displacement but with an initial velocity given at x
by g ( x). The boundary value problem for the dis-
placement function is
22
2
22
for0,0
yy
axlt
tx
∂∂
=≤≤>
∂∂
(12.126)
(
0,
)
0
(
,
)
for0
y
t
y
ltt==>
(12.127, 12.128)
zero initial
displacement:
(,0)0
for0
(,0)()
yx
xl
y
xyx
t
=
≤≤
⎫
⎪
∂
⎬
=
⎪
∂
⎭
prescribed
initial v
elocity: (12.129, 12.130)
By the method of separation of variables w
e set
y (x, t) =x (t) T (t) and obtain ordinary differential
equations
222
0and0XXTaT+=+=
′′′′
ll
(12.131, 12.132)
The boundar
y conditions are same as before and
hence w
e obtain eigen
v
alues
22
2
2
n
n
l
=
p
l
(12.133)
and the corresponding eigenfunctions ...
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