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Phys. 7268
Assignment 5
Due: 3/3/11
Problem 1
Hexagonal patterns.
Consider the superposition of modes
u
(
x,y
) =
e
i
(
q
1
·
x
+
φ
1
)
+
e
i
(
q
2
·
x
+
φ
2
)
+
e
i
(
q
3
·
x
+
φ
3
)
+
c.c.
with
x
= (
x,y
) and the wave vectors
q
1
,
q
2
, and
q
3
forming an equilateral triangle
q
1
+
q
2
+
q
3
= 0
,

q
i

= 1
and where the
φ
i
are the possible phases of the modes. Since
q
sets the scale of the pattern, let’s choose
q
= 1, and
orient our axes so that
q
1
= (1
,
0).
(a) Show that by redeﬁning the origin of coordinates we may set
φ
1
and
φ
2
to zero, so that the from of the pattern
(outside of translations and rotations) is determined by a single phase variable
φ
3
=
φ
.
(b) Using Mathematica, Matlab, or some other convenient plotting environment make some contour or density plots
of the ﬁeld
u
(
x,y
) for various choices of
φ
. Do you get hexagonal patterns for an arbitrary choice of
φ
?
(c) As we will discuss later, the amplitude equation analysis near the onset of the instability shows that the sum
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This note was uploaded on 08/25/2011 for the course PHYS 7268 taught by Professor Staff during the Spring '11 term at Georgia Institute of Technology.
 Spring '11
 Staff

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