Matakuliah
Tahun
: K0124 / Matematika Teknik II
: 2006/2007
PERTEMUAN 15
VIBRATING STRING EQUATION
(finding a solution)
1
A string of length L is strecthed between points (0, 0)
and (L, 0) on the x axis.
At time t = 0 it has a shape given by f(x), 0 < x< L
and it is released from rest.
2
Find the displacement of the string at any later time
y
Y =(x, t)
L

3
The equation of the vibrating string is
2
2 y
2  y
a
2
t
x 2
0<
x < L, t > 0. where y(x, t) = displacement from x
axis at time t.
To solve this boundy-value problem , let y = XT as
usual.
"
2
"
"
2
"
XT

a
X
T
or
T
a
T

X
X
Then
4
Calling the separation constant
 2
, we have
T "  2 a 2T  0, X "  2 X  0

The result is
m x 
m x
m at
2 L
y ( x, t )     f ( x) sin  dx  sin  cos 
0
L
L
L

m 1  L
which
can be verified as the solution.
5
TERIMA KASIH
6