1. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 12: Sound Sound Waves The Speed of Sound Amplitude & Intensity of Sound Waves Standing Sound Waves Beats The Doppler Effect Shock Waves Echolocation

2. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 § 12.1 Sound Waves Sound waves are longitudinal. They can be represented by either variations in pressure (gauge pressure) or by displacements of an air element.

3. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 The middle of a compression (rarefaction) corresponds to a pressure maximum (minimum).

4. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 § 12.2 The Speed of Sound Waves The speed of sound in different materials can be determined as follows: In fluids In thin solid rods B is the bulk modulus of the fluid and  its density. Y is the Young’s modulus of the solid and  its density.

5. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 In ideal gases Here v 0 is the speed at a temperature T 0 (in kelvin) and v is the speed at some other temperature T (also in kelvin). where T c is the air temperature in  C. For air, a useful approximation to the above expression is

6. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 Materials that have a high restoring force (stiffer) will have a higher sound speed. Materials that are denser (more inertia) will have a lower sound speed.

7. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 Example (text problem 12.8): A copper alloy has a Young’s Modulus of 1.1  10 11 Pa and a density of 8.92  10 3 kg/m 3. What is the speed of sound in a thin rod made of this alloy? The speed of sound in this alloy is slightly less than the value quoted for copper (3560 m/s) in table 12.1.

8. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 Example (text problem 12.1): Bats emit ultrasonic sound waves with a frequency as high as 1.0  10 5 Hz. What is the wavelength of such a wave in air of temperature 15.0  C? The speed of sound in air of this temperature is 340 m/s.

9. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 Example (text problem 12.10): A lightning flash is seen in the sky and 8.2 seconds later the boom of thunder is heard. The temperature of the air is 12.0  C. (a) What is the speed of sound in air at that temperature? The speed of sound in air of this temperature is 338 m/s. (b) How far away is the lightning strike?

10. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 Example continued: The speed of light is 3.00  10 5 km/s. How long does it take the light signal to reach the observer?

11. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 § 12.3 Amplitude & Intensity of Sound Waves For sound waves: p 0 is the pressure amplitude and s 0 is the displacement amplitude. The intensity of sound waves also follow an inverse square law.

12. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 Loudness of a sound is measured by the logarithm of the intensity. The threshold of hearing is at an intensity of 10 -12 W/m 2. Sound intensity level is defined by dB are decibels

13. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 Example (text problem 12.12): The sound level 25 m from a loudspeaker is 71 dB. What is the rate at which sound energy is being produced by the loudspeaker, assuming it to be an isotropic source? Solve for I, the intensity of a sound wave: Given:

14. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 The intensity of an isotropic source is defined by: Example continued:

15. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 Example: Two sounds have levels of 80 dB and 90 dB. What is the difference in the sound intensities? Subtracting:

16. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 § 12.4 Standing Sound Waves Consider a pipe open at both ends: The ends of the pipe are open to the atmosphere. The open ends must be pressure nodes (and displacement antinodes).

17. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 The distance between two adjacent antinodes is ½. Each pair of antinodes must have a node in between. The fundamental mode (it has the fewest number of antinodes) will have a wavelength of 2L.

18. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 The next standing wave pattern to satisfy the conditions at the ends of the pipe will have one more node and one more antinode than the previous standing wave. Its wavelength will be L. The general result for standing waves in a tube open at both ends is where n=1, 2, 3,… f 1 is the fundamental frequency.

19. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19

20. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Now consider a pipe open at one end and closed at the other. As before, the end of the pipe open to the atmosphere must be a pressure node (and a displacement antinode). The closed end of the pipe must be a displacement node (and a pressure antinode).

21. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 One end of the pipe is a pressure node, the other a pressure antinode. The distance between a consecutive node and antinode is one-quarter of a wavelength. Here, the fundamental mode will have a wavelength of 4L.

22. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 The next standing wave to satisfy the conditions at the ends of the pipe will have one more node and one more antinode than the previous standing wave. Its wavelength will be (4/3)L. The general result for standing waves in a tube open at one end and closed at the other is where n=1, 3, 5,…. n (odd values only!!) f 1 is the fundamental frequency.

23. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23

24. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 Example (text problem 12.22): An organ pipe that is open at both ends has a fundamental frequency of 382 Hz at 0.0 °C. What is the fundamental frequency for this pipe at 20.0 °C? At T c = 0.0 °C, the speed of sound is 331 m/s. At T c = 20.0 °C, the speed of sound is 343 m/s. The fundamental frequency is

25. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 The ratio of the fundamental frequencies at the two temperatures is: Example continued:

26. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 How long is this organ pipe? Using either set of v and f 1. Example continued:

27. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 § 12.5 Beats When two waves with nearly the same frequency are superimposed, the result is a pulsation called beats.

28. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 The beat frequency is Two waves of different frequency Superposition of the above waves

29. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 If the beat frequency exceeds about 15 Hz, the ear will perceive two different tones instead of beats.

30. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 § 12.6 The Doppler Effect When a moving object emits a sound, the wave crests appear bunched up in front of the object and appear to be more spread out behind the object. This change in wave crest spacing is heard as a change in frequency. The results will be similar when the observer is in motion and the sound source is stationary and also when both the sound source and observer are in motion.

31. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31

32. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 The Doppler Effect formula f o is the observed frequency. f s is the frequency emitted by the source. v o is the observer’s velocity. v s is the source’s velocity. v is the speed of sound. Note: take v s and v o to be positive when they move in the direction of wave propagation and negative when they are opposite to the direction of wave propagation.

33. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 Example (text problem 12.39): A source of sound waves of frequency 1.0 kHz is stationary. An observer is traveling at 0.5 times the speed of sound. (a) What is the observed frequency if the observer moves toward the source? f o is unknown; f s = 1.0 kHz; v o = -0.5v; v s = 0; and v is the speed of sound.

34. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 (b) Repeat, but with the observer moving in the other direction. f o is unknown; f s = 1.0 kHz; v o = +0.5v; v s =0; and v is the speed of sound. Example continued:

35. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 § 12.7 Shock Waves If a plane were traveling at the speed of sound, what would the wave crests looks like? They would be bunched up in front of the aircraft and an observer (to the right) would measure =0.

36. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 If the source moves with a speed greater than that of sound, then the wave crests pile up on top of each other forming a cone-shaped shock wave.

37. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 § 12.8 Echolocation Sound waves can be sent out from a transmitter of some sort; they will reflect off any objects they encounter and can be received back at their source. The time interval between emission and reception can be used to build up a picture of the scene.

38. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 Example (text problem 12.47): A boat is using sonar to detect the bottom of a freshwater lake. If the echo from a sonar signal is heard 0.540 s after it is emitted, how deep is the lake? Assume the lake’s temperature is uniform and at 25  C. The signal travels two times the depth of the lake so the one-way travel time is 0.270 s. From table 12.1, the speed of sound in freshwater is 1493 m/s.

39. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 Example (text problem 12.49): A bat emits chirping sounds of frequency 82.0 kHz while hunting for moths to eat. If the bat is flying toward a moth at a speed of 4.40 m/s and the moth is flying away from the bat at 1.20 m/s, what is the frequency of the wave reflected from the moth as observed by the bat? Assume T = 10.0  C. The speed of sound in air of this temperature is 337 m/s.

40. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40 The flying bat emits sound of f =82.0 kHz that is received by a moving moth. The frequency observed by the moth is: Example continued:

41. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 41 Example continued: Some of the sound received by the moth will be reflected back toward the bat. The moth becomes the sound source (f = 82.8 kHz) and the bat is now the observer.

42. Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 42 Summary Sound is a longitudinal wave. The speed of sound depends on material properties such as “stiffness”, density, and temperature. Sound Intensity Level Standing Waves in Pipes (both ends open & one end open/one end closed) The Doppler Effect Shock Waves Echolocation