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Lecture 31: Strings II (13 Nov 09)
Move Set VIII to Nov 23?
A. Continuum string
1. The Lagrangian density for transverse (
u
) vibrations of a 1D string
with mass density
σ
and spatially varying tension
τ
(
x
) is
L
=
1
2
σ
(
x
)(
∂u
∂t
)
2

1
2
τ
(
x
)(
∂u
∂x
)
2
2. The EulerLagrange equation gives a “wave equation”
∂
∂t
∂
L
∂
(
∂u/∂t
)
+
∂
∂x
∂
L
∂
(
∂u/∂x
)

∂
L
∂u
= 0
→
σ
(
x
)
∂
2
u
∂t
2
=
∂
∂x
τ
(
x
)
∂u
∂x
3.
σ
(
x
) =
σ
;
τ
(
x
) =
τ
(constants)
→
“elementary” wave equation:
1
c
2
∂
2
u
∂t
2
=
∂
2
u
∂x
2
;
c
2
=
τ/σ
4. Normal modes: separate time and space variables
u
(
x,t
) =
ρ
(
x
) cos(
ωt
+
δ
)
and deﬁne
k
2
=
ω
2
/c
2
. The equation for
ρ
(
x
) is
d
2
ρ
dx
2
+
k
2
ρ
= 0
5. Discrete values of
k
(and hence of
ω
) are set by boundary conditions
on a ﬁnite length 0
< x < ‘
. Fixed ends give (normalized solutions)
ρ
n
(
x
) =
q
2
/‘
sin
k
n
x
;
k
n
=
nπ/‘
1
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View Full Document6. The solution for general (
x,t
) can then be constructed from the initial
position
u
(
x,
0), here using the fourier sine series.
u
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 Fall '09
 BRUCH
 Mass

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