Homework 3 (Due by July 22nd)
Problem 1 (4.4.3 (b))
Consider a slightly damped vibrating string that satisfies
ρ0
∂2u
∂2u
∂u
=
T
−β
0
2
2
∂t
∂x
∂t
Determine the solution (by separation of variables) that satisfies the boundary conditions
u(0, t) = 0 and u(L, t) = 0
and the initial conditions
u(x, 0) = f (x) and
∂u
(x, 0) = g(x).
∂t
You can assume that this frictional coefficient β is relatively small (β 2 < 4π 2 ρ0 T0 /L2 ).
Hint: the general solution of the equation pT 00 + qT 0 + rT = 0, where q 2 < 4pr, is
p
p
q 4pr − q 2
4pr − q 2 − 2p
t
a cos
T (t) = e
t + b cos
t
2p
2p
Problem 2 (5.3.5)
For the Sturm-Liouville eigenvalue problem
dφ
dφ
d2 φ
+ λφ = 0 with
(0) = 0 and
(L) = 0,
2
dx
dx
dx
verify the following general properties:
(a) There is an infinite number of eigenvalues with a smallest but no largest.
(b) The nth eigenfunction has n − 1 zeros.
(c) The eigenfunctions are complete and orthogonal.
(d) What does the Rayleigh quotient say concerning negative and zero eigenvalues?
Problem 3 (5.4.5)
Consider
∂2u
∂2u
=
T
+ αu,
0
∂t2
∂x2
where ρ(x) > 0, α(x) < 0, and T0 is a constant, subject to
ρ
u(0, t) = 0, u(L, t) = 0
∂u
(x, 0) = g(x).
∂t
Assume that the approriate eigenfunctions are known. Solve the initial value problem.
u(x, 0) = f (x),