Chapter 7 Fourier Series (Fourier 급수) Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 7 Fourier Series (Fourier 급수) Lecture1 Periodic function Lecturer: Lee, Yunsang (Physics) Baird-Hall 01318 ylee@ssu.ac.kr 02-820-0404 1

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. 2

2. Simple harmonic motion and wave motion: periodic functions (단순 조화운동과 파동운동 ; 주기함수) 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. The back and forth motion of Q  simple harmonic motion For a constant circular motion, y coordinate of Q (or P): 3

2) Using complex number (복소수의 사용) 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 4

3) Periodic function (함수의 주기) i) Functional form of the simple harmonic motion: cf. phase difference or different choice of the origin Displacement Time 5

a. Time (simple harmonic motion) ii) Graph a. Time (simple harmonic motion) period: Displacement Time amplitude Kinetic energy: Total energy (kinetic+ potential = max of kinetic E) = 6

c. Arbitrary periodic function (like wave) b. Distance (wave) distance Wavelength: λ c. Arbitrary periodic function (like wave) 7

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). 8

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. 9

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4. Average value of a function (함수의 평균값) 1) average value of a function With the interval 11

2) Average of sinusoidal functions (사인함수의 평균) 12

Graph of sin2 nx 13

Chapter 7 Fourier Series 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@ssu.ac.kr Tel: 02-820-0404 14

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(nt) and cos(nt).] - Given a function f(x) of period 2, We need to determine the coefficients!! 15

In order to find formulas for an and bn, we need the following integrals on (-, ) 16

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i-1) To find a_o, we calculate the average on (-,) 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 (-,) 19

i-2) To find a_1, we multiply cos x (n=1) and calculate the average on (-,). 20

i-3) To find a_2, we multiply cos 2x (n=2) and calculate the average on (-,). 21

i-4) To find a_n, we multiply cos nx and calculate the average on (-,). 22

ii-1) To find b_1 and b_n, (cf 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 (-,). 23

## Fourier series expansion 24

Example 1. 25

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9% overshoot : Gibbs phenomenon 27

Example 2. - case i 1 - case ii 1 28

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 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. 29

7. Complex form of Fourier series (Fourier 급수의 복소수 형태) 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. 30

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Expanding f(x) with the e^inx series, Example. Expanding f(x) with the e^inx series, 33

The same with the results of Fourier series with sines and cosines!! Then, The same with the results of Fourier series with sines and cosines!! 34

Chapter 7 Fourier Series 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@ssu.ac.kr 02-820-0404 35

8. Other intervals (그 밖의 구간) 1) (-, ) and (0, 2 ). - Same Fourier coefficients for the interval (-, ) and (0, 2 ). 36

(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 ). 37

2) period 2 vs. 2l - Other period 2l [(0, 2l) or (-l, l)], not 2 [(0, 2) ] 38

For f(x) with period 2l, i) sinusoidal ii) complex 39

Using the complex functions as Fourier series, Example. - period 2l Using the complex functions as Fourier series, 40 40

Then, 41

9. Even and odd functions (짝함수, 홀함수) 1) definition 42

- 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. 43

2) Integration Integral over symmetric intervals like (-, ) or (-l, l) 44

- 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!! 45

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. 46

- 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!! 47

(a) odd function (period 2)  Fourier sine series. 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 48 48

(a) Fourier sine series (using odd function with period 2, l = 1) 49

(b) Fourier sine series (using odd function with period 2, l = 1) 50

(c) Ordinary Fourier series i) exponential 51

ii) sine-cosine 52

10. Application to sound (소리에 대한 응용) - odd function - period = 1/262 53

- Second harmonics is dominant!! - 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 225 1/9 1/25 25 1/49 1/81 - Second harmonics is dominant!! 54

Chapter 7 Fourier Series 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@ssu.ac.kr 02-820-0404 55

11. Parseval’s theorem (completeness relation) 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. Similarly, - Parseval’s theorem or completeness relation - The set of cos nx and sin nx is a complete set!! 56

12. Fourier Transforms (Fourier 변환) - Periodic function  Fourier series with discrete frequencies - What happens for ‘non-periodic function’?  Fourier transform with continuous frequencies cf. Fourier series vs. Fourier transform 57

- Conversion of the Fourier series to the Fourier transform 58

- Fourier sine/cosine transforms 59

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Example 1. 61

Example 2. 62 62

- Parseval’s Theorem for Fourier integrals 63

- Various Fourier transforms 64

- Michelson interferometer 65

HW Chapter 7 2-3, 9, 13, 18 (G1) 5-1, 7 (G2) 7-1 (G3) 9-1,6,7 (G4) 66