
8.5. Consider a uniform string fixed at both ends. It has mass per unit length m, length L, and tension
T. A spring with stiffness k is attached to the string at x = L/2. Find the eigenfunctions and eigenvalues.
Solution: Let us divide the string into two segments. Let y
L
be the displacement for the segment to
the left of the spring and y
2
for the segment to the right of the spring. y
L
is defined between x
1
= 0 and
x
1
= L/2 and y
2
between x
2
= 0 and x
2
= L/2 as shown in Figure 31. The equations of motion are given by
m T
,
L
/2
1
Figure 31: String configuration
T
1
y
1
−m¨y
1
= 0 for 0 < x
1
< L/2
T
2
y
2
−m¨y
2
= 0 for 0 < x
2
< L/2.
For now, it is assumed that the tension ...