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Unformatted text preview: ximated by P = −2
1
0
·
0 1
−2
1
·
0 0
1
−2
·
· ···
···
···
···
··· n2 P u,
0
0
·
·
−2 . Finally, the coeﬃcient (T /ρ(x)) can be represented by the diagonal matrix 0
0
···
0
T /ρ(x1 ) 0
T /ρ(x2 )
0
···
0 .
0
0
T /ρ(x3 ) · · ·
·
Q= ·
·
·
···
·
0
0
·
· · · T /ρ(xn−1 )
Putting all this together, we ﬁnd that our wave equation with variable mass density is approximated by a second order homogeneous linear system of ordinary
diﬀerential equations
d2 u
= Au,
dt2 where A = n2 QP. The eigenvalues of low absolute value are approximated by the eigenvalues of A,
while the eigenfunctions representing the lowest frequency modes of oscillation
are approximated eigenvectors corresponding to the lowest eigenvalues of A.
For example, we could ask the question: What is the shape of the lowest
mode of oscillation in the case where ρ(x) = 1/(x + .1)? To answer this question, 113 we could utilize the following Mathematica program (which is quite similar to
the one presented in...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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