AljalalPhys 102 20 Feb 2006 Ch 17 page

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 1 Chapter 17 Waves - II 17 -1 Sound Waves 17 -2 The Speed of Sound 17 -3 Traveling Sound Waves 17 -4 Interference 17 -5 Intensity and Sound Level 17 -6 Sources of Musical Sound (Sound Waves Resonances) 17 -7 The Doppler Effect

17 -1 Sound Waves Introduction Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 2 Sound waves are Mechanical waves • Need medium to travel Longitudinal waves • Oscillations are parallel to the direction of travel Motion of the wave Motion of particles We use sound waves to Speak and listen Probe Earth’s crust for oil Detect underwater objects Image fetuses

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 3 17 -1 Sound Waves Definitions A wavefront is an imaginary surface over which the displacements of particles are the same. A point source emits in all directions. Spherical waves A ray is a directed line perpendicular to wavefronts and indicates the direction of travel of the wavefronts. Away from a point source, we can approximate the wavefronts as planes. Wavefronts Ray l Planar waves are waves with planar wavefronts.

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 4 17 -2 The Speed of Sound Formula - Speed of Sound Speed of sound Bulk modulus of the medium Density of the medium

17 -2 The Speed of Sound Bulk modulus Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 5 Change in pressure V P Original volume Change in volume The bulk modulus always positive SI unit N/m 2 = Pascal (Pa) V+DV The bulk modulus of a medium is the property that determines the change in volume of an element of the medium when the pressure on it changes. A higher bulk modulus indicates a less compressible medium. Steel has higher bulk modulus than air. P+DP

17 -2 The Speed of Sound Table Speed of string wave Speed of sound Medium Air(00 C) Air(200 C) Water(200 C) Steel Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 6 Speed (m/s) 331 343 1482 5941 Speed of sound depends on temperature and pressure

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 7 17 -2 The Speed of Sound Example 1 A sound source which is very far such that wavefronts at ears are approximately planar. q Ear D Find an expression for the time delay Dt between the arrival of the sound at the left and the right ears. Solution Wavefronts From the time delay Dt, your brain will find the direction of the source. q L d q D R

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 8 17 -2 The Speed of Sound Example 2 Water Ear Suppose you are submerged in water at 200 C. Based on the time delay, at what angle from the forward direction does the source seem to be? D Solution Apparent position Real position

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 9 17 -3 Traveling Sound Waves Displacement No sound wave Air particles Sound wave Piston s(x, t) Displacement from equilibrium Displacement along x axis is represented along y axis. Displacement s(x, t) Displacement amplitude

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 10 17 -3 Traveling Sound Waves Pressure is proportional to density No sound wave Sound wave Piston Lower density Lower pressure Expansion Higher density Higher pressure Compression

Pressure variation Displacement s(x, t) Dp(x, t) 17 -3 Traveling Sound Waves Displacement and pressure variation Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 11 Displacement s(x, t) and pressure variation Dp(x, t) are p/2 out of phase. Pressure variation Pressure amplitude

Positive Dp. Compression. Pressure is higher than normal pressure. Pressure variation Dp(x, t) 17 -3 Traveling Sound Waves Displacement and pressure amplitudes Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 12 Negative Dp. Expansion. Pressure is less than normal (atmospheric) pressure. Pressure amplitude Speed of sound Density Displacement amplitude Angular frequency

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 13 17 -3 Traveling Sound Waves Loudest and faintest sound At 1000 Hz and at room temperature Dpm sm Loudest sound you can tolerate 30 Pa 10 mm Faintest sound you can hear 30 m. Pa 10 pm Atmospheric pressure 105 Pa Wavelength Human ears can hear sounds with frequencies between 20 Hz to 20 k. Hz. They are most sensitive for sounds with frequencies around 1000 Hz. 0. 34 m micro = 10 -6 pico = 10 -12

