1. Physics 1B03summer-Lecture 10 1)Identical waves in opposite directions: “standing waves” 2)2 waves at slightly different frequencies: “beats” 3)2 identical waves, but not in phase: “interference” Superposition of Waves

2. Physics 1B03summer-Lecture 10 Practical Setup: Fix the ends, use reflections. node L (“fundamental mode”) We can think of travelling waves reflecting back and forth from the boundaries, and creating a standing wave. The resulting standing wave must have a node at each fixed end. Only certain wavelengths can meet this condition, so only certain particular frequencies of standing wave will be possible. example:

3. Physics 1B03summer-Lecture 10 λ2λ2 Second Harmonic Third Harmonic.... λ3λ3

4. Physics 1B03summer-Lecture 10 In this case (a one-dimensional wave, on a string with both ends fixed) the possible standing-wave frequencies are multiples of the fundamental: f 1, 2f 1, 3f 2, etc. This pattern of frequencies depends on the shape of the medium, and the nature of the boundary (fixed end or free end, etc.).

5. Physics 1B03summer-Lecture 10 Problem 8mm y 1.2 m f = 150 Hz x a)Write out y(x,t) for the standing wave. b)Write out y 1 (x,t) and y 2 (x,t) for two travelling waves which would produce this standing wave. wave at t=0

6. Physics 1B03summer-Lecture 10 Quiz When the mass m is doubled, what happens to a) the wavelength, and b) the frequency of the fundamental standing-wave mode? What if a thicker (thus heavier) string were used? m

7. Physics 1B03summer-Lecture 10 Example a) m = 150g, f 1 = 30 Hz. Find μ (mass per unit length) b) Find m needed to give f 2 = 30 Hz c) m = 150g. Find f 1 for a string twice as thick, made of the same material. m 120 cm

8. Physics 1B03summer-Lecture 10 Standing sound waves Sound in fluids is a wave composed of longitudinal vibrations of molecules. The speed of sound in a gas depends on the temperature. For air at room temperature, the speed of sound is about 340 m/s. At a solid boundary, the vibration amplitude must be zero (a standing wave node). node antinode Physical picture of particle motions (sound wave in a closed tube) graphical picture

9. Physics 1B03summer-Lecture 10 Standing sound waves in tubes – Boundary Conditions -there is a node at a closed end -less obviously, there is an antinode at an open end (this is only approximately true) node antinode graphical picture

10. Physics 1B03summer-Lecture 10 L Pipe with one closed end, one open end

11. Physics 1B03summer-Lecture 10 Exercise: Sketch the first three standing-wave patterns for a pipe of length L, and find the wavelengths and frequencies if: a)both ends are closed b)both ends are open

12. Physics 1B03summer-Lecture 10 Example You blow across the end of a straw (both ends open), and produce a 600 Hz whistling sound. 1)If you close one end of the straw with your thumb, what frequency would you get? 2)What other frequencies (in addition to the fundamental) could you produce in each case? 3)If you filled the straw with helium (v sound = 1000 m/s), how would this affect the wavelength and frequency of the sound inside and outside the straw?

13. Physics 1B03summer-Lecture 10 Beats (section 21.8) Two waves of different frequencies arriving together produce a fluctuation in power or amplitude. Since the frequencies are different, the two vibrations drift in and out of phase with each other, causing the total amplitude to vary with time. y time 1 beat

14. Physics 1B03summer-Lecture 10 time t in phase 180 o out of phase in phase

15. Physics 1B03summer-Lecture 10 Same amplitudes, different frequencies: Trigonometry: cos a + cos b = 2 cos [(a-b)/2] cos [(a+b)/2] Result: slowly-varying amplitude SHM at average frequency The math:

16. Physics 1B03summer-Lecture 10 Note maximum power when “amplitude” part is 2A 2 beats per cycle of # beats/second = The beat frequency (number of beats per second) is equal to the difference between the frequencies:

17. Physics 1B03summer-Lecture 10 Quiz Two guitar strings originally vibrate at the same 400- Hz frequency. If you hear a beat of 5Hz, what are the other possible frequencies ?