PHY II – Waves

1. PHY II – Waves • The Wave Model • Speed of Waves on a String • Sinusoidal Waves • Spherical Waves, Plane Waves

2. Transverse versus Longitudinal

3. Snapshot Graph

4. History Graph

5. A string of beads are connected by a set of tiny springs. At the instant the clock starts (t=0), a pulse is moving to the right on the beads and the snapshot graph looks like this: Which of the following history graphs tracks the position for the bead marked with a red arrow as a function of time?

6. The graph at the top is the history graph at x = 4 m of a wave traveling to the right at a speed of 2 m/s. Which is the history graph of this wave at x = 0 m? (A) (B) (C)

7. Sinusoidal Waves

8. Quiz 2 What is the frequency of this traveling wave? A. 10 Hz B. 5 Hz C. 2 Hz D. 0.2 Hz E. 0.1 Hz

9. Quiz 3 What is the phase difference between the maximum of a wave and the adjacent minimum? (crest to trough) A. 0 B. π/4 C. π/2 D. 3 π/2 E. π

10. Sinusoidal Wave Snapshot Graph k = 2π/λis the wave number

11. Sinusoidal Wave History Graph ω=2π/T is the angular frequency

14. PHY II – Waves • Sound and Light • Medical Applications of Ultrasound • Power and Intensity • The Doppler Effect

15. Sound and Light • Sound is a pressure wave in a gas, liquid or solid. Speed depends on material. • Light is one type of electromagnetic wave. • In a vacuum, all electromagnetic waves (including light) travel at c = 3×108 m/s. • In transparent media, light slows down. Index of Refraction is n > 1. • This reduces the wavelength, but does not change the frequency!

16. A light wave travels through three transparent materials of equal thickness. Rank is order, from the largest to smallest, the indices of refraction n1, n2, and n3. A.n2 > n1 > n3B.n3 > n1 > n2C.n1 > n2 > n3 D.n3 > n2 > n1E.n1 = n2 = n3

17. Reflection of Transverse Wave Pulse • A pulse traveling to the right on a heavy string attached to a lighter string • Speed suddenly increases

18. Reflection of Transverse Wave Pulse • A pulse traveling to the right on a light string attached to a heavier string • Speed suddenly decreases

19. Physics of Ultrasound • Speed of sound in bone, flesh and blood are all different • When the speed of any wave suddenly changes, there is a reflection and transmission • Ultrasound images are formed from reflected high frequency sound • Image resolution is set by wavelength, λ • λ=v/f, so higher frequency yields smaller λ, and better resolution

20. Speed of sound in humans

21. Power and Intensity • The Power, P, of any wave source is how much energy per second is radiated as waves [units = Watts] • The Intensity, I, is the energy rate per area. This determines how loud (sound) or bright (light) the wave is. • I=P/a, where a is an area perpendicular to the wave direction. • At a distance r from a spherically symmetric source, the intensity is I=P/(4πr2)

22. Chapter 20, Problem 34 • The sound intensity from a jack hammer breaking concrete is 2 W/m2 at a distance of 2 m from the point of impact. This is sufficiently loud to cause permanent hearing damage if the operator doesn’t wear ear protection. What is the sound intensity for a person watching from 50 m away?

23. Doppler Effect

24. Which statement is true? Valerie is standing in the middle of the road, as a police car approaches her at a constant speed, v. The siren on the police car emits a “rest frequency” of f0. • The frequency she hears rises steadily as the police car gets closer and closer. • The frequency she hears steadily decreases as the police car gets closer and closer. • The frequency she hears does not change as the police car gets closer.

25. Which statement is true? Valerie is standing still as a police car approaches her at a constant speed, v. Daniel is in his car moving at the same constant speed, v, toward an identical police car which is standing still. Both hear a siren. Actual Answer! Check eq.20.38 and 20.39! • The frequency Daniel hears is lower than the frequency Valerie hears. • The frequency Daniel hears is higher than the frequency Valerie hears. • The frequencies that Daniel and Valerie hear are exactly the same. (Almost correct)

26. PHY II – Waves • The Principle of Superposition • Standing Waves • The Guitar: Stringed Instruments • The Trumpet: Wind Instruments

27. Principle of Superposition • If two or more waves combine at a given point, the resulting disturbance is the sum of the disturbances of the individual waves. • Two traveling waves can pass through each other without being destroyed or even altered!

28. Some Results of Superposition: • Two waves, same wavelength and frequency, opposite direction: Standing Wave • Two waves, same wavelength and frequency, similar direction, different phase: Interference • Two waves, same direction, slightly different frequency and wavelength: Beats!

29. Standing Wave: The superposition of two 1-D sinusoidal waves traveling in opposite directions.

30. Standing Waves • Are a form of “resonance” • There are multiple resonant frequencies called harmonics • The boundary conditions and speed of waves determine which frequencies are allowed. • The ends of the resonant cavity have forced nodes or antinodes • With a wave on a string, it is possible to force an intermediate node

31. OMGWTFBBQ Standing Wave in Pressure Forces the Flames higher at antinodes!

32. Harmonic frequencies Transverse standing wave on a string clamped at both ends: there are nodes in displacement at both ends. Standing sound wave in a tube open at both ends: there are nodes in pressure both ends.

33. Harmonic frequencies Standing sound wave in a tube closed at one end: there is a node in pressure at the open end, and an anti-node at the closed end.

34. PHY II – Waves • Constructive and Destructive Interference • Interference Patterns • Beats

35. Wave Interference • Two waves moving in the same direction with the same amplitude and same frequency form a new wave with amplitude: where a is the amplitude of either of the individual waves, and is their phase difference.

40. Beat frequency • Beats are loud sounds separated by soft sounds • The beat frequency is the difference of the frequencies of the two waves that are being added: • The frequency of the actual sound is the average of the frequencies of the two waves that are being added: