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# Therefore we obtain w w w dw dw w dw w f

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Unformatted text preview: · v + v · �u) u ˆΩ ¯ ¯ = (� · [uv] − v · �u + � [uv] − u� · v) ¯ ¯ Ω ‹ ˆ = −�Dw, w� + 2� (uv)dA, ¯ dΩ where the surface integral comes from the divergence theorem and � denotes the real part (since z + z = 2�z ). ¯ Therefore, we obtain: ‹ ‹ ∂ ˆ ˆ ˆ ˆ �w, w� = �w, Dw� + �Dw, w� = −�Dw, w� + F · dA + �Dw, w� = F · dA, ∂t dΩ dΩ where F = 2�(¯v). u Thus, −F can be interpreted as a ﬂux of energy per unit area per unit time. (Note the sign: the loss of ∂ energy from Ω is − ∂ t �w, w�.) ˆ (b) Plugging w(x, y, z, t) = wk (x, y )ei(kz−ωt) into the eigenequation Dw = −iω w, cancelling e−iωt from both sides, and moving eikz to the left side, we obtain: ˆ ˆ Dk wk = e−ikz Deikz wk = −iω wk , ˆ where as in class this transformation of D turns each � in...
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