Lecture 19. Superposition, interference, standing waves

Transcription

1 ecture 19 Superposition, interference, standing waves

2 Today s Topics: Principle of Superposition Constructive and Destructive Interference Beats Standing Waves

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. Constructive Destructive

4 Constructive and Destructive Interference When two sound waves always meet condensation-to-rarefaction, they are said to be exactly out of phase and to exhibit destructive interference. When two sound waves always meet condensation-to-condensation and rarefaction-to-rarefaction, they are said to be exactly in phase and to exhibit constructive interference.

5 Conditions for Constructive and Destructive Interference For two wave sources vibrating in phase, a difference in path lengths that is a) zero or an integer number (1, 2, 3,.. ) of wavelengths leads to constructive interference; b) a half-integer number (½, 1 ½, 2 ½,..) of wavelengths leads to destructive interference. d 1 d2 DEMO: 2 speakers If d 1 d 2 n 1 2 λ (with integer n) Destructive Interference

6 ACT: Interference Speakers A and B emit sound waves of λ 1 m, which interfere constructively at a donkey located far away (say, 200 m). What happens to the sound intensity if speaker A steps back 2.5 m? a) intensity increases b) intensity stays the same c) intensity goes to zero d) impossible to tell If λ 1 m, then a shift of 2.5 m corresponds to 2.5λ, which puts the two waves out of phase, leading to destructive interference. The sound intensity will therefore go to zero. A B

7 Example: Interference A speaker generates a continuous tone of 440 Hz. In the drawing, sound travels into a tube that splits into two segments, one longer than the other. The sound waves recombine before being detected by a microphone. The speed of sound in air is 343 m/s. What is the minimum difference in the lengths of the two paths for sound travel if the waves arrive in phase at the microphone? (a) 0.10 m (b) 0.39 m (c) 0.78 m (d) 1.11 m (e) 1.54 m For constructive interference, T h i s theminimum path difference l

8 Reflected Waves A pulse travels through a rope towards the end that is fixed to a vertical pole and gets inverted on reflection. If the rope is connected to a ring that can slide up and down without friction along the pole (ie, free end), the pulse remains upright. Fixed boundary condition Free boundary condition

9 Standing Waves v > 0 v < 0 Superposition of a right-going wave and a left-going wave (for example from reflection) causes a large amplitude standing wave to develop. Standing wave

10 Standing waves on a vibrating string occur at well-defined frequencies Integer numbers of ½ wavelengths are allowed String fixed at both ends DEMO: standing waves f n v n λ n v 2 n 1,2,3,4,

11 ongitudinal Standing Waves The basis for wind instruments! Integer numbers of ½ wavelengths are allowed Tube open at both ends f n æ v ö nç n è 2 ø 1,2,3,4,!

12 Or only one end open Only odd numbers of ¼ wavelengths are allowed Tube open at one end f n æ v ö nç n è 4 ø 1,3,5,!

13 ACT: Soda bottle pipe If you blow across the opening of a partially filled soda bottle, you hear a tone. If you take a big sip of soda and then blow across the opening again, how will the frequency of the tone change? a) frequency will increase b) frequency will not change c) frequency will decrease By drinking some of the soda, you have effectively increased the length of the air column in the bottle. A longer pipe means that the standing wave in the bottle would have a longer wavelength. Because the wave speed remains the same, and we know that v f λ, then we see that the frequency has to be lower.

14 ACT: Open and Closed Pipes You blow into an open pipe and produce a tone. What happens to the frequency of the tone if you close the end of the pipe and blow into it again? a) depends on the speed of sound in the pipe b) you hear the same frequency c) you hear a higher frequency d) you hear a lower frequency In the open pipe, of a wave fits into the pipe, and in the closed pipe, only 1 4 of a wave fits. Because the wavelength is larger in the closed pipe, the frequency will be lower. 1 2

15 Example Standing Waves A rope of length is clamped at both ends. Which one of the following is not a possible wavelength for standing waves on this rope? (a) /2 (b) 2/3 (c) (d) 2 (e) ) 2 2 ) ) ) 2 2 ) l l l l l l l l l l Þ Þ Þ Þ Þ e d c b a

16 Example Standing Waves A string with a linear density of kg/m and a mass of kg is clamped at both ends. Under what tension in the string will it have a fundamental frequency of 110 Hz? (a) 270 N (b) 410 N (c) 550 N (d) 680 N (e) 790 N f n æ v ö nç n è 2 ø 1,2,3,4,!

17 So we can change the frequency of a standing wave on a string by changing the tension in the string. How do we change the frequency (e.g. pitch) of a sound in a resonant cavity? Sound travels through gases, liquids, and solids at considerably different speeds. Mythbusters: Helium and Sulfurhexafluoride

18 Beats et s move from the spatial domain to the time domain What happens when two tones with slightly different frequencies interfere? The beat frequency is the difference between the two sound frequencies. DEMO: Tune forks

19 ACT: Beats The traces below show beats that occur when two different pairs of waves interfere. For which case is the difference in frequency of the original waves greater? a) pair 1 b) pair 2 c) same for both pairs d) impossible to tell by just looking Recall that the beat frequency is the difference in frequency between the two waves: f beat f 2 f 1. Pair 1 has the greater beat frequency (more oscillations in same time period), so pair 1 has the greater frequency difference. Pair 1 Pair 2

20 Example - Beats Two timpani (tunable drums) are played at the same time. One is correctly tuned so that when it is struck, sound is produced with wavelength of 2.20 m. The second produces sound with a wavelength of 2.08 m. If the speed of sound is 343 m/s, what beat frequency is heard? (a) 7 Hz (b) 9 Hz (c) 11 Hz (d)13 Hz (e) 15 Hz Beats v 343m/s f1 165 Hz l 2.08 m v 343m/s f2 156 Hz l 2.20 m Beats 165 Hz -156 Hz Beats f Hz f 2