1 chapter 7 fourier series (fourier 급수 ) mathematical methods in the physical sciences 3rd...
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1
Chapter 7 Fourier Series (Fourier 급수 )
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture1 Periodic function
Lecturer: Lee, Yunsang (Physics)Baird-Hall 01318 [email protected]
2
1. Introduction
Problems involving vibrations or oscillations occur frequently in physics and engineering. You can think of examples you have already met: a vibrating tuning fork, a pendulum, a weight attached to a spring, water waves, sound waves, alternating electric currents, etc. In addition, there are many more examples which you will meet as you continue to study physics. On the other hand, Some of them – for example, heat conduction, electric and magnetic fields, light – do not appear in elementary work to have anything oscillatory about them, but will turn out in your advanced work to involve the sine and cosines which are used in describing simple harmonic motion and wave motion. It is why we learn how to expand a certain function with Fourier series consisting of ‘infinite’ sines and cosines.
3
2. Simple harmonic motion and wave motion: periodic functions ( 단순 조화운동과 파동운동 ; 주기함수 )
locityangular ve:)22
( , fT
t
y coordinate of Q (or P): tAAy sinsin
The back and forth motion of Q simple harmonic motion
1) Harmonic motion ( 단순 조화 운동 )
- P moves at constant speed around a circle of radius A.- Q moves up and down in such a way that its y coordinate is always equal to that of P.
For a constant circular motion,
4
tAytAx sin ,cos
tiAe
titAiyxz
)sin(cosIn the complex plane,
imaginary part velocity of Q
)sin(cos)( titAieAiAedt
d
dt
dz titi
2) Using complex number ( 복소수의 사용 )
The x and y coordinates of P:
Then, it is often convenient to use the complex notation.
(Position of Q: imaginary part of the complex z)
Velocity:
5
)sin( ,cosor sin tAtAtA
cf. phase difference or different choice of the origin
i) Functional form of the simple harmonic motion:
3) Periodic function ( 함수의 주기 )
Time
Displacement
6
Displacement
Time
ii) Graph
amplitude
period:
a. Time (simple harmonic motion)
2
T
tBtAdt
dy
tAy
coscos
sin
Kinetic energy: tmBdt
dym 22
2
cos2
1
2
1
Total energy (kinetic+ potential = max of kinetic E) =
22222
2
1fAAmB
7
x
Ay2
sin
t
vxAvtxAy
22
sin)(2
sin
Wavelength: λ
c. Arbitrary periodic function (like wave)
ngth)(or wavele period :
)()(
p
xfpxf
b. Distance (wave)
distance
Tf
vTcf
1,.
8
3. Applications of Fourier Series (Fourier 급수의 응용 )
- Fundamental (first order):
- Higher harmonics (higher order):
- Combination of the fundamental and the harmonics complicated periodic
function. Conversely, a complicated periodic function the combination of the
fundamental and the harmonics (Fourier Series expansion).
tt cos ,sin
)cos( ),sin( tntn
9
-What a-c frequencies (harmonics) make up a given signal and in what proportions? We can answer the above question by expanding these various periodic functions with Fourier Series.
ex) Periodic function
10
0 5 10 15 200
0 5 10 15 200
Inte
nsity
0 5 10 15 200
0 5 10 15 200
0 5 10 15 200
0 5 10 15 200
xxx 3sin,2sin,sin xxx 3sin2sinsin3
1
xsin
xx 2sinsin2
1
xxx 3sin2sinsin3
1
xxxx 10sin3sin2sinsin10
1
11
4. Average value of a function ( 함수의 평균값 )
1) average value of a function
xn
xxfxfxfxf
n
xfxfxfxf
baxf
nn
)()()()()()()()(
),(on )( of average e'Approximat'
321321
ab
dxxfbaxf
xnabxn
b
a
)(),(on )( of Average
n)integratio ofconcept (,0&, ,When
With the interval n
abx
12
nxdxnxdx
dxdxnxnxnxnx
nxdxnxdx
n
xdxxdx
22
2222
22
22
cossin
2cossin ,1cossin Using
cossin
,)0(for Similarly
.cossin
2
1cos
2
1sin
2
1
)cos ( sin of period) a(over valueAverage
22
22
nxdxnxdx
nxornx
2) Average of sinusoidal functions ( 사인함수의 평균 )
2
2cos1cos,
2
2cos1sin
sincos2cos .
