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With this equation in place the boundary condition
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@y x; 0 0, 0 < x < L can be considered. @t @y h n ih n n n n i c1 sin x c3 cos t c4 sin t @t L L L L L h n i n 0 c1 sin x c3 L L
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At t 0
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Since c1 6 0 and sin
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n x is not identically zero, it follows that c3 0 and that L h n ih n n i x c4 cos t y c1 sin L L L y x; 0 m Lx x2 ; 0 < x < L
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The remaining initial condition is
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When it is imposed m Lx x2 c1 c4 n n sin x L L
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However, this relation cannot be satis ed for all x on the interval 0; L . Thus, the preceding extensive analysis of the problem of the vibrating string has led us to an inadequate form n n n y c1 c4 sin x cos t L L L and an initial condition that is not satis ed. At this point the power of Fourier series is employed. In particular, a theorem of di erential equations states that any nite sum of a particular solution also is a solution. Generalize this to in nite sum and consider y
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1 X n 1
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bn sin
n n x cos t L L
with the initial condition expressed through a half range sine series, i.e.,
1 X n 1
bn sin
n x m Lx x2 ; L
According to the formula of Page 338 for coe cient of a half range sine series L L nx bn Lx x2 sin dx 2m L 0 That is L b 2m n L Lx sin
nx dx L
L x2 sin
nx dx L
Application of integration by parts to the second integral yields L L L nx L3 L nx bn L x sin dx cos 2x dx cos n 2m L n L n 0 0 When integration by parts is applied to the two integrals of this expression and a little algebra is employed the result is bn 4L2 1 cos n n 3
FOURIER SERIES
[CHAP. 13
Therefore, y with the coe cients bn de ned above.
1 X n 1
bn sin
n n x cos t L L
ORTHOGONAL FUNCTIONS Two vectors A and B are called orthogonal (perpendicular) if A B 0 or A1 B1 A2 B2 A3 B3 0, where A A1 i A2 j A3 k and B B1 i B2 j B3 k. Although not geometrically or physically evident, these ideas can be generalized to include vectors with more than three components. In particular, we can think of a function, say, A x , as being a vector with an in nity of components (i.e., an in nite dimensional vector), the value of each component being speci ed by substituting a particular value of x in some interval a; b . It is natural in such case to de ne two functions, A x and B x , as orthogonal in a; b if b
A x B x dx 0
A vector A is called a unit vector or normalized vector if its magnitude is unity, i.e., if A A A2 1. Extending the concept, we say that the function A x is normal or normalized in a; b if b
fA x g2 dx 1
10
From the above it is clear that we can consider a set of functions fk x g; k 1; 2; 3; . . . ; having the properties b b
m x n x dx 0
m 6 n
11 12
fm x g2 dx 1
m 1; 2; 3; . . .
In such case, each member of the set is orthogonal to every other member of the set and is also normalized. We call such a set of functions an orthonormal set. The equations (11) and (12) can be summarized by writing b m x n x dx mn
13
where mn , called Kronecker s symbol, is de ned as 0 if m 6 n and 1 if m n. Just as any vector r in three dimensions can be expanded in a set of mutually orthogonal unit vectors i; j; k in the form r c1 i c2 j c3 k, so we consider the possibility of expanding a function f x in a set of orthonormal functions, i.e., f x
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