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Unformatted text preview: 3. The wave equation utt − c2 uxx = 0 can be written as a system of two ﬁrst order PDEs
∂v
∂v
−c
=0
∂t
∂x
and ∂u
∂u
+c
= v.
∂t
∂x
Solving the ﬁrst for v , by rewriting it as a system of ODEs
dv
=0
dt
dx
= −c
dt The characteristic equation is solved
x = −ct + x0
and then
v (x, t) = v (x0 , 0) = V (x + ct)
where V is the initial solution for v . Now use this solution in the second PDE rewritten as
a system of ODEs
du
= V (x + ct)
dt
dx
=c
dt
The characteristic equation is solved
x = ct + x0
and then du
= V (x + ct) = V (x0 + 2ct)
dt Integrating
u(x0 , t) = t
0 V (x0 + 2cτ )dτ + K (x0 ) Change variables
z = x0 + 2cτ
then
dz = 2cdτ 278 ...
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 Fall '08
 BELL,D

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