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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
and higher modes which have frequencies which are exact integer multiples of
Step II consists of determining the constants an and bn in (4.22) so that the
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
(x, t) = [
a1 sin(cπt/L) +
b1 cos(cπt/L)] sin(πx/L)
[a2 sin(2cπt/L) +
b2 sin(2cπt/L)] cos(2πx/L) + . . . ,
and setting t = 0 yields
+ h2 (x) = ∂u
(x, 0) =
b1 sin(πx/L) +
b2 sin(2πx/L) + . . . .
L We conclude that
bn = the n-th coeﬃcient in the Fourier s...
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- Winter '14