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SOUND WAVES waves_04 1 flute clarinet click for sounds.

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2 SOUND WAVES waves_04 1 flute clarinet click for sounds

3 waves_04: MINDMAP SUMMARY - SOUND WAVES 2 Sound waves, ultrasound, compressional (longitudinal) waves, pressure, particle displacement, medical imaging, superposition principle, stadning waves in air columns, constructive interference, destructive interference, boundary conditions, pipe – open and closed ends, nodes, antinodes, speed of sound in air, period, frequency, wavelength, propagation constant (wave number), angular frequency, normal modes of vibrations, natural frequencies of vibration, fundamental, harmonics, overtones, harmonic series, frequency spectrum, radian, phase, sinusoidal functions, wind musical instruments, beats, beat frequency, Doppler Effect, Doppler radar, shock waves

4 3 SOUND WAVES IN AIR Longitudinal wave through any medium which can be compressed: gas, liquid, solid Frequency range 20 Hz - 20 kHz (human hearing range) Ultrasound: f > 20 kHz Infrasound: f < 20 Hz Atoms/molecules are displaced in the direction of propagation about there equilibrium positions For medical imaging f = 1-10 MHz Why so large?

5 4 0 max  0 max  0 displacement min 0 max 0 min pressure Displacement of molecules Pressure variation SOUND WAVES IN AIR

6 5 Ultrasonic sound waves have frequencies greater than 20 kHz and, as the speed of sound is constant for given temperature and medium, they have shorter wavelength. Shorter wavelengths allow them to image smaller objects and ultrasonic waves are, therefore, used as a diagnostic tool and in certain treatments. Internal organs can be examined via the images produced by the reflection and absorption of ultrasonic waves. Use of ultrasonic waves is safer than x-rays but images show less details. Certain organs such as the liver and the spleen are invisible to x-rays but visible to ultrasonic waves. Physicians commonly use ultrasonic waves to observe fetuses. This technique presents far less risk than do x-rays, which deposit more energy in cells and can produce birth defects. ULTRASOUND

7 What is the physics of this image? CP 495 6

8 7 Flow of blood through the placenta ULTRASOUND

9 The speed of a sound wave in a fluid depends on the fluid’s compressibility and inertia.  Speed of sound wave in a fluid B : bulk modulus of the fluid  : equilibrium density of the fluid  Speed of sound wave in a solid rod Y : Young’s modulus of the rod  : density of the fluid  Speed of sound wave in air v = 343 m.s -1 at T = 20 o C Above formulae not examinable 8

10 The loudest tolerable sounds have intensities about 1.0x10 12 times greater than the faintest detectable sounds.  Intensity level in decibel Energy and Intensity of Sound waves The sensation of loudness is approximately logarithmic in the human ear. Because of that the relative intensity of a sound is called the intensity level or decibel level, defined by: I 0 = 1.0x W.m -2 : the reference intensity the sound intensity at the threshold of hearing Threshold of pain Threshold of hearing 9 Not examinable

11 Problem 1 A noisy grinding machine in a factory produces a sound intensity of 1.00x10 -5 W.m -2. (a) Calculate the intensity level of the single grinder. (b) If a second machine is added, then: (c) Find the intensity corresponding to an intensity level of 77.0 dB. 10 Not examinable

12 11 Problem 2 A point source of sound waves emits a disturbance with a power of 50 W into a surrounding homogeneous medium. Determine the intensity of the radiation at a distance of 10 m from the source. How much energy arrives on a little detector with an area of 1.0 cm 2 held perpendicular to the flow each second? Assume no losses. Solution P = 50 W r = 10 m I = ? W.m -2 A = 1.0 cm 2 = 1.0×10 -4 m 2 W = ? J I = P / (4  r 2 ) = 4.0×10 -2 W.m -2 W = I A = 4.0×10 -6 J

13 A small source emits sound waves with a power output of 80.0 W. (a) Find the intensity 3.00 m from the source. ( b) At what distance would the intensity be one-fourth as much as it is at r = 3.00 m? (c) Find the distance at which the sound level is 40.0 dB? Problem 3 12

