CHAPTER THREE FOURIER SERIES After completing these tutorials, students should be able to: determine whether the given function is even, odd or neither. compute the Fourier Series for the given function find the Fourier Series for the given function determine the Fourier Cosine Series for the given function on the indicated interval determine the Fourier Sine Series for the given function on the indicated interval find the half-range Cosine and Sine expansions of the given function apply the Fourier Series in solving engineering problems
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CHAPTER THREE FOURIER SERIESdetermine the Fourier Sine Series for the given function on the indicated interval find the half-range Cosine and Sine expansions of the given function
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CHAPTER THREE
FOURIER SERIES
After completing these tutorials, students should be able to:
determine whether the given function is even, odd or neither.
compute the Fourier Series for the given function
find the Fourier Series for the given function
determine the Fourier Cosine Series for the given function on the indicated interval
determine the Fourier Sine Series for the given function on the indicated interval
find the half-range Cosine and Sine expansions of the given function
apply the Fourier Series in solving engineering problems
51
Question 1
In problem I throught III:
a) Sketch the graph of the given function.
b) Determine whether the given function is even, odd or neither.
c) Compute the Fourier Series for the given function.
I ( )f x x ; x
( 2 ) ( )f x f x
Solution:
(a) The graph:
(b) ( )f x is symmetric with respect to the origin
- 0dd function
- Fourier Sine Series
(c) 0
2( )sin
L
n
n xb f x dx
L L
0
0
2
0
2sin
2sin
2 cos sin
2 cos0 (0 0)
2cos
n xx dx
x nxdx
x nx nx
n n
n
n
n
52
2( 1) ; 1,3,5,...
2(1) ; 2,4,6,...
2;
2;
nn
nn
n oddn
n evenn
1
( ) sinn
n
n xf x b
1
1
1
1
2( 1) sin
2( 1)sin
n
n
n
n
n x
n
nxn
II 1( )f x
x
;
;
2 0
0 2
x
x
( 4) ( )f x f x
Solution:
(a) The graph:
(b) ( )f x is neither even nor an odd function.
(c) 0
0
1( )
L
a f x dxL
2
2
2
2 0
1( )
2
11
2
n
f x dx
dx xdx
53
22
0
2
0
1
2 2
1(0 2 2 0)
2
2
x
x
xx
1
( )cos
L
n
L
n xa f x dx
L L
2
2
0 2
2 0
0 2
2 0
2
1( )cos
2 2
11cos cos
2 2 2
1 2 2 2sin sin cos
2 2 2 2
1 2 2 2. (0 0) 0 cos 0
2
2(cos 1)
2
n xf x dx
n x n xdx x dx
n x n x n xx x
n n n
nn n n
nn
2
2
2
( 1 1) ;
2(1 1) ;
2( 1) 1n
n oddn
n evenn
n
1
( )sin
L
n
L
n xb f x dx
L L
2
2
0 2
2 0
1( )sin
2 2
11sin sin
2 2 2
n xf x dx
n x n xdx x dx
54
20 2
2 0
1 2 2 2cos cos sin
2 2 2 2
1 2. 1 cos( ) 2cos 0 0 0
2
1( 1 cos )
n x n x n xx
n n n
n nn
nn
1( 1 ( 1)) ;
1( 1 1) ;
n oddn
n evenn
11 11 ( 1) ( 1 ( 1) )n n
n n
2 1
21
2 1( ) 1 ( 1) 1 cos ( 1 ( 1) )sin
( ) 2 2
n
n
n x n xf x
n n
III 2( )f t t ; 1 1t
( 2) ( )f t f x
Solution:
(a) The graph:
(b) ( )f x is symmetrical with y-axis
- Even function
- Cosine Fourier Series
(c) 2
0
0
2
1
L
a x dx
13
0
23
2(1 0)
3
x
55
1
2
0
2cos
1 1n
n xa x dx
12
0
1
2
2
0
2
2
sin 2 sin2
1 2 cos sin2 sin
( )
22 1cos 0
( )
4(cos )
x n x x n xdx
nx nx
x n x n xx n x
nx nx n n
nn
nn
2
2
2
4( 1) ; 1,3,5,...
4(1) ; 2,4,6,...
4( 1)n
nn
nn
n
2
1
1 4( ) ( 1) cos
3
n
n
f x n xn
Question 2
Find the Fourier Series for the given function.