22
22
xx
xx
xxxcf
13
-0.5
0.0
0.5
1.0
sin2x
-0.5
0.0
0.5
1.0
sin22x
-0.5
0.0
0.5
1.0
sin23x
-0.5
0.0
0.5
1.0sin24x
-0.5
0.0
0.5
1.0sin25x
Graph of sin2 nx
14
Chapter 7 Fourier Series
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 2 Basic of Fourier series
Lecturer: Lee, Yunsang (Physics)Baird-Hall 01318 Email: [email protected]: 02-820-0404
15
5. Fourier coefficients (Fourier 계수 )
We want to expand a given periodic function in a series of sines and cosines.[First, we start with sin(nx) and cos(nx) instead of sin(nt) and cos(nt).]
,3sin2sinsin
3cos2coscos2
1)(
321
3210
xbxbxb
xaxaxaaxf
- Given a function f(x) of period 2,
We need to determine the coefficients!!
16
integer) :(0sincf.
0sinsin2
1
2
1
cossin2
1
integer) :,(period) a(over cossin of valueAverage 1)
ndxnx
dxxnmxnm
nxdxmx
nmnxmx
coscos2
1sinsin
coscos2
1coscos
sinsin2
1sincos
sinsin2
1cossin
In order to find formulas for an and bn, we need the following integrals on (-, )
17
.0interger for,0coscf.
0 ,0
0 ,2
1
,0
coscos2
1
2
1
sinsin2
1
period) a(over sinsin of valueAverage 2)
ndxnx
nm
nm
nm
dxxnmxnm
nxdxmx
nxmx
coscos2
1sinsin
coscos2
1coscos
sinsin2
1sincos
sinsin2
1cossin
18
0 ,1
0 ,2
1
,0
coscos2
1
2
1
coscos2
1
period) a(over coscos of valueAverage 3)
nm
nm
nm
dxxnmxnm
nxdxmx
nxmx
coscos2
1sinsin
coscos2
1coscos
sinsin2
1sincos
sinsin2
1cossin
19
Using the above integrals, we can find coefficients of Fourier series by calculating the average value.
,3sin2sinsin
3cos2coscos2
1)(
321
3210
xbxbxb
xaxaxaaxf
i-1) To find a_o, we calculate the average on (-,)
xdxbxdxb
xdxaxdxadxa
dxxf
2sin2
1sin
2
1
2cos2
1cos
2
1
2
1
2)(
2
1
21
210
21
0)(1
adxxf
20
i-2) To find a_1, we multiply cos x (n=1) and calculate the average on (-,).
.cos)(1
cos2sin2
1cossin
2
1
cos2cos2
1coscos
2
1cos
2
1
2cos)(
2
1
1
21
210
axdxxf
xdxxbxdxxb
xdxxaxdxxaxdxa
xdxxf
xcos
2
1
0 ,1
0 ,2
1
,0
coscos2
1.
nm
nm
nm
nxdxmxcf
21
.2cos)(1
2cos2sin2
12cossin
2
1
2cos2
12coscos
2
12cos
2
1
22cos)(
2
1
2
21
221
0
axdxxf
xdxxbxdxxb
xdxadxxaxdxa
xdxxf
i-3) To find a_2, we multiply cos 2x (n=2) and calculate the average on (-,).
x2cos
2
1
0 ,1
0 ,2
1
,0
coscos2
1.
nm
nm
nm
nxdxmxcf
22
i-4) To find a_n, we multiply cos nx and calculate the average on (-,).
.cos)(1
cossin2
1
cos2
1cos)(
2
1
1
2
n
n
anxdxxf
nxdxxb
nxdxanxdxxf
nxcos
2
1
0 ,1
0 ,2
1
,0
coscos2
1.
nm
nm
nm
nxdxmxcf
23
ii-1) To find b_1 and b_n, (cf. n=0 term is zero), we multiply the sin x (n=1) or sin nx and calculate the average on (-,).
nbnxdxxf
bxdxxf
xdxxbxdxb
xdxxadxxaxdxa
xdxxf
sin)(1
Similarly,
.2
1sin)(
2
1
sin2sin2
1sin
2
1
sin2cos2
1sincos
2
1sin
2
1
2sin)(
2
1
1
22
1
210
nxsin
2
1
0 ,0
0 ,2
1
,0
sinsin2
1.
nm
nm
nm
nxdxmxcf
24
. sin1
,cos1
.3sin2sinsin
3cos2coscos2
1)(
321
3210
nxdxxfbnxdxxfa
xbxbxb
xaxaxaaxf
nn
## Fourier series expansion
25
Example 1.