14 Loud-soft-loud modulations of intensity are produced when waves of slightly different frequencies are superimposed. The beat frequency is equal to the difference frequency f beat = | f 1 - f 2 | 1 beat Used to tune musical instruments to same pitch BEATS CP 52 13

15 BEATS two interfering sound waves can make beat Two waves with different frequency create a beat because of interference between them. The beat frequency is the difference of the two frequencies. 14

16 Superimpose oscillations of equal amplitude, but different frequencies Modulation of amplitude Oscillation at the average frequency frequency of pulses is | f 1 -f 2 | BEATS CP

17 BEATS – interference in time Consider two sound sources producing audible sinusoidal waves at slightly different frequencies f 1 and f 2. What will a person hear? How can a piano tuner use beats in tuning a piano? If the two waves at first are in phase they will interfere constructively and a large amplitude resultant wave occurs which will give a loud sound. As time passes, the two waves become progressively out of phase until they interfere destructively and it will be very quite. The waves then gradually become in phase again and the pattern repeats itself. The resultant waveform shows rapid fluctuations but with an envelope that various slowly. The frequency of the rapid fluctuations is the average frequencies = The frequency of the slowly varying envelope = Since the envelope has two extreme values in a cycle, we hear a loud sound twice in one cycle since the ear is sensitive to the square of the wave amplitude. The beat frequency is CP

18 CP 527 f 1 = 100 Hz f 2 = 104 Hz f rapid = 102 Hz T rapid = 9.8 ms f beat = 4 Hz T beat = 0.25 s (loud pulsation every 0.25 s) 17 click for sound

19 CP 527 f 1 = 100 Hz f 2 = 110 Hz f rapid = 105 Hz T rapid = 9.5 ms f beat = 10 Hz T beat = 0.1 s (loud pulsation every 0.1 s) 18 click for sound

20 CP 527 f 1 = 100 Hz f 2 = 120 Hz f rapid = 110 Hz T rapid = 9.1 ms f beat = 20 Hz T beat = 0.05 s (loud pulsation every 0.05 s) click for sound

21 DOPPLER EFFECT - motion related frequency changes Doppler 1842, Buys Ballot trumpeters on railway carriage Source (s) Observer (o) Applications: police microwave speed units, speed of a tennis ball, speed of blood flowing through an artery, heart beat of a developing fetous, burglar alarms, sonar – ships & submarines to detect submerged objects, detecting distance planets, observing the motion of oscillating stars, weather (Doppler radar), plaet detection CP 495 note: formula is very different to textbook formula different to textbook 20


23 DOPPLER EFFECT Consider source of sound at frequency f s, moving speed v s, observer at rest (v o = 0) Speed of sound v What is frequency f o heard by observer? On right - source approaching source catching up on waves wavelength reduced frequency increased On left - source receding source moving away from waves wavelength increased frequency reduced CP 495 v = f 22

24 CP Source frequency f s = 1000 Hz click for sounds f o > 1000 Hz f o < 1000 Hz

25 source v s observer v o observed frequency f o stationary = f s stationaryreceding< f s stationaryapproaching> f s recedingstationary< f s approachingstationary> f s receding < f s approaching > f s approachingreceding? approaching? CP

26 SHOCK Waves – supersonic waves CP bullet travelling at Mach 2.45

27 CP 506 Shock Waves – supersonic waves 26

28 27 DOPPLER EFFECT Stationary Sound Source Source moving with v source < v sound ( Mach 0.7 ) Source moving with v source = v sound ( Mach 1 - breaking the sound barrier ) Source moving with v source > v sound (Mach supersonic)

29 Problem 4 A train whistle is blown by the driver who hears the sound at 650 Hz. If the train is heading towards a station at 20.0 m.s -1, what will the whistle sound like to a waiting commuter? Take the speed of sound to be 340 m.s Solution f s = 650 Hz v s = 20 m.s -1 v o = 0 m.s -1 v = 340 m.s -1 f o = ? Hz (must be higher since train approaching observer).