(a)
2
0( )f x
x
;
;
0
0
x
x
Solution:
0
1( )a f x dx
0
2
0
3
0
3 2
10
1
3
1
3 3
dx x dx
x
56
1( )cosn
n xa f x dx
0
2
0
2
00
2
0 0
2
0
10cos cos
1 sin 2sin
1 2 cos 1sin cos
1 2 cos 1 sinsin
nxdx x nxdx
x nxx nxdx
n n
x nxn nxdx
n n n n
x nx nxn
n n n n n
2
2 2
1 2 cos 2sinsin
n nn
n n n
2
2
2
1 2 cos
2cos
2( 1)n
n
n
nn
n
1( )sinn
n xb f x dx
0
2
0
2
00
2
0 0
2
2 2
0
2
2
10sin sin
1 cos 2cos
1 2 sin sincos
1 2 2 coscos sin
1 2cos s
nxdx x nxdx
x nxx nxdx
n n
x nx nxn dx
n n n n
nxn n
n n n n
nn n
3 3
3 3
2 cos 2in
2( 1) 2( 1)
nn
nn
n n
n n n
2 1
2 31
2 ( 1) 2( ) ( 1) cos ( 1) 1 sin
6
nn n
n
f x nx nxn n n
57
Question 3
Determine the Fourier Cosine Series for the following functions on the indicated interval.
(a)
0( )
1f x
;
;
0
2
x
x
Solution:
2
0
0
2( )
2a f x dx
2
0
2
10 1
1
dx dx
x
12
1
2
0
2( )cos
2n
n xa f x dx
2
0
2
10cos 1.cos
2 2
1 2sin
2
1 2 2sin sin
2
1 2sin
2
nx nxdx dx
nx
n
nn
n n
n
n
1
1 2( ) sin cos
2 2 2n
n nxf x
n
1
1 2sin cos
2 2 2n
n nx
n
(b)
( ) xf x e ; 0 1x
Solution: 1
0
0
2( )
1a f x dx
1
0
2 xe dx
58
1
02 xe
1 0
1
2( )
12 1
e e
e
1
0
2( )cos
1 1n
n xa f x dx
1
0
2 cosxe n xdx
11 1
0 00
cos sin sinx x
x e ee n xdx n x n xdx
n n
1 1 1 1
0 0
1cos cos sin cos
xx e e e
e n xdx n xdx n nn n n n
1 1
0
1 11 cos cosx e
e n xdx nn n n
1 1
0
1 ( 1)cos
1
nx e
e n xdxn
1
0
2 cosx
na e n xdx
12 2 ( 1)
1
ne
n
1
1
1 2 2 ( 1)( ) 1 cos
1
n
n
ef x n x
e n
1
1
1 1 ( 1)1 2 cos
1
n
n
en x
e n
xu e ' cosv n x
' xu e sin n x
vn
xu e ' sinv n x
' xu e cos n x
vn
59
Question 4
Determine the Fourier Sine Series for the following functions on the indicated interval.
(a) ( ) 1f x x ; 0 1x
Solution:
1
0
2( )sin
1 1n
n xb f x dx
1
0
1 1
0 0
1 11
0 00
1
2
0
2 ( 1)sin
2 sin sin
cos cos cos2
cos sin cos 12
x n xdx
x n xdx n xdx
x n x n x n x
n n n
n n x n
n n nn
1 2cos
2n
n n
1
2( ) 1 2( 1) sinn
n
f x n xn
(b) ( )
2
xf x
x
;
; 0 1
1 2
x
x
Solution:
2
0
2( )sin
2 2n
n xb f x dx
1 2
0 1
1 2 2
0 1 1
2 31
sin (2 )sin2 2
sin 2sin sin2 2 2
case casecase
n nx xdx x xdx
n n nx xdx xdx x xdx
' 1
u x
u
' sin2
2cos
2
n xv
n xv
n
60
Case 1:
11 1
00 0
2 2sin cos cos
2 2 2
n n x n xx xdx dx
n n
2 1
0
2
2 2cos sin
2 2
2 4cos sin
2 2
n x n x
n n
n x n
n n
Case 2: 22
11
22sin 2 cos
2 2
n n xxdx
n
4
cos cos2
nn
n
Case 3: 2 2 22
1 11
2 2sin cos sin
2 2 2
n n x n xx xdx
n n
2
4 2 4cos cos sin sin
2 2
n nn n
n n n
2
2 4 4 4 4 2cos sin cos cos cos cos
2 2 2 2n
n n n nb n n
n n n n nn
2 2
4 4sin sin
2
nn
n n
2
8sin
2
n
n
2
1
8( ) sin sin
2 2n
n n xf x
n
u x ' sin2
n xv
' 1u 2
cos2
n xv
n
61
Question 5
Expand the given function in an appropriate Cosine or Sine Series.
