ECE215A/Materials206A Winter 2008
Prof. Brown/ECE Dept/UCSB
1
Homework 4
1.
Monatomic linear lattice
.
Consider a longitudinal wave
( )
cos
s
uu
t
s
K
a
ω
=−
which propagates in
a monatomic linear lattice of atoms of mass M, spacing
a
, and nearestneighbor interaction C.
(a) Show that the total energy of the wave is given by
()
2
2
11
1
22
,
ss
s
E
M
du
dt
C
u
u
+
=+
−
∑
∑
where s runs over all atoms.
(b) By substitution of
,
s
u
in this expression, show that the timeaverage total energy per atom is
( )
2
1
42
2
1c
o
s
,
M
uC
K
a
u
M
u
ωω
+−
=
where in the last step we have used the longitudinalwave dispersion relation.
2.
Diatomic chain.
Consider the normal modes of linear chain in which the force constants between
nearestneighbor atoms are alternately C and 10C.
Let the masses be equal, and let the nearest
neighbor separation be
2
a
.
Find
ω
(k) at
k = 0
and k =
π
/a.
3.
Show that for long wavelengths the latticewave equation of motion reduces to the continuum elastic
wave equation
2
,
v
tx
∂
∂
=
∂
∂
where v
is the velocity of sound.
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 Winter '08
 Brown
 Thermodynamics, Heat, Gate, Fundamental physics concepts, monatomic linear lattice

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