30 Problem 5 The speed of blood in the aorta is normally about m.s -1. What beat frequency would you expect if MHz ultrasound waves were directed along the blood flow and reflected from the end of red blood cells? Assume that the sound waves travel through the blood with a velocity of 1540 m.s

31 Solution 5 Doppler Effect Beats 30

32 Blood is moving away from source  observer moving away from source  f o < f s Wave reflected off red blood cells  source moving away from observer  f o < f s Beat frequency = | 4.00 – |  10 6 Hz = 1558 Hz In this type of calculation you must keep extra significant figures. 31

33 An ambulance travels down a highway at a speed of 33.5 m.s -1, its siren emitting sound at a frequency of 4.00x10 2 Hz. What frequency is heard by a passenger in a car traveling at 24.6 m.s -1 in the opposite direction as the car and ambulance: (a) approach each other and (b) pass and move away from each others? Speed of sound in air is 345 m.s -1. (a) (b) Problem 6 Solution 6 32

34 Problem 7 An ultrasonic wave at  10 4 Hz is emitted into a vein where the speed of sound is about 1.5 km.s -1. The wave reflects off the red blood cells moving towards the stationary receiver. If the frequency of the returning signal is  10 4 Hz, what is the speed of the blood flow? What would be the beat frequency detected and the beat period? Draw a diagram showing the beat pattern and indicate the beat period

35 34 Solution f s = 8.000×10 4 Hz f o = 8.002×10 4 Hz v = 1.5×10 3 m.s -1 v b = ? m.s -1 Need to consider two Doppler shifts in frequency – blood cells act as observer and than as source. Red blood cells (observer) moving toward source Red blood cells (source) moving toward observer f beat = |f 2 -f 1 | = ( )×10 4 Hz = 20 Hz T beat = 1/f beat = 0.05 s t beat

36 35 If we try to produce a traveling harmonic wave in a pipe, repeated reflections from an end produces a wave traveling in the opposite direction - with subsequent reflections we have waves travelling in both directions The result is the superposition (sum) of two waves traveling in opposite directions The superposition of two waves of the same amplitude travelling in opposite directions is called a standing wave STANDING WAVES IN AIR COLUMNS (PIPES)

37 36 L Closed end: displacement zero (node), pressure max (antinode) Open end: displacement max (antinode), pressure zero (node) CP 516 Closed at both ends Closed at one end open at the other Open at both ends Flute & clarinet same length, why can a much lower note be played on a clarinet? STANDING WAVES IN AIR COLUMNS (PIPES)

38 37 Organ pipes are open at both ends

39 Sound wave in a pipe with one closed and one open end (stopped pipe) 38 STANDING WAVES IN AIR COLUMNS (PIPES)


41 40 Search google or YouTube for Rubens or Rubins tube


43 Normal modes in a pipe with an open and a closed end (stopped pipe) 42 STANDING WAVES IN AIR COLUMNS (PIPES) Fundamental 3 rd harmonic or 2 nd overtone

44 43 CP 523

45 44 Musical instruments – wind An air stream produced by mouth by blowing the instruments interacts with the air in the pipe to maintain a steady oscillation. All brass instruments are closed at one end by the mouth of the player. Flute and piccolo – open at atmosphere and mouth piece (embouchure) – covering holes L     f  Trumpet – open at atmosphere and closed at mouth – covering holes adds loops of tubing into air stream L     f  Woodwinds – vibrating reed used to produce oscillation of the air molecules in the pipe. CP 516

46 45 Woodwind instruments are not necessarily made of wood eg saxophone, but they do require wind to make a sound. They basically consist of a tube with a series of holes. Air is blow into the top of the tube, either across a hole or past a flexible reed. This makes the air inside the tube vibrate and give out a note. The pitch of the note depends upon the length of the tube. A shorter tube produces a higher note, and so holes are covered. Blowing harder makes a louder sound. To produce deep notes woodwind instruments have to be quite long and therefore the tube is curved. Brass instruments (usually made of brass) consist of a long pipe that is usually coiled and has no holes. The player blows into a mouthpiece at one end of the pipe, the vibration of the lips setting the air column vibrating throughout the pipe. The trombone has a section of pipe called a slide that can be moved in and out. To produce a lower note the slide is moved out. The trumpet has three pistons that are pushed down to open extra sections of tubing. Up to six different notes are obtained by using combinations of the three pistons. CP 516

