1. Group Work 1.Add these two waves together.

2. Waves Part 2: resonance and more dimensions

3. Objectives Evaluate combinations of waves from confined systems. Explain the refraction of waves. Explain the behavior of ocean waves near the shore.

4. Combinations of Waves

5. Standing Waves Sum of waves of equal amplitude and wavelength traveling in opposite directions Half-wavelength divides exactly into the available space Wave pattern has locations of minimum and maximum variation (nodes and antinodes) (standing longitudinal waves)standing longitudinal waves

6. Resonance Objects have characteristic frequencies at which standing waves are sustained Lowest frequency = fundamental Higher frequencies = overtones Sustained motion is a combination of normal modes

7. Vibrational Modes: Clamped String Source: Griffith, The Physics of Everyday Phenomena, Figure 15.13

8. Poll Question Bowing a ‘cello string 1/4 of the way along will excite which harmonic the MOST? A.The fundamental. B.The first overtone. C.The second overtone. D.The third overtone.

9. Group Work Add together: 2.a fundamental and second overtone.

10. Group Work Result Add together: 1.a fundamental and first overtone.

11. Group Work Result Add together: 2.a fundamental and second overtone.

12. Combinations of Harmonics Characteristic sounds arise from combining particular harmonics in specific ratios fluteoboesaxophone

13. “Closed” and “Open” Tube Modes Source: Halliday, Resnick, and Walker, Fundamentals of Physics, 2003, p 419.

14. Poll Question Which has the lowest fundamental tone? A.A closed tube of length L. B.An open tube of length L. C.Both have the same fundamental tone. D.Not enough information.

15. Sequence of Harmonics Western musical scale and harmonies are based on overtone series (sound files)

16. Musical Tones Octave higher = 2  the frequency Octave + fifth: 3  the frequency Even-tempered scale = compromise to facilitate transposition –12 (half-) steps per octave –Half-step: 2  the frequency 12

17. Two-Dimensional Waves Ocean waves Earthquake surface waves Animation

18. Wave Refraction Source: University of Southampton, Institute for Sound and Vibration ResearchUniversity of Southampton, Institute for Sound and Vibration Research Direction changes to keep wave front continuous

19. Wave Refraction Speed u 1 Speed u 2 11 22 Snell’s Law: u 2 sin  1 = u 1 sin  2

20. Poll Question Waves approach a slow zone at an angle. The direction the waves travel slow fast A B C A.does not change. B.bends toward the interface. C.bends away from the interface.

21. Refraction Example u = 1 u = 3/4 u = 1/2

22. Water Waves Restoring force is gravity Fastest in deep water, slow in shallow water Breaking and refractive behavior in shallow water Speed depends on wavelength Very complicated behavior!

23. Poll Question Ocean waves approaching a shore will A.bend so that the wave fronts make a greater angle with the shore. B.bend so that the wave fronts are more parallel to the shore. C.veer away from the shore. D.not change their direction.

24. Ocean Wave Refraction

25. Group Work 3.Sketch how the train of waves would move into the region of slow speed. fastslow

26. Breaking Wave

27. Shoaling and Breaking Waves slow in shallows by shore Wavelength shortens and amplitude rises Troughs move slower than crests Crests fall forward

28. Poll Question When waves approach the shore, their increases and decreases. A.speed, mass B.force, mass C.wavelength, period D.amplitude, frequency E.amplitude, wavelength F.frequency, wavelength

29. Tsunami

30. Reading for Next Time Gravity Big Ideas: –Inverse-square force law –Gravitational fields –Orbital dynamics