17 -3 Traveling Sound Waves Checkpoint 1 Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 14 A sinusoidal sound wave is traveling rightward in a fluid. At the moment an oscillating fluid element is moving to the right through the point of zero displacement, is the pressure in the element (a) at its equilibrium value, (b) just beginning to increase, or (c) just beginning to decrease? Solution Fluid element fluid Pressure variation Displacement s(x, t) Dp(x, t) (c) just beginning to decrease

17 -3 Traveling Sound Waves Example 3 Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 15 The maximum pressure amplitude Dpm that the human ear can tolerate in loud sounds is about 28 Pa. What is the displacement amplitude sm for such a sound in air? Density r = 1. 21 kg/m 3 Frequency =1000 Hz Sound speed = 343 m/s Solution

17 -4 Interference Phase difference and path length difference Wave 1 L 1 S 2 Wave 2 L 2 Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 16 P The distance between the point P and the sound sources L 1 and L 2 is much greater than the distance between the sources so that we can approximate the waves as traveling in the same direction at P. At point P Two point sources S 1 and S 2 emit sound waves that are in phase and of identical Phase difference wavelength l. The two sources are in phase. between wave 1 and wave 2 Path length difference DL= L 2 -L 1

17 -4 Interference Constructive and destructive interference Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 17 P Wave 1 L 1 S 1 Wave 2 L 2 At point P S 2 Constructive interference Destructive interference The two waves are in phase The two waves are out of phase Phase difference between wave 1 and wave 2 Path length difference

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 18 17 -4 Interference Example 4 Two point sources S 1 and S 2, which are in phase and separated by a distance D = 1. 5 l, emit sound waves of identical wavelength l. (a) What kind of interference at point P 1? (b) What kind of interference at point P 2? (c) What is the number of points on the circle at which the interference is fully constructive? Solution (a) At point P 1, DL=L 2 -L 1=0. The interference is fully constructive at P 1. (b) At point P 2, DL=L 2 -L 1=1. 5 l. The interference is fully destructive at P 2. (c) There are six points on the circle where the interference is fully constructive. DL=1. 5 l P 2 DL=1. 0 l S 1 DL=0 l P 1 DL=0 l D=1. 5 l S 2 DL=1. 0 l DL=1. 5 l

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 19 17 -4 Interference Checkpoint 2 In previous sample problem, if D = 4 l (a) What would occur at point P 1? (b) What would occur at point P 2? Solution (a) At point P 1, DL=L 2 -L 1=0. The interference is fully constructive at P 1. (b) At point P 2, DL=L 2 -L 1=4. 0 l. The interference is fully constructive at P 2. DL=4 l P 2 S 1 P 1 DL=0 l D=4 l S 2

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 20 17 -5 Intensity and Sound Level Formula The intensity I of a sound wave at a surface is the average rate per unit area at which energy is transferred by the wave into the surface Intensity Surface with area A Average Power Area Intensity SI unit is W/m 2 Angular frequency Intensity Energy transmitted in one period Displacement amplitude Speed of sound Density of the medium For string waves Sound

17 -5 Intensity and Sound Level Intensity of a point source Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 21 For a real source, the intensity varies, often in a complex way, with the distance from the source. For a point source, the sound is emitted isotropically, that is with equal intensity in all directions. In this case, the intensity can be found easily. All energy emitted by the source must pass through the sphere. Intensity Power of the source Distance from the point source The intensity of sound from isotropic point source decreases with the square of the distance from the source. Power of the source PS r S Imaginary Sphere

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 22 17 -5 Intensity and Sound Level Checkpoint 3 The three patches receive the same energy. Rank them, greatest first, according to (a) the intensity of sound on them, and (b) their area. 3 S 2 1 Imaginary Spheres Solution Intensity 1 and 2 tie 3 Area 3 1 and 2 tie

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 23 17 -5 Intensity and Sound Level Sound level Intensity Sound level Unit to measure sound level d. B = decibell Deci related to 10 Bell Alexander Graham Bell Dpm At 1000 Hz and at room temperature sm I 0=10 -12 W/m 2 Reference intensity (lowest you can hear) I b Pressure Amplitude Displacement Amplitude Intensity Sound level Loudest sound you can tolerate 30 Pa 10 mm 1 W/m 2 120 d. B Faintest sound you can hear 30 m. Pa 10 pm 10 -12 W/m 2 0 d. B b is a simpler way to express intensity

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 24 17 -5 Intensity and Sound Level Example 5 An electric spark of length L = 10 m emits a pulse of sound radially outward. The power of the emission is Ps = 1. 6 x 104 W. (a) What is the intensity of the sound at a distance r = 12 m from the spark? (b) At what rate Pd is sound energy intercepted by a detector of area Ad = 2. 0 cm 2 and located at r = 12 m? Solution (a) (b) All energy emitted by the source must pass radially through the cylinder. Linear sound source (spark) Detector Area Ad L r

17 -5 Intensity and Sound Level Example 6 Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 25 Two sources with sound levels b 1 = 92 d. B and b 2 = 120 d. B. What is the ratio of their intensities? Solution

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 26 17 -6 Sources of Musical Sound (Sound Waves Resonances) Air columns Musical sounds are produced by resonances of different oscillating bodies Strings Membranes Air columns Pipes with two open ends At resonant frequencies, the string oscillates with large amplitudes pushing against the surrounding air and generating a noticeable sound. At certain frequencies, reflections at both ends set up oscillating modes in the air column Resonant frequencies Pipes with one open end

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 27 17 -6 Sources of Musical Sound (Sound Waves Resonances) Pipes with two open ends At an open end, there is some reflection. The reflection is not complete. Sound source After the 1 st reflection from the left end After the 1 st reflection from the right end After the 2 nd reflection from the right end At resonance, the right-going waves are in phase as well as the left-going waves.

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 28 17 -6 Sources of Musical Sound (Sound Waves Resonances) Pipes with one open end At the closed end, the wave is reflected back and it is inverted. Sound source After the 1 st reflection from the open end After the 1 st reflection from the closed end After the 2 nd reflection from the closed end At resonance, the right-going waves are in phase as well as the left-going waves.

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 29 17 -6 Sources of Musical Sound (Sound Waves Resonances) Standing waves in pipes with one open end No wave Air particles Displacement s(x, t) Standing wave Representing a standing sound wave in a pipe. Pipe An open end of a pipe is like the end of a string attached to a freely moving ring. A wave is not inverted upon reflection from an open end. Nodes cannot occur at open ends. At resonance, there must be an antinode at the open ends.

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 30 17 -6 Sources of Musical Sound (Sound Waves Resonances) Standing waves in pipes with two open ends No wave Air particles Representing a standing sound wave in a pipe. Displacement s(x, t) Standing wave Pipe A closed end of a pipe is like the end of a string attached to a fixed point. A wave is inverted upon reflection from a closed end. Air cannot move back and forth and the displacement is always zero at closed ends. At resonance, there must be an antinode at the open end a node at the closed end.

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 31 17 -6 Sources of Musical Sound (Sound Waves Resonances) Oscillation modes Harmonic number n =1 Fundamental 1 st harmonic n =2 2 nd harmonic n =3 3 rd harmonic n =4 4 th harmonic n =5 5 th harmonic nth harmonic Oscillation mode L Missing n = integer n = odd integer

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 32 17 -6 Sources of Musical Sound (Sound Waves Resonances) Resonant frequencies of pipes with two open ends The distance between two consecutive left-going waves is 2 L For constructive interference, the phase difference between the two waves is After 3 rd reflection from the right end After 4 th reflection from the right end Resonance occurs when Round trip distance = 2 L Resonant frequencies

