# We then present two derivations of the wave equation

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Unformatted text preview: on the fact that for f, g ∈ V0 , L(f ), g = f , L(g ) , so that if we thought of L as represented by a matrix, the matrix would be symmetric. This identity can be veriﬁed by integration by parts; indeed, b L(f ), g = b ρ(x)σ (x)L(f )(x)g (x)dx = a a b =− K (x) a d dx K (x) df (x) g (x) dx df dg (x) (x)dx = · · · = f , L(g ) , dx dx where the steps represented by dots are just like the ﬁrst steps, but in reverse order. It follows that if fi (x) and fj (x) are eigenfunctions corresponding to distinct eigenvalues λi and λj , then λi fi , fj = L(f ), g = f , L(g ) = λj fi , fj , and hence (λi − λj ) fi , fj = 0. Since λi − λj = 0, we conclude that fi and fj are perpendicular with respect to the inner product ·, · , as claimed. Thus to determine the cn ’s, we can use exactly the same orthogonality techniques that we have used before. Namely, if we normalize the eigenfunctions so that they have unit length and are orthogonal to each other with respect to , ·, · , then cn = h, fn , or equivalently, c...
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## This document was uploaded on 01/12/2014.

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