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Unformatted text preview: l to the sum of its Fourier series?
y
1 FIGURE 3
The triangular wave function _1 0 1 2 x F OURIER SERIES ■ 7 SOLUTION We ﬁnd the Fourier coefﬁcients by putting L
1
2 a0 y Notice that a 0 is more easily calculated as
an area. 1 ■■ 1
4 and for n 1
2 x dx 1 x2 0 1
4 1 x2 y 0 1
0 1
2 x dx 1 1 in (9): y 1 0 x dx 1
2 1, y an 1 1 2 y x cos n x d x x cos n x d x 1 0 because y
x cos n x is an even function. Here we integrate by parts with u
and dv cos n x d x. Thus,
x
2
sin n x
n an 2
n 0 Since y 1 2
n 0 y 1 0 1 cos n x
n sin n x d x
2
n2 0 x 2 cos n 1 x sin n x is an odd function, we see that y bn 1 x sin n x d x 1 0 We could therefore write the series as
1
2
But cos n n1 2 cos n
n2 1 if n is even and cos n n2 cos n x 1 if n is odd, so
0 2 an 1
2 cos n 2 if n is even 1 4
n2 if n is odd 2 Therefore, the Fourier series is
1
2 4
2 cos 1
2 4 x 9 2 cos 3 x 4
n1 2k 1 2 2 cos 2k 4
25
1 2 cos 5 x x The triangular wave function is continuous everywhere and so, according to the Fourier
Convergence Theorem, we have
fx 1
2 4
n1 2k 1 2 2 cos 2k 1 x for all x 8 ■ FOURIER SERIES In particular,
1
2 x 4
2k k1 1 2 2 cos 2k 1 x for 1 x 1 Fourier Series and Music
One of the main uses of Fourier series is in solving some of the differential equations that
arise in mathematical physics, such as the wave equation and the heat equation. (This is
covered in more advanced courses.) Here we explain brieﬂy how Fourier series play a role
in the analysis and synthesis of musical sounds.
We hear a sound when our eardrums vibrate because of variations in air pressure. If a
guitar string is plucked, or a bow is drawn across a violin string, or a piano string is struck,
the string starts to vibrate. These vibrations are ampliﬁed and transmitted to the air. The
resulting air pressure ﬂuctuations arrive at our eardrums and are converted into electrical
impulses that are processed by the brain. How is it, then, that we can distinguish between
a note of a given pitch produced by two different musical...
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This note was uploaded on 12/05/2011 for the course MATH 41 taught by Professor Bray,c during the Spring '08 term at Duke.
 Spring '08
 Bray,C
 Math, Fourier Series

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