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 33 17 -6 Sources of Musical Sound (Sound Waves Resonances) Resonant frequencies of pipes with one open end The distance between two consecutive left-going waves is For constructive interference, the phase difference between the two waves is After 3 rd reflection from the right end After 4 th reflection from the right end At the closed end, the wave will be inverted. To do this, the wave moves an extra half wavelength. Resonance occurs when Resonant frequencies

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 34 17 -6 Sources of Musical Sound (Sound Waves Resonances) Size and resonant frequency When a musical instrument is played, the fundamental and one or more higher harmonics are generated at the same time Resonant frequencies To produce low frequency sounds, you need to use larger musical instruments. Violin Brass High frequency Low frequency

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 35 17 -6 Sources of Musical Sound (Sound Waves Resonances) Checkpoint 4 Which harmonic of pipe B has the same frequency of the fundamental of pipe A? Pipe A Pipe B LA 2 LA f = fundamental f = which harmonic? Solution The second harmonic Pipe A Pipe B

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 36 17 -6 Sources of Musical Sound (Sound Waves Resonances) Example 7 Background noises from a room setup the fundamental standing wave in a cardboard tube Use speed of sound 343 m/s. What frequency do you hear? L =67. 0 cm Solution cardboard tube If you jam your ear against one end, what fundamental frequency do you hear? Solution The tube is closed from one end L =67. 0 cm

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 37 17 -7 The Doppler Effect Definition The Doppler effect is the change in the detected frequency because of a relative motion between the source of sound and the detector. Detector of sound Ear Microphone Source of sound Siren Loudspeaker S vs Speed of the source Frequency emitted by the source D v. D Speed of the detector Frequency detected by the detector v. S and v. D are measured relative to the air. If air is stationary, these will be the same as that measured relative to the ground.

17 -7 The Doppler Effect Moving detector Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 38 Detector intercepts f wavefronts per second Detector intercepts more than f wavefronts per second Detector intercepts less than f wavefronts per second

17 -7 The Doppler Effect Moving source Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 39 Detector intercepts f wavefronts per second Detector intercepts more than f wavefronts per second Detector intercepts less than f wavefronts per second

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 40 17 -7 The Doppler Effect Relative motion and frequency S D v. D S S v. D D When they move towards each other, the frequency is shifted up. When they move away form each other, the frequency is shifted down.

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 41 17 -7 The Doppler Effect Formula Frequency at detector Frequency at source Speed of detector Speed of source Speed of sound Choose the signs such that when the source and detector move towards each other you get higher frequency, and when they move away from each other you get lower frequency

17 -7 The Doppler Effect Choosing the correct signs Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 42 Detector moves away from the source Detector moves towards the source Source moves towards the detector Source moves away from the detector Choose the signs such that when the source and detector move towards each other you get higher frequency, and when they move away from each other you get lower frequency

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 43 17 -7 The Doppler Effect Checkpoint 5 Solution Is the detected frequency greater than or less than the emitted frequency, or can’t we tell without more information about the actual speeds? Source Detector (a) ● 0 speed (a) greater (b) ● 0 speed (b) less (c) cannot tell (d) cannot tell (e) greater (f) less

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 44 17 -7 The Doppler Effect Example 8 Detector Rocket Speed = 242 m/s. Emitting sound waves at frequency f = 1250 Hz What frequency f′ is measured by the detector? Solution Pole

Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 45 17 -7 The Doppler Effect Example 9 Rocket Detector Speed = 242 m/s. Pole Echo Sound reflected from the pole What frequency f′′ does a detector on the rocket detect for the echo? Solution

17 -7 The Doppler Effect Example 10 Rocket Air speed = 20 m/s Aljalal-Phys. 102 -20 Feb 2006 -Ch 17 -page 46 Pole Speed = 242 m/s. In the two previous examples, What value of the source speed v. S should be used? What value of the detector speed v. D should be used? Solution v. S and v. D is measured relative to the air. This is the speed of the rocket relative to the air. v. S = 242 m/s – 20 m/s = 222 m/s. v. D = 242 m/s – 20 m/s = 222 m/s.