3.5 Estimation of Eigenvalues 237
see that the frequencies of vibration areλnc/ 2 π,n= 1 , 2 , 3 ,....Thuswemust
find the eigenvaluesλ^2 nin order to identify the frequencies of vibration.
Consider the following Sturm–Liouville problem:
(
s(x)φ′
)′
−q(x)φ+λ^2 p(x)φ= 0 , l<x<r, (1)
φ(l)= 0 ,φ(r)= 0 , (2)wheres,s′,q,andpare continuous andsandpare positive forl≤x≤r.(Note
that we have a rather general differential equation but very special boundary
conditions.)
Ifφ 1 is the eigenfunction corresponding to the smallest eigenvalueλ^21 ,then
φ 1 satisfies Eq. (1) forλ=λ 1 .Alternatively,wecanwrite
−(
sφ′ 1)′
+qφ 1 =λ^21 pφ 1 , l<x<r.Multiplying through this equation byφ 1 and integrating fromltor,weobtain
∫r
l−
(
sφ 1 ′)′
φ 1 dx+∫rlqφ^21 dx=λ^21∫rlpφ^21 dx.If the first integral is integrated by parts, it becomes
−sφ′ 1 φ 1∣∣r
l+∫rlsφ′ 1 φ 1 ′dx.Butφ 1 (l)=φ 1 (r)=0, so the first term vanishes and we are left with the equal-
ity
∫r
ls[
φ′ 1] 2
dx+∫rlqφ^21 dx=λ^21∫rlpφ^21 dx.Becausep(x)is positive forl≤x≤r, the integral on the right is positive and
we may defineλ^21 as
λ^21 =∫r
ls[φ
1 ′]^2 dx+∫r
lqφ
12 dx
∫r
lpφ
12 dx =N(φ 1 )
D(φ 1 ). (3)
It can be shown that, ify(x)is any function with two continuous derivatives
(l≤x≤r)that satisfiesy(l)=y(r)=0, then
λ^21 ≤N(y)
D(y). (4)
By choosing any convenient functionythat satisfies the boundary conditions,
we obtain from the ratioN(y)/D(y)an upper bound onλ^21. Usually this bound
is quite a good estimate. One should keep in mind that the graph of the eigen-
functionφ 1 (x)does not cross thex-axis betweenlandr,sothegraphofy(x)
should not cross the axis either.