Physics Standing Waves. Tues. 4/18, and Thurs. 4/20

Transcription

1 Physics Standing Waves Tues. 4/18, and Thurs. 4/20

2 A long string is firmly connected to a stationary metal rod at one end. A student holding the other end moves her hand rapidly up and down to create a pulse towards the rod. What picture best describes what the pulse on the string looks like after it has completely reflected from the wall. A.) B.) C.) D.) E.) Not enough information.

3 Two pulses travel toward each other along a long stretched spring as shown. Pulse A is wider than pulse B, but not as high. What is the speed of pulse A compared to pulse B? The speed of pulse A is the speed of pulse B. A.) B.) C.) D.) Larger than Smaller than Equal to Not enough information

4 Mathematical Description Two sine waves travelling in opposite directions with the same amplitude and same wave speed: y Right x, t = A o sin (kx ωt) y Left x, t = A o sin (kx + ωt) What happens when we add them together? y total x, t = y Right x, t + y Left x, t y total x, t = A o sin kx ωt + A o sin kx + ωt

5 Mathematical Description Adding together Right- and Left-moving waves y total x, t = y Right x, t + y Left x, t y total x, t = A o sin kx ωt + A o sin kx + ωt Using trig identity: sin A ± B = sinacosb ± cosasinb y total x, t = A o [sin kx cos ωt cos kx sin ωt ] + A o [sin kx cos ωt + cos kx sin ωt ] y total x, t = 2A o sin kx cos ωt

6 Mathematical Description Adding together Right- and Left-moving waves creates a standing wave pattern: y total x, t = 2A o sin kx cos ωt Same period, same wavelength, but twice the amplitude! How can we visualize this equation compared to the travelling wave equation?

7 Standing Waves A standing wave is produced when a wave that is traveling is reflected back upon itself. There are two important features of a standing wave: Antinodes Areas of MAXIMUM AMPLITUDE Nodes Areas of ZERO AMPLITUDE. Demo Time: Standing wave on a rope!

8 A string is stretched so that it is under tension and is tied at both ends so that the endpoints don't move. A mechanical oscillator then vibrates the string so that a standing wave is created. All of the strings have the same length but may not have the same mass. The number of nodes and antinodes in the standing wave is the same in Cases A and D. The tensions in the strings (T) and the standing wave frequencies (f) are given in each figure. A.) Rank the speeds of the wave in the string. B > C > D > A B.) C > B = A > D C.) C = B > A = D D.) D > A > B > C E.) The speeds are the same for all cases

9 Announcements Online survey about attitudes toward science is active, you received a link via Sakai about this earlier. Must be completed by 11:59pm on Monday May 1 st to be eligible for 1% credit. Paper post-test will be held in class on 4/26 + 4/27 next week, you must do this and the online survey to get the 1% credit. SIRS Course Surveys are open, please take ~5 min and thoughtfully fill it out (constructive feedback is helpful!)

10 Mathematical Description Adding together Right- and Left-moving waves y total x, t = y Right x, t + y Left x, t y total x, t = A o sin kx ωt + A o sin kx + ωt Using trig identity: sin A ± B = sinacosb ± cosasinb y total x, t = A o [sin kx cos ωt cos kx sin ωt ] + A o [sin kx cos ωt + cos kx sin ωt ] y total x, t = 2A o sin kx cos ωt

11 Both ends of a string that are held fixed How do wave speeds compare in each case?

12 Standing Waves Standing waves are an example of RESONANCE. Resonance when a FORCED vibration matches an object s natural frequency energy transfer is optimized and the waves interfere constructively, and build up causing very large amplitude vibrations One example of this involves shattering a wine glass by hitting a musical note that is on the same frequency as the natural frequency of the glass. (Natural frequency depends on the size, shape, and composition of the object in question.) Because the frequencies resonate, or are in sync with one another, maximum energy transfer is possible. The Tacoma Narrows Bridge collapse is likely Demo Time: Wine Glass, Resonating Bowl not caused by resonance, but complex winds above and below the bridge.

13 Sound Waves The production of sound involves setting up a wave in air. To set up a CONTINUOUS sound you will need to set a standing wave pattern. Three LARGE CLASSES of instruments Stringed - standing wave is set up in a tightly stretched string Percussion - standing wave is produced by the vibration of solid objects Wind - standing wave is set up in a column of air that is either OPEN or CLOSED

14 Guitar Strings guitar-strings-vibrate/ -XI&t=0m50s

15 Open Pipes OPEN PIPES- have an antinode on BOTH ends of the tube. What is the SMALLEST length of pipe you can have to hear a sound? You will get your FIRST sound when the length of the pipe equals one-half of a wavelength.

