The Principle of Linear Superposition and Standing Waves

1. The Principle of Linear Superposition and Standing Waves

2. The principle of linear superposition http://bcs.wiley.com/he-bcs/Books?action=mininav&bcsId=4678&itemId=0470223553&assetId=160341&resourceId=15300 Simulation #25, Ch. 17, Cutnell

3. THE PRINCIPLE OF LINEAR SUPERPOSITION When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves.

4. Superposition of 2 pulses The drawing shows a graph of 2 pulses traveling toward each other at t = 0s. Each pulse has a constant speed of v = 1 cm/s. When t = 3.0s, what is the amplitude of the resultant pulse at a) x = 5 cm? a. +2 cm b. -2cm c. -3 cm d. 0 cm

6. 17.2 Constructive and Destructive Interference of Sound Waves Pilots use noise-canceling headsets like the ones shown below. The headset “hears” the exterior cockpit noise, which is of fairly consistent frequency, and emits a sound wave that is exactly out of phase. Sounds transmitted through the headset, such as radio transmissions, or music, are not “canceled”.

7. Two identical waves coming from two sources • Path length (d1) between observer and left speaker equals path length (d2) between observer and right speaker • Δd = 0, so observer hears wave in phase, resulting in constructive interference.

8. Constuctive Interference For two wave sources vibrating in phase, a difference in path lengths that is zero or an integer number such that: Δd = mλ (m=0,1,2….) leads to constructive interference, and a sound wave will sound louder:

9. Two identical waves coming from two sources • Path length (d1) and (d2) between observer and right speaker differ by a half wavelength. • Δd = .5λ, so observer hears wave out of phase, resulting in destructive interference.

10. Destructive Interference For two wave sources vibrating in phase, a difference in path lengths that is a half-integer number of wavelengths, such that Δd = (m +1/2)λ (m=0,1,2….) leads to destructive interference.

11. Wave interference logic • Find values for Δd and λ • Compare Δd and λ as follows: x = Δd /λ • IF x = m (integer), THEN constructive. • IF x = (m + ½), THEN destructive. • IF x ≠ m AND x ≠ (m + ½), THEN neither constructive nor destructive. • If the problem asks for the smallest wavelength, set m = 0 in either 3 or 4 above. This logic is only for 2 sources which are vibrating in phase. Speakers are usually set up to vibrate in phase.

12. What Does a Listener Hear? • Two in-phase loudspeakers, A and B, are separated by 3.20 m. A man is listening at point C, as shown. Both speakers are playing identical 142.9-Hz tones. The speed of sound is 343 m/s. • Does the listener hear • Constructive interference, • Destructive interference • C. Neither

13. Two overlapping waves with slightly different frequencies gives rise to the phenomena of beats.

14. The beat frequency is the difference between the two sound frequencies: fbeat = | f1 - f2 | http://bcs.wiley.com/he-bcs/Books?action=mininav&bcsId=4678&itemId=0470223553&assetId=160341&resourceId=15300

15. Beats Two pure tones are sounded together. The drawing shows the pressure variations of the two sound waves, measured with respect to atmospheric pressure, where t1 = 0.0206 s and t2 = 0.0250 s. What is the beat frequency in Hz? A. .0044 B. 40 C. 48.54 D. 8.54

16. http://www.walter-fendt.de/ph14e/stwaverefl.htm Uncheck “resultant standing wave” (3rd box) • How long is λ in in terms of string length L? λ = ___L • Once the standing wave is completely generated, can you find at least one place along the string where the cumulative wave amplitude is always 0? Where?

17. http://www.walter-fendt.de/ph14e/stwaverefl.htm Now check “resultant standing wave” (3rd box). • What happens at a node? Antinode? • How far apart are the nodes? • What is the largest possible λ(in terms of L for a standing wave to occur? λfundamental= ___L

18. http://bcs.wiley.com/he-bcs/Books?action=mininav&bcsId=4678&itemId=0470223553&assetId=160341&resourceId=15300http://bcs.wiley.com/he-bcs/Books?action=mininav&bcsId=4678&itemId=0470223553&assetId=160341&resourceId=15300 Or googleCutnell Johnson 8th edition, click on student companion site, concept simulations, ch. 17 # 27. 7. Wave speed is 4 m/s. String length is 1 m 8. Start at f=4 Hz and work your way down to 1Hz. Do standing waves occur at that frequency? • Re-answer #6: λfundamental= ___L

19. Standing Waves on a String For a string of fixed length L, the conditions can be satisfied only if the wavelength has one of the values A standing wave can exist on the string only if its wavelength is one of the values shown above. Because λf = v, the frequency corresponding to wavelength λm is

20. Transverse standing wave patterns

22. m/L

23. Standing Sound Waves • A long, narrow column of air, such as the air in a tube or pipe, can support a longitudinal standing sound wave. • sound is a pressure wave rather than a displacement wave. The pressure oscillates around its equilibrium value. • Musical instruments use longitudinal standing waves in tubes that are either open at both ends (e.g. flute, organ) or open at one end (e.g. clarinet, trumpet). • An open end acts as an antinode (maximum vibration) while a closed end acts as a node, just like for a wave on a string.

24. Tube open at both ends This is the same as for a transverse standing wave on a string, except that the antinodes are at the ends.

25. The length of an organ pipe vsound = 343 m/s An organ pipe is open at both ends

26. The length of an organ pipe

27. Check your understanding A tube is found to have standing waves at 390, 520 and 650 Hz and no frequencies in between. The behavior of the tube at frequencies higher/lower than that was not tested. The speed of sound is 343 m/s. How long is the tube, in meters? a. 1.76 b. 1.32 c. 0.879 d. 0.440

28. Tube open at one end • The open end is an antinode and the closed end is a node. • The distance between node and antinode is λ/4. • Therefore the tube length (L) must be equal to an odd number of quarter wavelengths:

29. The notes on a clarinet in a tropical country

30. The notes on a clarinet

31. The notes on a clarinet