Velocity

1. Velocity • The velocity of the wave is the measurement of how fast a crest is moving from a fixed point. • The speed of sound is about 1000 feet/second. The speed of light is 186,000 miles/second. • Velocity = Wavelength x Frequency

2. Velocity = frequency x wavelength v =גּ f m/s m Hz (1/s) “distance” “time” m•Hz = m•1/s = m/s

3. Frequency • Frequency: The number of times the wavelength occurs in one second. Measured in kilohertz (Khz), or cycles per second. The faster the sound source vibrates, the higher the frequency. • Higher frequencies are interpreted as a higher pitch. For example, when you sing in a high-pitched voice you are forcing your vocal chords to vibrate quickly. • measured in Hertz (Hz) • 1 Hz = 1/second • The shorter the wavelength, the higher the frequency

4. Simple Harmonic Motion period, T: time to complete one cycle (s) # of cycles per second (Hz) frequency, f:

5. wall equilibrium position frictionless floor Dx = 0 Felas = 0 Dxmax v = 0 Dx = max Felas = max –Dxmax period of a mass-spring system: m = mass (kg) k = spring constant (N/m)

6. A 5.5 kg cat is attached to a fixed horizontal spring of stiffness 22.8 N/m and is set in motion on a frictionless surface. Find the period of motion of… …the cat. …a 240 g mouse, with the same spring and surface.

7. What stiffness must a spring have so that the period of the mouse’s motion is the same as that of the cat? Ballpark answer: Need a “less stiff” spring; k < 22.8 N/m.

8. A 1275 kg car carries two passengers with a combined mass of 153 kg. The car has four shock absorbers, each with a spring constant of 2.0 x 104 N/m. Find the frequency of the vehicle’s motion after it hits a pothole.

9. Frestore Dx restoring force: acts to move an object back to equilibrium simple harmonic motion (SHM): As displacement increases, so does Frestore. And when Dx = 0… Frestore = 0. For a mass-spring system, Hooke’s law applies: Frestore = Felas = k Dx

10. energy: the ability to do work kinetic energy, KE: energy of mass m having velocity v KE = ½ m v2 m (kg); v (m/s) KE = max. at eq. pos.; KE = 0 at Dxmax.

11. potential energy, PE: stored energy For a spring with spring constant k and “stretch” Dx: PEelas = ½ k (Dx)2 k (N/m); Dx (m) For m-s sys., PEelas is max at Dxmax and 0 at eq. pos. For a mass m at a height h above a reference line: PEg = m g h m (kg); h (m); g = 9.81 m/s2 For pend., PEg is max at Dxmax and min. at eq. pos.

12. A A Energy (J) PEg amplitude, A: maximum displacement from equilibrium frictionless Period T is not affected by amplitude A. Energy of a Mass-Spring System total energy PEelas KE

13. length L amplitude Q mass m of bob Energy (J) PEelas The Pendulum For Q < 15o, a simple pendulum approximates SHM. Energy of a Simple Pendulum total energy PEg KE

14. period of a simple pendulum: Period T is independent of mass and amplitude. The period of a pendulum is 5.2 s. Find… A. …its length B. …the mass of the bob

15. vibrations moving through space and time Waves Waves transmit energy, not matter. medium: the matter through which the energy of mechanical waves moves

16. transverse waves: particles of medium move to direction of wave travel Energy Amplitude2 crest amplitude A trough wavelengthl For a transverse wave:

17. particles of medium move // to direction of wave travel longitudinal (compressional) wave: compression rarefaction l pulse wave: a single vibration periodic wave: rhythmic, repeated vibrations

18. A A v v v v • Wave Reflection fixed boundary free boundary waves are reflected and inverted waves are reflected and upright

19. http://www.acs.psu.edu/drussell/Demos/reflect/reflect.html

20. A1 + A2 A1 – A2 A1 A1 A2 A2 Wave Interference Two waves (unlike two objects) can occupy the same place at the same time. This condition is called interference. constructive interference: destructive interference: displacements are in same direction displacements are in opposite directions A1 A2 A1 A2

21. Standing Waves incident and reflected waves interfere so that antinodes have a max. amplitude, while nodes have zero amplitude On a string, nodes remain motionless; antinodes go from max. (+) to max. (–) displacement.

