The free vibration equation of a bar (length ) along the axial direction can be expressed as
 2u
 2u
c 2 2 ( x, t )  2 ( x, t )
x
t
The bar is fixed at x  0 and free at x   and the boundary conditions can be expressed as
u(0, t)  0, t  0
u
(, t)  0, t  0
x
The initial conditions can be stated as u0(x) and u0 (x). Using the method of separation of
variables u(x, t)  U(x)T(t) to find the eigenfunctions un(x, t) and eigenvalues n of this vibrating
bar and the general free vibration solution. (98清大動機)
Solution: Substitution of u(x, t)  U(x)T(t) in the governing equation, there follows
T  U 
c2U T  UT   0  2 
 2 U  + 2U  0, T  + c22T  0
cT U
Boundary conditions u(0, t)  0  U(0)T(t)  0  U(0)  0
u
(, t)  0  U ()T(t)  0  U ()  0
x
First we consider the problem U + 2U  0, U(0)  0, U ()  0
The solution is
U(x)  c1cosx + c2sinx
U(0)  0  c1  0
U ()  0  c2cos  0
For nontrivial solution we must have cos  0, and find the eigenvalues n 
( 2n  1) 
2
and eigenfunctions Un  sinnx.
Next we consider the equation of T, the solution of which is
Tn  ancoscnt + bnsincnt
and
un(x, t)  (ancoscnt + bnsincnt)sinnx

The solution u( x, t )   ( a n cos c n t  bn sin c n t ) sin  n x
n 1
u
( x, t )   ( c n a n sin c n t  c n bn cos c n t ) sin  n x
t
n 1

Applying the initial conditions, we obtain


2 
2
u0   c n bn sin  n x  c n bn   u0 sin  n xdx  bn 
u0 sin  n xdx

 0
c n  0
n 1

u0   a n sin  n x  a n 
n 1
2 
2 
u0 sin  n xdx  a n   u0 sin  n xdx

0

 0
Wave equations 1