1. Frans Pretorius University of Alberta

2. 동일 진동수 빛의 중첩 PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 2 두 빛의 중첩 ( 더하기 )

3. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 3 두 빛이 더해진 밝기의 표현 식 계산을 간단히 하기 위해서 E 01 =E 02 라고 가정하자. 그리고 E 01 2 =E 02 2 =I o 라 하자

4. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 4 Standing waves  Standing waves are waves that look stationary, but have an amplitude that changes with time. Several situations can produce standing waves, including  the superposition of left and right moving waves on a string  the “natural” modes of vibration of a string fixed at both ends (stringed instruments work like this)  sound waves in a tube open at one or both ends (wind instruments work likes this)  sustained 40mph winds set up standing waves in the Tacoma Narrows Bridge in 1940, causing it to collapse:

5. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 5 Standing waves on a string fixed at both ends

6. 광물리학 빛의 중첩, 회절 간섭 6 공명 (resonance) 서로 반대 방향으로 진행하는, 같은 진동수의 빛이 만나면 제자리에서 진동하는 파동이 생긴다. 마디와 마시 사이 거리는 파장의 절반이다. node antinode 정상파 (Standing

7. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 7 Beats  When two tones of similar frequency f 1 and f 2 are added together, interference will create what is called a beat frequency at the difference between the two frequencies : f b =f 1 -f 2 Example: A 200hz tone: A 200hz + 201hz tone: (beat frequency is 1 hz … this is a 5 second sample, so we should hear ~5 beat cycles) A 200hz + 210hz tone: (beat frequency is 10 hz …should hear ~50 beat cycles)

8. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 8 Interference of two waves sources vibrating in phase  Two wave sources, S 1 and S 2, are emitting waves in phase, and of exactly the same frequency and amplitude. Consider a point p that is a distance d 1 from source 1, and a distance d 2 from source 2.  If where n is a non-negative integer and is the wavelength, then p will be a point of complete constructive interference  If then p will be a point of complete destructive interference d1d1 d2d2 p S1S1 S2S2

9. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 9 Example A  Consider the configuration of loudspeakers and listener shown to the right. Assume both loudspeakers are playing the exact same music. What set of frequencies will the listener not be able to hear at all? Image courtesy John Wiley & Sons, Inc.

10. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 10 Diffraction  diffraction is the bending of a wave as it moves past edges or obstacles

11. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 11 Single Slit Diffraction  With single slit diffraction, we have a sound wave of wavelength passing through an opening of width D. On the other side of the opening there will be interference between parts of the wave, and at an angle given by there will be complete destructive interference (the so-called first- minimum)  Note: the above formula only works if D>>

12. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 12 Standing waves on a string fixed at both ends  Since both ends of the string are fixed, the only possible set of wavelengths are  n=1 gives the first or fundamental harmonic  n=2 gives the second harmonic or first overtone, n=3 the third harmonic or second overtone, etc.  Given the relationship f=v, the set of frequencies corresponding to these wavelengths are

13. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 13 Standing waves in a tube  A resonance can be used to set up standing sound waves in a tube  this is a longitudinal standing wave (compared to the transverse standing wave on a string)  If both ends are open, the possible set of natural frequencies are (as with the string) :

14. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 14 Standing waves in a tube  If only one end is open, the following set of resonant frequencies are possible:

15. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 15 Example B: (ch 17, prob. 44)  A tube, open at one end, is cut into two shorter, unequal length pieces. The piece that is open at one end has a fundamental frequency of 675hz, while the piece that is open at both ends has a fundamental frequency of 425hz. What was the fundamental frequency of the original tube?

16. PHYS 124, Section A2, Chapter Chapter 17.1-17.6: Principle of Linear Superposition and Interference 16 Example B  Answer: 162Hz