1. Wave Interference Superposition Principle – when two or more waves encounter each other while traveling through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point.

5. Sound Wave Interference When the path difference is an integer multiple of the wavelength of the sound, there is constructive interference. In this case a maximum in sound intensity is detected at the receiver. When the path difference is a half-integer multiple of the wavelength of the sound, there is destructive interference. In the case of completely destructive interference, no sound is detected at the receiver. Path difference  r = |r 2 - r 1 |

6. Mathematics of Interference Two waves, traveling in the same direction, with a phase difference  we can rewrite this using the following trigonometry rule:

7. Mathematics of Interference (cont.) Two identical waves, traveling in opposite directions using the same trig rule as before we get

8. Standing Waves

9. Standing waves can occur mathematically, as seen in the previous slides, and they can also occur in ropes and strings. This is how stringed musical instruments (violins, guitars, harps) produce sound. This effect also explains how you can change the sound produced by a string by tightening or loosening the string.

10. Standing Waves on a String There are only certain stable patterns that will occur on a particular string. These are called the normal modes of oscillation. The properties of the string ( , L, T) determine the normal modes of oscillation. The motion of the string at one of these normal modes of oscillation is a standing wave. If the string is driven at a frequency that is not one of the normal frequencies, then the string will not exhibit a stable pattern (it will not produce a pleasant sound)

11. Standing Waves - String

12. Standing Wave - string terminology n = 1Fundamental1 st harmonic n = 2First overtone2 nd harmonic n = 3Second overtone 3 rd harmonic n = 4Third overtone4 th harmonic

13. Standing Waves - string We can look at the standing wave patterns to determine a relationship between L and. Fundamental 2 nd harmonic 3 rd harmonic nth harmonic

14. Standing Waves - string Rearrange that last equation: and since v=f  Replace the velocity of a wave on a string with the equation from last chapter,

15. Standing Waves in a Pipe Just like how standing waves are formed on a string by the interference between two oppositely directed transverse waves, standing waves in pipes are the result of interference between two longitudinal sound waves traveling in opposite directions The interference is between the original wave sent into the pipe and its reflection. This is how musical instruments like flutes and pipe organs produce sound.

16. Standing Waves - pipe There are particular harmonics for sound waves in pipes. At each of the harmonics, the pipes produce a “clean” sound. The harmonics are dependent on the length of the pipe.

17. Pipe - open at both ends

18. Standing Wave – open pipe terminology n = 1Fundamental1 st harmonic n = 2First overtone2 nd harmonic n = 3Second overtone 3 rd harmonic n = 4Third overtone4 th harmonic

19. Pipe - open at one end, closed at the other end

20. Standing Wave – closed pipe terminology n = 1Fundamental1 st harmonic n = 3First overtone3 rd harmonic n = 5Second overtone 5 th harmonic n = 7Third overtone7 th harmonic

21. Resonance All of the possible harmonic frequencies are also called resonance frequencies. If you add energy to the system at a frequency equal to one of the resonance frequencies, you will continually add to the amplitude of the vibration (motion) of the system. Eventually, the system will break. Exs: Tacoma Narrows Bridge, a shattered wine glass (from a high note), a building that collapses during an earthquake

22. Beats

23. Beats (cont.) There are a number of frequencies present in the previous equation. One is the frequency of the resultant sound wave, one is the frequency of the amplitude. Another frequency we can pull out of that equation is the beat frequency. The beat frequency is the number of beats you hear per second. The maximum our ears can detect is about 20 beats per second. If the beat frequency is larger than that, the two sounds blend together.