Chapter 7 Fourier Series (Fourier ??)

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Mathematical methods in the physical sciences
3rd edition Mary L. Boas
Chapter 7 Fourier Series (Fourier ??)
Lecture1 Periodic function
Lecturer Lee, Yunsang (Physics) Baird-Hall
01318 ylee_at_ssu.ac.kr 02-820-0404
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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.
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2. Simple harmonic motion and wave motion
periodic functions (?? ????? ???? ????)
1) Harmonic motion (?? ?? ??)
- 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.
The back and forth motion of Q ? simple
harmonic motion
For a constant circular motion,
y coordinate of Q (or P)
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2) Using complex number (???? ??)
The x and y coordinates of P
Then, it is often convenient to use the complex
notation.
In the complex plane,
(Position of Q imaginary part of the complex z)
Velocity
imaginary part ? velocity of Q
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3) Periodic function (??? ??)
i) Functional form of the simple harmonic motion
cf. phase difference or different choice of the
origin
Displacement
Time
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ii) Graph
a. Time (simple harmonic motion)
period
Displacement
Time
amplitude
Kinetic energy
Total energy (kinetic potential max of
kinetic E)
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b. Distance (wave)
distance
Wavelength ?
c. Arbitrary periodic function (like wave)
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3. Applications of Fourier Series (Fourier ??? ??)
- 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).
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ex) Periodic function
-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.
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4. Average value of a function (??? ???)
1) average value of a function
With the interval
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2) Average of sinusoidal functions (????? ??)
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Graph of sin2 nx
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Mathematical methods in the physical sciences
3rd edition Mary L. Boas
Chapter 7 Fourier Series
Lecture 2 Basic of Fourier series
Lecturer Lee, Yunsang (Physics) Baird-Hall
01318 Email ylee_at_ssu.ac.kr Tel [phone]
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5. Fourier coefficients (Fourier ??)
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(n?t) and
cos(n?t).
- Given a function f(x) of period 2?,
We need to determine the coefficients!!
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In order to find formulas for an and bn, we need
the following integrals on (-?, ?)
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Using the above integrals, we can find
coefficients of Fourier series by calculating the
average value.
i-1) To find a_o, we calculate the average on
(-?,?)
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i-2) To find a_1, we multiply cos x (n1) and
calculate the average on (-?,?).
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i-3) To find a_2, we multiply cos 2x (n2) and
calculate the average on (-?,?).
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i-4) To find a_n, we multiply cos nx and
calculate the average on (-?,?).
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ii-1) To find b_1 and b_n, (cf. n0 term is
zero), we multiply the sin x (n1) or sin nx and
calculate the average on (-?,?).
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Fourier series expansion
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Example 1.
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Example 2.
- case i
- case ii
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6. Dirichlet conditions (Dirichlet ??)
convergence problem (?? ??)
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
discontinuities 4) 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.
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7. Complex form of Fourier series (Fourier ???
??? ??)
Using these relations, we can get a series of
terms of the forms einx and e-inx from the
forms of sin nx and cos nx.
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Example.
Expanding f(x) with the einx series,
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Then,
The same with the results of Fourier series with
sines and cosines!!
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Mathematical methods in the physical sciences
3rd edition Mary L. Boas
Chapter 7 Fourier Series
Lecture 3 Fourier series
Lecturer Lee, Yunsang (Physics) Baird-Hall
01318 ylee_at_ssu.ac.kr 02-820-0404
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8. Other intervals (? ?? ??)
1) (-?, ?) and (0, 2 ?).
- Same Fourier coefficients for the interval (-?,
?) and (0, 2 ?).
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(Caution)
Different periodic functions made from the same
function, - same function f(x) x2 -
different periodic with respect to the
intervals, (-?, ?) and (0, 2 ?).
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2) period 2? vs. 2l
- Other period 2l (0, 2l) or (-l, l), not 2?
(0, 2?)
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For f(x) with period 2l,
i) sinusoidal
ii) complex
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Example.
- period 2l
Using the complex functions as Fourier series,
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Then,
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9. Even and odd functions (???, ???)
1) definition
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- 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.
ex.
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2) Integration
Integral over symmetric intervals like (-?, ?) or
(-l, l)
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- 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!!
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3) Fourier series
- 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.
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- How to represent a function on (0, 1) by
Fourier series 1) sine-cosine or exponential
(ordinary method) (period 1, l1/2) 2) odd or
even function (period 2, l1) (caution)
different period!!
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Example
(a) odd function (period 2) ? Fourier sine
series.
(b) even function (period 2) ? Fourier cosine
series.
(c) original function (period 1) ? Ordinary
sine-cosine, or
exponential
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(a) Fourier sine series (using odd function with
period 2, l 1)
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(b) Fourier sine series (using odd function with
period 2, l 1)
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(c) Ordinary Fourier series
i) exponential
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ii) sine-cosine
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10. Application to sound (??? ?? ??)
- odd function - period 1/262
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- 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!!
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Mathematical methods in the physical sciences
3rd edition Mary L. Boas
Chapter 7 Fourier Series
Lecture 4 Fourier Transform
Lecturer Lee, Yunsang (Physics) Baird-Hall
01318 ylee_at_ssu.ac.kr 02-820-0404
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11. Parsevals theorem (completeness relation)
(Parsevals ?? ??? ??)
Hint 1) average of (1/2a_0)2 (1/2a_0)2 2)
average of (a_n cos nx)2) a_n2 1/2 3)
average of (b_n sin nx)2 b_n2 1/2. 4)
average of all cross product terms, a_nb_mcos
nxsin mx, 0.
Similarly,
- Parsevals theorem or completeness relation -
The set of cos nx and sin nx is a complete set!!
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12. Fourier Transforms (Fourier ??)
- Periodic function ? Fourier series with
discrete frequencies
- What happens for non-periodic function?
? Fourier transform with continuous frequencies
cf. Fourier series vs. Fourier transform
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- Conversion of the Fourier series to the Fourier
transform
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- Fourier sine/cosine transforms
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Example 1.
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Example 2.
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- Parsevals Theorem for Fourier integrals
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- Various Fourier transforms
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- Michelson interferometer
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HW Chapter 7 2-3, 9, 13, 18 (G1) 5-1, 7
(G2) 7-1 (G3) 9-1,6,7 (G4)
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