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Solved Problems

Exercise 162. If the striking hammer is not perfectly rigid, then its effect must
be included as a time dependent forcing term of the form:

( π(x−ξ)
v cos sin πt , for |x − ξ| < d, 0 < t < δ,
s(x, t) = 2d δ
0 otherwise.

Find the motion of the string for t > δ. Discuss the effects of the width of the
hammer and duration of the blow with regard to the energy in overtones.
Solution: In mathematical notation, the problem is

2
u − c u xx = s(x, t), 0 < x < L, t > 0,
tt
u(0, t) = u(L, t) = 0,
u(x, 0) = u (x, 0) = 0.
t

Since this is an inhomogeneous partial differential equation, we will expand the
solution in a series of eigenfunctions in x for which the coefficients are functions of
t. The solution for u has the form,

∞

X nπx
u(x, t) = u (t) sin .
n
L
n=1
Substituting this expression into the inhomogeneous partial differential equation will
give us ordinary differential equations for each of the u .
n
∞
2
X 00 2 nπ nπx
u + c u n sin = s(x, t).
n
n=1 L L
We expand the right side in a series of the eigenfunctions.

∞

X nπx
s(x, t) = s (t) sin .
n
L
n=1
For 0 < t < δ we have
Z L
2 nπx
s (t) = s(x, t) sin dx
n
L 0 L
2 Z L π(x − ξ) πt nπx
= v cos sin sin dx
L 0 2d δ L

8dLv nπd nπξ πt
= cos sin sin .
2 2
2
π(L − 4d n ) L L δ
For t > δ, s (t) = 0. Substituting this into the partial differential equation yields,
n

(
8dLv nπd nπξ πt
nπc 2 2 2 2 cos sin sin , for t < δ,
00
u + u = π(L −4d n ) L L δ
n
n
L 0 for t > δ.
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