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

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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 = ´ ¯ v , 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 w , D w + D ˆ w , w D ˆ w , w + D ˆ w , w = 0 . (a) Suppose we don’t have any boudnary conditions; in this case integration by parts gives w , D ˆ w = ˆ u � · v + v ¯ · � u ) Ω = ˆ ( � · u v ] v · � u ¯ + [ u v ¯] u � · v ¯) Ω = −� D ˆ w , w + 2 d Ω u v ) d A ,
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