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Unformatted text preview: e equation ,
ρ ∂x2 or
= c2 2 ,
∂x where c2 = T /ρ.
Just as in the preceding section, we could approximate this partial diﬀerential
equation by a system of ordinary diﬀerential equations. Assume that the string
has length L = 1 and set xi = i/n and
ui (t) = u(xi , t) = the displacement of the string at xi at time t.
Then the function u(x, t) can be approximated by the vector-valued function u1 (t) u (t) u(t) = 2 ·
of one variable, just as before. The wave equation is then approximated by the
system of ordinary diﬀerential equations
= c2 n2 P u,
where P is the (n − 1) × (n − 1) matrix described in the preceding section. Thus
the diﬀerential operator
dx2 is approximated by the symmetric matrix n2 P, and we expect solutions to the wave equation to behave like solutions to a
mechanical system of weights and springs with a large number of degrees of
4.4.1.a. Show that if f : R → R is any well-behaved function of one variable,
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- Winter '14