The solution for the initial displacement given by (8.457) and zero
initial velocity is
1
2
1
(21)(21)
(1)
4
(,)sincos
(21)
m
m
mxmat
yxt
mll
∞
−
=
−−
−
=
−
∑
pp
p
(8.461)
8.13.7 Vibrating String with Given Initial Velocity and
Zero Initial Displacement
Next, we consider the case when the string is released from its horizon-
tal position with zero initial displacement but with an initial velocity
given at x by g ( x). The boundary value problem for the displacement
function is
22
2
22
for0,0
yy
axlt
tx
∂∂
=≤≤>
∂∂
(8.462)
(0,)0(,)for0yt
yltt==>
(8.463), (8.464)
zero initial
displacement:
(,0)0
for0
(,0)()
yx
xl
y
xgx
t
=
≤≤
⎫
⎪
∂
⎬
=
⎪
∂
⎭
prescribed ...
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