Lecture 30: String I (11 Nov 09)
review 9 Nov HW
HJ theory of oscillator
A. Review – continuum string
1. FW Secs. 25 and 38; G Sec. 13.2
2. 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
3. The EulerLagrange equations are obtained from
∂
∂t
∂
L
∂
(
∂u/∂t
)
+
∂
∂x
∂
L
∂
(
∂u/∂x
)

∂
L
∂u
= 0
and for the string
L
are
σ
(
x
)
∂
2
u
∂t
2
=
∂
∂x
τ
(
x
)
∂u
∂x
4. Hamiltonian formulation G Sec.
13.4 (FW Sec.
45), using notation
˙
u
=
∂u/∂t
:
π
=
∂
L
/∂
˙
u
;
H
=
π
˙
u
 L
here:
π
=
σ
(
x
)
∂u
∂t
;
H
=
Z
dx
[
1
2
σ
(
x
)(
∂u
∂t
)
2
+
1
2
τ
(
x
)(
∂u
∂x
)
2
]
5. The case
σ
(
x
) =
σ
and
τ
(
x
) =
τ
(constants) gives rise to the elementary
form of the wave equation with constant speed
c
2
=
τ/σ
:
1
c
2
∂
2
u
∂t
2
=
∂
2
u
∂x
2
1
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6. 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
7. Discrete allowed values of
k
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 Fall '09
 BRUCH
 Mass, Boundary value problem, Partial differential equation, Fundamental physics concepts, wave equation, ∂t ∂X ∂X, FW Sec.

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