated from the second initial condition.
It should now be clear that, while the computation of the solution to the
original problem is possible in theory, it will be very painful in practice. Worse
yet, the final form of the solution Eq. (25) does not give a clear idea of whatu
looks like. All is not wasted, however. We can say, from an examination of the
λ’s, that the tone produced is not musical — that is,uis not periodic int.Also
we can sketch some of the fundamental modes of vibration of the membrane
corresponding to some low eigenvalues (Fig. 9). The curves represent points
for which displacement is zero in that mode (nodal curves).
EXERCISES
- Verify that each of the functions in the series in Eq. (10) satisfies Eqs. (1),
(2), and (8). - Derive the formulas for thea’s andb’s of Eq. (10).