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Solved Problems

This gives us differential equations for X(x), Y (y) and T(t).

00
X = −µX, X(0) = X(a) = 0
00
Y = −(ν − µ)Y, Y (0) = Y (b) = 0
00
2
T = −c νT
First we solve the problem for X.

mπ mπx
2
µ = , X = sin
m
m
a a
Then we solve the problem for Y .
mπ nπ nπy
2 2
ν m,n = + , Y m,n = sin
a b b

Finally we determine T.

!
r
n
cos m 2 2
T m,n = cπ + t
sin a b
The modes of oscillation are

!
r
mπx nπy cos m n
2 2
u m,n = sin sin cπ + t .
a b sin a b
The frequencies are
r
m n
2 2
ω m,n = cπ + .
a b

Exercise 161.
(a) A piano string of length L is struck, at time t = 0, by a flat hammer of width
2d centered at a point ξ, having velocity v. Find the ensuing motion, u(x, t),
of the string for which the wave speed is c.

(b) Suppose the hammer is curved, rather than flat as above, so that the initial
velocity distribution is

( π(x−ξ)
v cos , |x − ξ| < d
u (x, 0) = 2d
t
0 |x − ξ| > d.
Find the ensuing motion.

(c) Compare the kinetic energies of each harmonic in the two solutions. Where
th
should the string be struck in order to maximize the energy in the n harmonic
in each case?

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