Lecture 19: Continuum string (16 Oct 09)
Midterm exam, takehome, handed out in class today, due Monday 19 Oct.
at 5 PM at 5338 Chamberlin or LB mailbox.
No class on Monday.
A. continuum approx – discrete chain
1. Chain of spacing
a
,
N
atoms in repeat unit (or in chain with specified
ends). “Shortest wavelength” is
λ
=
a
, longest is
λ
=
Na
.
2. Wavenumber is defined by
k
= 2
π/λ
, long wavelength, smooth varia
tion has
ka <<
1.
3. Chain with successive displacements having relative phase exp(
iq
);
q
≡
ka
;
x
coordinate
x
j
=
ja
4. The equations of motion for the chain were
m
¨
η
j
=

K
[2
η
j

η
j
+1

η
j

1
]
with phased displacements
η
j
∝
exp(
ijq
) having
ω
(
q
) =
q
4
K/m
sin(
q/
2).
5. Go to a notation
η
j
→
ξ
(
x
j
, t
) and use finite differences for the second
derivative:
ξ
(
x
j
+1
, t
) +
ξ
(
x
j

1
, t
)

2
ξ
(
x
j
, t
)
’
(
δx
)
2
∂
2
ξ/∂x
2
6. Then the equations of motion become, with density
σ
=
m/a
and
tension
τ
=
Ka
σ
∂
2
ξ
∂t
2
=
τ
∂
2
ξ
∂x
2
7. The “smooth variation” of
ξ
corresponds to
q <<
1 or
ω
(
q
)
’
q
q
K/m
=
k
q
τ/σ.
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 Fall '09
 BRUCH
 Equations, Derivative, Lagrangian mechanics, C. Continuum Lagrangian

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