22. http://www.physicsclassroom.com/Class/waves/h4.gif

23. Standing Waves in Open Tubes L n = 1 n = 2 n = 3 l1 = 2 L l2 = L l3 = 2/3 L wavelength of nth harmonic of an open tube: (n = 1,2,3,…)

24. L n = 1; 1st harmonic l1 = 2 L (fundamental) n = 2; 2nd harmonic l2 = L (1st overtone) n = 3; 3rd harmonic l3 = 2/3 L (2nd overtone) n = 4; 4th harmonic l4 = ½ L (3rd overtone) wavelength of nth harmonic on a string: (n = 1,2,3,…)

25. Waves travel along a 96.1 cm guitar string at 492 m/s. Find the fundamental frequency of the string. Find the frequency of the 5th harmonic. frequency of the nth harmonic:

26. 1.24 m Find the fundamental frequency for an open tube of length 1.24 m and a velocity of 343m/s

27. Closed Tubes L n = 1 n = 2 n = 3 l1 = 4 L l2 = 2 L l3 = 4/3 L wavelength of nth harmonic of a closed tube: (n = 1,3,5,…) (even harmonics are not present)

28. Find fundamental frequency for a Closed tube in the third harmonic of length 1.24m and a Velocity of 350m/s .

29. 20 Hz 20,000 Hz Sound compression: high pressure / high density rarefaction: low pressure / low density audible frequencies (human hearing) infrasonic ultrasonic

30. Fundamental frequency determines pitch. high pitch high f = = short l low pitch = long l low f =

31. f1 f2 f3 f4 f1 f2 f3 f4 f1 f2 f3 f4 f1 f2 f3 f4 Number and intensity of an instrument’s harmonics give it its unique sound quality, or ________. timbre

32. The Doppler Effect Relative motion between wave source and observer causes a change in the ____________ frequency. observed v = 0 femitted fobserved (higher) femitted femitted fobserved (lower)

33. Sun R O Y G B V most stars Other examples of Doppler effect: race cars police radar (“red-shifted”) dolphins (echolocation) expansion of universe

34. wave barrier • Traveling Very Fast vbug = 0 vbug < vwave bow wave vbug > vwave vbug = vwave

35. supersonic: “faster than sound” (vs. subsonic) shock wave: a 3-D bow wave sonic boom: caused by high-pressure air, not roaring engine lion tamer’s whip cracking bullets The Matrix

36. Sound Intensity If a piano’s power output is 0.302 W, find the sound intensity at a distance of… A. …1.0 m B. …2.0 m

37. Intensity is related to volume (or relative intensity): -- how loud we perceive a sound to be -- measured in decibels (dB) A difference of 10 dB changes the sound intensity by a factor of 10 and the volume by a factor of 2. 50 dB40 dB 60 dB90 dB 8X louder half as loud 1/10 as intense 1000X more intense

38. 1 second elapses 1 second elapses Beats alternating loud-and-soft sounds resulting from interference between two slightly- different frequencies Equation: f1 = 16 Hz f1 = 16 Hz f2 = 17 Hz f2 = 18 Hz fbeat = 1 Hz fbeat = 2 Hz

39. Forced Vibrations and Resonance natural frequency: the frequency at which an object most easily vibrates forced vibration: a vibration due to an applied force resonance: occurs when a force is repeatedly applied to an object AT the object’s natural frequency large amplitude -- result of resonance =

40. Examples: swing shattering crystal wine glasses Tacoma Narrows Bridge (1940) British regiment (Manchester, 1831) aeolian harps “The wind in the wires made a tattletale sound, as a wave broke over the railing…”

41. vsound = 331 + 0.6Ta Frestore = Felas = k Dx KE = ½ m v2 PEelas = ½ k (Dx)2 fn = n f1 PEg = m g h