.0 ,1
,0 ,0)(
x
xxf
.0for 11
0for 0sin11
cos1
cos1cos01
cos)(1
0
0
0
0
n
nnxn
nxdx
nxdxnxdx
nxdxxfan
26
. oddfor 2
even for 0
1)1(1cos1
sin1
sin1sin01
sin)(1
00
0
0
nn
n
nn
nxnxdx
nxdxnxdx
nxdxxfb
n
n
.5
5sin
3
3sin
1
sin2
2
1)(
xxxxf
27
.0 ,1
,0 ,0)(
x
xxf .
5
5sin
3
3sin
1
sin2
2
1)(
xxxxf
9% overshoot: Gibbs phenomenon
28
Example 2.
.5
5sin
3
3sin
1
sin2
2
1)(
xxxxf
1
1
- case i
- case ii
5
5sin
3
3sin
1
sin4
12
xxx
xfxg
5
5cos
3
3cos
1
cos2
2
1
525sin
323sin
12sin2
2
1
2
xxx
xxx
xfxh
29
6. Dirichlet conditions (Dirichlet 조건 ) : convergence problem ( 수렴 문제 )
Does a Fourier series converge or does it converge to the values of f(x)?
-Theorem of Dirichlet: If f(x) is 1) periodic of period 2 2) single valued between - and
3) a finite number of Max., Min., and discontinuities4) integral of absolute f(x) is finite,
then, 1) the Fourier series converges to f(x) at all points where f(x) is continuous. 2) at jumps (e.g. discontinuity points), converges to the mid-point of the jump.
30
7. Complex form of Fourier series (Fourier 급수의 복소수 형태 )
.2
cos ,2
sininxinxinxinx ee
nxi
eenx
Using these relations, we can get a series of terms of the forms e^inx and e^-inx from the forms of sin nx and cos nx.
31
ixixixix
ixixixixixix
ixixixixixix
ei
bae
i
bae
i
bae
i
baa
i
eeb
i
eeb
i
eeb
eea
eea
eeaa
xbxbxb
xaxaxaaxf
22222211110
33
3
22
21
33
3
22
210
321
3210
222222222
1
222
2222
1
3sin2sinsin
3cos2coscos2
1)(
n
n
inxn
ixixixix
ec
ececececc 22
22110
32
.0cos ifonly zero-non is integral This
)sin()cos(
)(2
1 Here,
.)(
)(
xnmnm
xdxnmixnmdxedxee
dxexfc
ecxf
xnmiinximx
inxn
n
n
inxn
nxdxxfbnxdxxfacf nn sin)(
1 ,cos)(
1 .
33
Example.
.0 ,1
,0 ,0)(
x
xxf
.2
1
2
1
2
1
0,even ,0
odd ,1
12
1
2
1
12
10
2
1
00
0
0
dxdxxfc
n
nine
inin
e
dxedxec
ininx
o
inxinxn
Expanding f(x) with the e^inx series,
dxexfc inx
n )(2
1
34
Then,
xxi
ee
i
ee
eee
i
eee
iecxf
ixixixix
ixixixixixixinx
n
3sin3
1sin
2
2
1
23
1
2
2
2
1
functions) sinusoidal with converting(
531
1
531
1
2
1)(
33
5353
The same with the results of Fourier series with sines and cosines!!
35
Chapter 7 Fourier Series
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 3 Fourier series
Lecturer: Lee, Yunsang (Physics)Baird-Hall 01318 [email protected]
36
8. Other intervals ( 그 밖의 구간 )
2
0
2
0
2
0
)(2
1)(
2
1
sin)(1
sin)(1
cos)(1
cos)(1
dxexfdxexfc
nxdxxfnxdxxfb
nxdxxfnxdxxfa
inxinxn
n
n
- Same Fourier coefficients for the interval (-, ) and (0, 2 ).
1) (-, ) and (0, 2 ).
.2ith function w periodic :cos),(
cos)(cos)( Here,
cos)(cos)(cos)(
cos)(cos)(cos)(
Proof)(
20
2
0
2
0
0
0
nxxf
nxdxxfnxdxxf
nxdxxfnxdxxfnxdxxf
nxdxxfnxdxxfnxdxxf
37
(Caution)
Different periodic functions made from the same function, - same function f(x) = x^2 - different periodic with respect to the intervals, (-, ) and (0, 2 ).
