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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