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Unformatted text preview: x/L)
+[a2 cos(2cπt/L) + b2 sin(2cπt/L)] sin(2πx/L) + . . . . (4.21)
(4.22) The vibration of the string is a superposition of a fundamental mode which has
frequency
T /ρ
cπ 1
c
=
=
,
L 2π
2L
2L
and higher modes which have frequencies which are exact integer multiples of
this frequency.
Step II consists of determining the constants an and bn in (4.22) so that the
initial conditions
u(x, 0) = h1 (x) and ∂u
(x, 0) = h2 (x)
∂t are satisﬁed. Setting t = 0 in (4.22) yields
h1 (x) = u(x, 0) = a1 sin(πx/L) + a2 sin(2πx/L) + . . . ,
so we see that the an ’s are the coeﬃcients in the Fourier sine series of h1 .
If we diﬀerentiate equation(4.22) with respect to t, we ﬁnd that
∂u
−cπ
cπ
(x, t) = [
a1 sin(cπt/L) +
b1 cos(cπt/L)] sin(πx/L)
∂t
L
L
100 −2cπ
cπ
[a2 sin(2cπt/L) +
b2 sin(2cπt/L)] cos(2πx/L) + . . . ,
L
L
and setting t = 0 yields
+ h2 (x) = ∂u
2cπ
cπ
(x, 0) =
b1 sin(πx/L) +
b2 sin(2πx/L) + . . . .
∂t
L
L We conclude that
ncπ
bn = the nth coeﬃcient in the Fourier s...
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 Winter '14
 Equations

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