(a) 2( )f x x ; 1 1x
Solution:
( )f x is an even function, 0nb
1
0
0
2( )
1a f x dx
1
2
0
13
0
2
23
12
3
2
3
x dx
x
1
0
2( )cos
1 1n
n xa f x dx
1
2
0
2 cosx n xdx
2u x ' cosv n x
' 2u x sin n x
vn
62
1 12
00
sin sin2 2n
x n x x n xa dx
n n
1 1
0 0
sin 2 cos cos2n
n x n x n xa
n n n n
1
2
0
2 2 cos sin2
n n
n n n n
2 3
2
2
2 2cos 2sin2
2 2( 1)2
4 ( 1) 1
n
n
n n
n n n
n n
n n
21
1 4 ( 1) 1( ) cos
3
n
n
f x n xn n
(b)
3( )f x x ; x
Solution:
( )f x is an odd function, 0 0na a
0
2( )sinn
n xb f x dx
u x ' sinv n x
' 1u cos n x
vn
63
3
0
2sin
nxx dx
32
00
2 cos 3cosn
x nxb x nxdx
n n
3 2
00
2 cos 3 sin 2 sinn
n x nx x nxb dx
n n n n
3 2
0
2 cos 3 sin 2sin
n nx nxdx
n n n n
32
2 2
0 0
2 cos 3 6 sin 1cos sinn
n x nxb n xdx
n n n n n
3
2 2 0
3
3 2 2
3
2 2
3
2 2
2 cos 6 sin 1cos
2 cos 6 sin cos 1
2 cos cos 1
2 ( 1) ( 1) 1n n
n nnx
n n n n
n n n
n n n n
n n
n n n
n n n
3
2 21
2 ( 1) ( 1) 1( ) sin( )
n n
n
f x nxn n n
3u x ' sinv nx
2' 3u x cos nx
vn
2u x ' cosv nx
' 2u x sin nx
vn
u x ' sinv nx
' 1u sin nx
vn
64
Question 6
Find the half-range Cosine and Sine expansions of the given function.
( )
x
f x
x
;
;
02
2
x
x
Solution:
Half-range cosine expansion:
0
0
2( )a f x dx
2
0
2
2( )x dx x dx
2 22
02
2 2 2 22
2
2
2 2
2
8 2 2 8
2
4 2
x
x xx
0
2( )cosn
n xa f x dx
2
0
2
2
0
2 21
2 3
2cos cos
2cos cos cos
case
case case
x nxdx x nxdx
x nxdx nxdx x nxdx
65
Case 1:
2 22
00 0
sin( ) sin( )cos
x nx nxx nxdx dx
n n
22 0
2
1sin cos( )
2 2
1sin cos 1
2 2 2
nnx
n n
n n
n n
Case 2:
2
2
cos sinnxdx nn
sin sin2
nn
n
Case 3:
2
2 2
sin 1cos sin
x nx nxdx n dx
n n
22
2 2
1sin sin cos
2 2
1 1sin sin cos cos
2 2 2
nn nx
n n n
n nn n
n n n n
2 2 2 2
2 1 1 1 1sin cos sin sin cos cos
2 2 2 2 2 2 2n
n n n n na n
n n n n n n n
2 2 2
2
2 2 1 1cos cos
2
2coscos ( ) 12 2
n
nn
n n n
n
n
21
2coscos ( ) 13 2( ) cos
4 4
n
n
n
f x nxn
66
Half-Range Sine Expansion:
0
2( )sinn
n xb f x dx
2
0
2
2
0
2 21
2 3
2sin( ) ( )sin
2sin( ) sin sin
case
case case
x nx dx x nx dx
x nx dx nx dx x nx dx
Case 1:
2 22
0 00
cos cossin( )
x n nx nx dx dx
n n
22 0
2
1cos sin( )
2 2
1cos sin
2 2 2
nnx
n n
n n
n n
Case 2:
0
2
2
sin( ) cosnx dx x nn
cos cos2
nn
n
67
Case 3:
2
2 2
cos cossin
x nx nxx nx dx dx
n n
2
2
2 2
cos cos sin2 2
sin2cos cos sin
2 2
n nxn
n n n
nn n
nn n n n
2 2 2
2 1 1 2 1cos sin cos cos cos cos cos sin
2 2 2 2 2 2 2n
n n n n nb n n n
n n n n n n n
2
2 2sin
2
n
n
21
4 1( ) sin sin( )
2n
nf x nx
n
Question 7
A mass-spring system, in which a piston is attached to the end of the spring and drives the
system.The driving force acts on the piston is represented by ( )f t .Find the Fourier series of
( )f t .
( )k
f tk
,
,
0
0
t
t
tk
tktf
0,
0,)(
( )f t
k
t
k
68
Fourier sines series
1
2 1 1( )( ) sin
n
n
kf t nt
np
Question 8
Find the Fourier series of the given input signal.