16 Open Pipes - Harmonics Since harmonics are MULTIPLES of the fundamental, the second harmonic of an open pipe will be ONE WAVELENGTH. The picture above is the SECOND harmonic or the FIRST OVERTONE. Two half-wavelengths fit into the tube length L

17 Open Pipes - Harmonics Another half of a wavelength would ALSO produce an antinode on BOTH ends. In fact, no matter how many halves you add you will always have an antinode on the ends The picture above is the THIRD harmonic or the SECOND OVERTONE. Three half-wavelengths fit into the tube length L CONCLUSION: Sounds in OPEN pipes are produced at ALL HARMONICS (all multiples of one-half wavelength, n λ 2 )!

18 How does the fundamental frequency for closed pipes compare to open pipes? Closed Pipes...have an antinode at one end and a node at the other. Each sound you hear will occur when an antinode appears at the top of the pipe. What is the SMALLEST length of pipe you can have to hear a sound? You get your first sound or encounter your first antinode when the length of the actual pipe is equal to a quarter of a wavelength. This FIRST SOUND is called the FUNDAMENTAL FREQUENCY or the FIRST HARMONIC. First harmonic: One quarter-wavelength fits into the tube length L

19 Closed Pipes...have an antinode at one end and a node at the other. Each sound you hear will occur when an antinode appears at the top of the pipe. Will there be a sound produced for the 2 nd harmonic (shown below)? No, there s a node at the pipe opening so sound production will be suppressed! In reality you may still hear faint sounds, but the sounds are MUCH louder with an antinode at the open end of the pipe. It s easy to hear when you have a standing sound wave in your air column.

20 Closed Pipes - Harmonics In a closed pipe you have an ANTINODE at the 3rd harmonic position, therefore SOUND is produced. CONCLUSION: Sounds in CLOSED pipes are produced ONLY at ODD HARMONICS! (odd multiples of the fundamental, which means odd multiples of ¼ wavelengths, ¼ λ, ¾ λ, etc.)

21 Closed Pipes - Harmonics Harmonics are MULTIPLES of the fundamental frequency. Three quarter-wavelengths fit into the tube length L In a closed pipe, you have a NODE at the 2nd harmonic position, therefore NO SOUND is produced

22 Closed Pipes - Harmonics

23 Both ends open Adding up how many ½ wavelengths fit into L 1 2 λ = Fundamental = sound fundamental One end open, one end closed Adding up how many ¼ wavelengths fit into L 1 4 λ = Fundamental = sound

24 Both ends open L Adding up how many ½ wavelengths fit into L 1 2 λ = Fundamental = sound 2 2 λ = Second harmonic = sound Second harmonic One end open, one end closed L Adding up how many ¼ wavelengths fit into L 1 4 λ = Fundamental = sound 2 4 λ = Second harmonic = NO sound

25 Both ends open L Adding up how many ½ wavelengths fit into L 1 2 λ = Fundamental = sound 2 2 λ = Second harmonic = sound 3 2 λ = Third harmonic = sound etc Third harmonic One end open, one end closed L Adding up how many ¼ wavelengths fit into L 1 4 λ = Fundamental = sound 2 4 λ = Second harmonic = NO sound 3 4 λ = Third harmonic = sound etc

26 Both ends open or both ends closed One end open, one end closed n=1, L = λ / 2, f = f 1 n=1, L = λ / 4, f = f 1 n=2, L = 2λ / 2, f 2 = 2f 1 n=2, L = 2λ / 4, f 2 = 2f 1 (no sound) n=3, L = 3λ / 2, f 3 = 3f 1 n=3, L = 3λ / 4, f 3 = 3f 1

27 Example The speed of sound waves in air is found to be 340 m/s. Determine the fundamental frequency (1st harmonic) of an open-end air column which has a length of 67.5 cm. v 2lf 340 2(0.675) f f HZ

28 Atmospheric Pressure (P 0 ) Ruben s (Flame) Tube From Bernoulli s: P T > P 0 Tube Pressure (P T ) Flame max height occurs at nodes of sound waves in the gas tube! Why? How does the average speed of the gas in the tube compare at nodes versus antinodes of the standing sound wave