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problemset10 - 18.303 Problem Set 10 Solutions Problem 1(10...

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± ± ± ± ² ³ ± 18.303 Problem Set 10 Solutions Problem 1: (10+10+10+10 points) u In class (and notes), we showed that we can turn the scalar wave equation b · ( a u ) = 2 ( a > 0 and b > 0 ) into ∂t 2 two coupled first-derivative equations: ∂u = b · v , v = a u by introducing a new (vector) unknown v ( x ,t ) . By ∂t ∂t defining w = ( u, v ) T , we obtained the form w = b · w = D ˆ w , ∂t a 1 where D ˆ was anti-Hermitian ( ˆ = D ) under the inner product ± w , w ² = ´ ² ¯ v ³ , for appropriate D ˆ uu + 1 ¯ v Ω b a · boundary conditions (e.g. = 0 ). This gave us conservation of “energy”: = ˆ = u | d Ω ∂t ± w , w ² ± w ,D w ² + ± D ˆ w , w ² ± D ˆ w , w ² + ± D ˆ w , w ² = 0 . (a) Suppose we don’t have any boudnary conditions; in this case integration by parts gives ± w ˆ w ² = ˆ u · v + v ¯ · u ) Ω = ˆ ( · u v ] v · u ¯ + [ u v ¯] u · v ¯) Ω = −± D ˆ w , w ² + 2 ³ d Ω u v ) d A , where the surface integral comes from the divergence theorem and ³ denotes the real part (since z + ¯ z = 2 ³ z ).
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problemset10 - 18.303 Problem Set 10 Solutions Problem 1(10...

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