47 46 Boundary conditions Reflection of sound wave at ends of air column: Open end – a compression is reflected as a rarefaction and a rarefaction as a compression (  phase shift). Zero phase change at closed end. Natural frequencies of vibration (open – closed air column) Speed of sound in air (at room temperature v ~ 344 m.s -1 ) v = f odd harmonics exit: f 1, f 3, f 5, f 7, … CP 516

48 47 Problem 8 A narrow glass tube 0.50 m long and sealed at its bottom end is held vertically just below a loudspeaker that is connected to an audio oscillator and amplifier. A tone with a gradually increasing frequency is fed into the tube, and a loud resonance is first observed at 170 Hz. What is the speed of sound in the room?

49 48 Solution 8 Pressure distribution in tube for fundamental mode pressure node pressure antinode f = 170 Hz L = 0.50 m v = ? m.s -1 speed of wave N A

50 49 Problem 9 What are the natural frequencies of vibration for a human ear? Why do sounds ~ (3000 – 4000) Hz appear loudest? Solution Assume the ear acts as pipe open at the atmosphere and closed at the eardrum. The length of the auditory canal is about 25 mm. Take the speed of sound in air as 340 m.s -1. L = 25 mm = m v = 340 m.s -1 For air column closed at one end and open at the other L = 1 / 4  1 = 4 L  f 1 = v / 1 = (340)/{(4)(0.025)} = 3400 Hz When the ear is excited at a natural frequency of vibration  large amplitude oscillations (resonance)  sounds will appear loudest ~ (3000 – 4000) Hz.

51 RESONANCE When we apply a periodically varying force to a system that can oscillate, the system is forced to oscillate with a frequency equal to the frequency of the applied force (driving frequency): forced oscillation. When the applied frequency is close to a characteristic frequency of the system, a phenomenon called resonance occurs. Resonance also occurs when a periodically varying force is applied to a system with normal modes. When the frequency of the applied force is close to one of normal modes of the system, resonance occurs. 50

52 51 Why does a clarinet play a lower note than a flute when both instruments are about the same length ? A flute is an open-open tube. A clarinet is open at one end and closed at the other end by the player’s lips and reed. open closed Problem 10

53 52 Solution 10 FLUTE is open at both ends  particle displacement antinodes at the open ends A A Fundamental (1 st harmonic) 2 nd harmonic (1 st overone) A All harmonics can be excited Particle displacement variations

54 53 Solution 10 CLARINET is open at one and closed at the other  pressure node at open end & antinode at closed end A N Fundamental (1 st harmonic) 3 nd harmonic (1 st overone) All odd harmonics can be excited pressure variations A N N

55 The sound waves generated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of L for which a peak occurs in the sound intensity is 90.0 mm. Problem 11 Resonance L smalles t = 9.00 cm (a)Find the frequency of the tuning fork. (b) Find the wavelength and the next two water levels giving resonance. 54

56 55 Solution 11 (alternative) 1 st resonance L 1 = 90.0 mm = 90.0×10 -3 m v = 343 m.s -1 f 1 = ? Hz = 4 L = m v = f f 1 = v / = (343)/(0.36) Hz = 958 Hz 2 nd and 3 rd resonances L 3 = 3 /4 = (3)(0.36)/(4) m = m L 5 = 5 /4 = (5)(0.36)/(4) m = m Tube closed at one end and open at the other  odd harmonics can be excited pressure variation The frequency of the tuning fork is fixed and hence the wavelength also has a fixed value. The length of the tube is variable. AN L 1 = /4 L 3 = 3 /4 L 5 = 5 /4 N A NN

57 56 Why does a chimney moan ? Chimney acts like an organ pipe open at both ends Pressure node N N A Fundamental mode of vibration Speed of sound in air v = 340 m.s -1 Length of chimney L = 3.00 m L = / 2 = 2 L v = f f = v / = 340 / {(2)(3)} Hz f = 56 Hz low moan Problem 12

58 Natural frequencies of a trumpet closed at mouth and open at the flared end Fundamental f 1 = 60 Hz 1 st overtone 163 Hz 2 nd overtone 247 Hz NOT a harmonic sequence for the natural frequencies of vibration

59 58 Simulation of the human voice tract – natural frequencies

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