38
- Other period 2l [(0, 2l) or (-l, l)], not 2 [(0, 2) ]
l
l
lxin
l
lel
xn
l
xn
nx
2
1
2
1
2),cosor (sin
2sin
/
2) period 2 vs. 2l
39
For f(x) with period 2l,
.sincos2
2sinsin
2coscos
2)(
1
0
21210
l
xnb
l
xna
al
xb
l
xb
l
xa
l
xa
axf
nn
.sin)(1
,cos)(1
l
ln
l
ln dxl
xnxf
lbdx
l
xnxf
la
l
l
lxinn dxexf
lc .)(
2
1 /.)( /
lxinnecxf
i) sinusoidal
ii) complex
40
Example.
lxl
lxxf
2 ,1
,0 ,0)( - period 2l
.2
1
2
1
, odd ,1
,0even ,01
2
1
2
1
/2
1
12
10
2
12
1
2
0
2
2/
2 /
0
2
0
/
l
l
in
inin
l
l
lxin
l
l
lxinl
l lxinn
dxl
cn
in
ne
in
eeinlin
e
l
dxel
dxl
dxexfl
c
Using the complex functions as Fourier series,
41
Then,
.3
sin3
1sin
2
2
1
3
1
3
11
2
1)( /3/3//
l
x
l
x
eeeei
xf lxilxilxilxi
42
9. Even and odd functions ( 짝함수 , 홀함수 )
)()( ifeven is )( xfxfxf
)()( if odd is )( xfxfxf
1) definition
43
- Even powers of x even function, and odd powers of x odd function.
- Any functions can be written as the sum of an even function and an odd function.
)()(2
1)()(
2
1)( xfxfxfxfxf
ex. odd)( sinheven)( cosh)(2
1)(
2
1xxeeeee xxxxx
44
Integral over symmetric intervals like (-, ) or (-l, l)
l
l
l xfdxxf
xfdxxf
0even. is )( if )(2
odd, is )( if 0)(
2) Integration
45
- In order to represent a f(x) on interval (0, l) by Fourier series of period 2l, we need to have f(x) defined on (-l, 0), too. - We can expand the function on (-l, 0) to be even or odd on (-l, 0). Anything is OK!!
46
1
0 sincos2
)(l
xnb
l
xna
axf nn
- Cosine function: even, Sine function: odd.
- If f(x) is even, the terms in Fourier series should be even. b_n should be zero.
- If f(x) is odd, the terms in Fourier series should be odd. a_n should be zero.
.sin)(2
0 odd, is )( If
0
l
n
n
dxl
xnxf
lb
axf
.0
,cos)(2
even, is )( If 0
n
l
n
b
dxl
xnxf
la
xf
3) Fourier series
47
- How to represent a function on (0, 1) by Fourier series 1) sine-cosine or exponential (ordinary method) (period 1, l=1/2) 2) odd or even function (period 2, l=1) (caution) different period!!
48
Example
1 ,0
0 ,1)(
21
21
x
xxf
(c) original function (period 1) Ordinary sine-cosine, or exponential
(b) even function (period 2) Fourier cosine series.
(a) odd function (period 2) Fourier sine series.
49
(a) Fourier sine series (using odd function with period 2, l = 1)
,4
0 ,
3
2 ,
2
4 ,
2
,4,3,2,1for ,0,1,2,112
cos
,12
cos2
cos2
sin2sin)(2
4321
2/1
0
2/1
0
1
0
bbbb
nn
n
nxn
n
dxxndxxnxfl
bn
6
6sin2
5
5sin
3
3sin
2
2sin2sin
2)(
xxxxxxf
.sin)(2
0 odd, is )( If
0
l
n
n
dxl
xnxf
lb
axf
50
(b) Fourier sine series (using odd function with period 2, l = 1)
5
5cos
3
3cos
1
cos2
2
1)(
.2
sin2
sin2
cos)(2
,12)(2
2/1
0
1
0
2/1
0
1
00
xxxxf
n
nxn
nxdxnxfa
dxdxxfa
n
.0
,cos)(2
even, is )( If 0
n
l
n
b
dxl
xnxf
la
xf
51
(c) Ordinary Fourier series
.
even. 0,
odd, ,1
2
)1(1
2
1
)(
21
2/1
00
2/1
0
21
0
2
dxc
n
nin
inin
e
dxedxexfc
nin
xinxinn
).3
6sin2(sin
2
2
1
)(1
2
1)( 6
316
3122
xx
eeeei
xf inininin
i) exponential
52
,0 ,3
2 ,0 ,
2
.)1(11
)cos1(1
2sin2
.02cos2
12)(2
4321
2/1
0
2/1
0
2/1
0
1
00
bbbb
nn
nxdxnb
xdxna
dxdxxfa
nn
n
ii) sine-cosine
53
10. Application to sound ( 소리에 대한 응용 )
- odd function- period = 1/262
7
)7524sin(
6
)6524sin(30
5
)5524sin(
3
)3524sin(
2
)2524sin(30
1
524sin
4
1)(
cos8
71
2cos
8
152 524sin)()524(2
524/1
0
ttt
ttttp
nn
ntdtntpbn
54
- Intensity of a sound wave is proportional to the average of the square of amplitude, A2.
n = 1 2 3 4 5 6 7 8 9 10
relative intensity
= 1 225 1/9 0 1/25 25 1/49 0 1/81 9
- Second harmonics is dominant!!
55
Chapter 7 Fourier Series
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 4 Fourier Transform
Lecturer: Lee, Yunsang (Physics)Baird-Hall 01318 [email protected]
56
11. Parseval’s theorem (completeness relation) (Parseval’s 정리 ; 완전성 관계 )
?)(2
1)( of Average
sincos)(
22
102
1
dxxfxf
nxbnxaaxf nn
Hint 1) average of (1/2a_0)^2 = (1/2a_0)^2 2) average of (a_n cos nx)^2) = a_n^2* 1/2 3) average of (b_n sin nx)^2 = b_n^2 * 1/2. 4) average of all cross product terms, a_n*b_m*cos nx*sin mx, = 0.
1
2
1
2202
12
2
1
2
1)( of Average nn baaxf
22)( of average ncxfSimilarly,
- Parseval’s theorem or completeness relation- The set of cos nx and sin nx is a complete set!!
57
12. Fourier Transforms (Fourier 변환 )
expansion) series(Fourier 2
1, //
l
l
lxinn
lxinn dxexf
lcecxf
ransform)(Fourier t.2
1,
dxexfgdegxf xixi
- Periodic function Fourier series with discrete frequencies
- What happens for ‘non-periodic function’?
Fourier transform with continuous frequencies
ddxll
nd
l
l
2
1
2
1,,
cf. Fourier series vs. Fourier transform
58
confusion) theavoid to(.22
1
),(,
l
l
uil
l
xin
nxi
n
dueufdxexfl
c
ll
necxf
nn
n
- Conversion of the Fourier series to the Fourier transform
.where,2
1
2
2
l
l
uxinn
l
l
uxi
xil
l
ui
dueufFFdueuf
edueufxf
nn
nn
.
,2
1
2
1
.2
1
2
1
2
1
degxf
dueufdxexfg
dueufdedudeufdFxf
dueufF
xi
uixi
uixiuxi
l
l
uxi
59
.sin2 Similarly,
. odd,
.sinsin2
1
, oddFor
sincos2
1
sincos
0
0
xdgidxegxf
ggg
xdxxfi
dxxixfg
xf
dxxixxfg
xixe
xi
xi
- Fourier sine/cosine transforms
60
.cos2
,cos2
Transform, CosineFourier ii)
.sin2
,sin2
Transform, SineFourier i)
0
0
0
0
xdxxfxg
xdgxf
xdxxfxg
xdgxf
cc
cc
ss
ss
61
Example 1.
,1,0
,11,1
x
xxf
.sin
2
1
2
1
2
1
2
1
1
1
1
1
i
ee
i
edxe
dxexfg
iixixi
xi
0
cossin2sincossin1sin
d
xdx
xixdxexf xi
62
Example 2.
,1,0
,11,1
x
xxf
.1for 0
1for 4
,1for 2
2
cossin0
x
xxxfd
x
.2
sin,0For
0
dx
63
- Parseval’s Theorem for Fourier integrals
.2
1
.~
2
1~
~
2
1~
2
1
.~
2
1~
.~
2
1~
22
2121
2121
2121
11
dxxfdg
dxxfxfdagg
dxxfxfdegdxxf
dgdxexfdgg
dxexfg
xi
xi
xi
22)( of averagecf. ncxf
64
- Various Fourier transforms
65
- Michelson interferometer
66
HW
Chapter 7
2-3, 9, 13, 18 (G1)5-1, 7 (G2)7-1 (G3)9-1,6,7 (G4)