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CH 14: Sound and Hearing FBB102 What is sound? A disturbance of matter that is transmitted from its source outward FBB102 How does sound originate? A vibrating source oscillates and transfers energy to the surrounding molecules....

CH 14: Sound and Hearing FBB102 What is sound? A disturbance of matter that is transmitted from its source outward FBB102 How does sound originate? A vibrating source oscillates and transfers energy to the surrounding molecules. Sound in a vacuum? Some of the energy is used to compress / expand the surrounding medium. fsound = fsource high pressure regions (compressions) low pressure regions (rarefactions) move through the medium as a longitudinal wave ⇒ sound FBB102 Sound and hearing SOUND physical phenomenon (wave) HEARING perception of this phenomenon FBB102 Speed of sound Sound travels at speed vs vs = λ f Perception of frequency ƒ ? PITCH FBB102 Speed of sound Is the speed of sound always the same? NO! It depends on the following 2 factors : medium through which the sound propagates temperature of the medium FBB102 How the speed of sound depends on the medium vs ∝ rigidity of medium (B) (1/compressibility in gases) solids B v sound = ρ vs ∝ 1 / density of medium (ρ) B = bulk modulus Did you expect that? (measure of rigidity) k vmax = ωA = A m ⇓ stiffer springs ⇒ bigger vmax FBB102 How the speed of sound depends on T In air at sea level: −1 T (K ) vs = 331 ms TK = 273+TC 273K GASES SPEED (m/s) Air (0 °C) 331 LIQUIDS SPEED (m/s) SOLIDS SPEED (m/s) Air (20 °C) 343 Water (0 °C) 1402 Aluminium 6420 Helium 965 Water (20 °C) 1482 Steel 5941 Hydrogen 1285 Seawater 1522 Granite 6000 NB: vs ≈ independent of f How do we know this? Listening to a band ! What causers thunder? FBB102 Categories of sound waves Can humans hear all frequencies? NO! EXAMPLE EXAMPLE earthquakes dog whistles medical ultrasound FBB102 English language frequencies FBB102 Hearing ranges of various animals FBB102  Sound levels - loudness Loudness is related to how energetically the source is vibrating how far away you are (i.e. the area over which the energy is spread out) Loudness ∝ Intensity I Intensity I = power/area (W/m2) FBB102 Sound levels - loudness Loudness ∝ Intensity I = P/A (W/m2) FBB102 Sound levels - threshold of hearing Faintest sound at 1 kHz : Patmos ~ 105 Pa Threshold intensity I0 = 1x10-12 W/m2 Increase in pressure ∆P = 3 x 10-5 Pa molecular diameter ~ 10-10m Air molecule max displacement = 10-11m The ear is extremely sensitive : can detect pressure variations as small as 3 parts in 1010 ≡ air molecules moving only 10% of their diameter FBB102 Sound levels - threshold of hearing Faintest sound at 1 kHz : Threshold intensity I0 = 1x10-12 W/m2 What is the power delivered to your ear drum at this threshold intensity? SOLUTION Power delivered to your ear drum : Area A of ear drum ≈ 1 cm2 = 1x10-4 m2 ⇒ P = I0 A = 10-16 W FBB102 I = power/area (W/m2) Sound levels - threshold of pain Loudest sound at 1 kHz : Intensity I = 1 W/m2 Increase in pressure ∆P = 29 Pa Air molecule max displacement = 10-5 m FBB102 Sound levels – intensity vs loudness 12 I loudest = 1×10 I faintest Does it sound 1 000 000 000 000 times louder? NO! sensation of loudness is logarithmic in the human ear measured in decibels (db)  I  β = 10 log  symbol = β  I0  Loudness ≡ Intensity level ≡ Decibel level FBB102 Sound levels – decibels x10 Alexander deci–bels Graham Bell Threshold of hearing I0 = P/A = 10-12 W/m2 What is the sound level β at the threshold of hearing SOLUTION At the threshold of hearing : β = 10 log (I/I0) Sound level = 10 log (I0/I0) = 10 log 1 = 0 dB FBB102 Comparison of noise levels Intensity vs loudness  I  change in loudness of 10 dB : β = 10 log   I0  β ′ = β + 10 + 10 β ′ 10 I ′ = I 0 ⋅10 10 log (I/I0) = β = I 0 ⋅10( β +10 ) 10 log (I/I0)= β/10 = I 0 ⋅10 β 10+1 (I/I0) =10(β/10) = I 0 ⋅10 β 10 ⋅10 I = I0 ⋅10(β/10) = 10 ⋅ I x 10 intensity ↑ by factor of 10 FBB102 Intensity vs Loudness Change in loudness of 30 dB I = I0 ⋅10(β/10) β ′ = β + 30 + 10 + 10 + 10 β ′ 10 ⇒ I ′ = I 0 ⋅10 ( β + 30 ) 10 = I 0 ⋅10 β 10 + 3 = I 0 ⋅10 β 10 3 = I 0 ⋅10 ⋅10 x 10 x 10 x 10 = 1000 ⋅ I Intensity ↑ by factor of 1000 FBB102 Comparison of noise levels INTENSITY SOURCE DECIBELS (W/m2) Threshold 1 x 10-12 0 x10x10x10 +10+10+10 Whisper 1 x 10-9 30 Vacuum Cleaner 1 x 10-5 70 Noisy factory 1 x 10-2 100 (damage from 8 hrs per day) Rock concert 1 x 100 120 (damage in seconds) Bursting eardrums 1 x 104 160 FBB102 22 Sound levels - Example  Sound levels - Example A noisy grinding machine in a factory produces a sound intensity of 1.00 x 10-5 W/m2. a) Calculate the decibel level of this machine.  I  β = 10 log   I0  b) Calculate the new intensity level when a second identical machine is added to the factory. FBB102  Sound levels - Example c) A certain number of additional such machines are put into operation alongside these 2 machines. When all the machines are running at the same time, the decibel level is 77.0 dB. Find the sound intensity.  I  β = 10 log   I0  5 x intensity of 1 machine we now have 5 machines Intensity x 5 Sound level + 7 FBB102 Doppler Effect Change in the observed frequency of sound due to motion of source and/or observer.  Doppler Effect observed source speed of sound in (stationary) medium frequency frequency  v + vo  + : motion of observer towards source f o = f s   - : motion of observer away from source  v − vs  - : motion of source away from observer + : motion of source towards observer Assumptions : Applies to all waves, including light the air is stationary redshift of galaxies moving away all measurements are made relative to from us this stationary medium Applications : radar speed traps (Doppler shift in radar (microwaves)) measuring blood velocity (Doppler shift in ultrasound)  Doppler effect - EXAMPLE  v + vo  f o = f s     v − vs  A train moving at a speed of 40.0 m/s sounds its whistle, which has a frequency of 5.00 x 102 Hz. Determine the frequency heard by a stationary observer as the train approaches the observer. The ambient temperature is 24°C. Doppler Effect : SONIC BOOM object passing through the air creates a series of pressure waves in front of it and behind it, similar to the bow and stern waves created by a boat these waves travel at the speed of sound as the speed of the object increases, the waves are forced together, or compressed, because they cannot "get out of the way" of each other eventually they merge into a single shock wave at the speed of sound this critical speed is known as Mach 1 and is approximately 1,225 km/h (761 mph) at sea level at room temperature sound associated with the shock waves is called a sonic boom generates enormous amounts of sound energy, sounding much like an explosion 31 FBB102 Standing Waves FBB102 Standing Waves - Example FBB102 Standing Waves 1st overtone nv fn = Harmonic 2L series 2nd overtone Standing Waves Standing Waves tension v= mass per unit length nv v fn = f1 = 2L 2L FBB102  Standing Waves - Tube open at one end Max particle movement (antinode) Min particle movement (node) λ ∴ 4L=λ 37 FBB102 Standing Waves - Tube open at one end =3f1 =5f1 1st harmonic 3rd harmonic 5th harmonic Harmonics: f = nvw / 4L (n = 1, 3, 5 ….) f = vw / λ FBB102 Standing Waves - Tube open at one end 39 FBB102  Standing Waves - Tube open at both ends f = vw / λ 1st harmonic 2nd harmonic =2f1 3rd harmonic =3f1 Harmonics: fm = mvw / 2L (m = 1, 2, 3 ….) fn = nvw / 4L (n = 2,4,6 ….) fn = nvw / 4L (n = 1,3,5 ….) = 2vw / 4L, 4vw / 4L, 6vw / 4L, … = vw / 4L, 3vw / 4L, 5vw / 4L, … Open at both ends Open at one end FBB102 Standing Waves - Tube open at both ends 41 FBB102 Standing Waves - Music Sound quality ≡ timbre FBB102 Standing Waves - Example vv f = = λ 2L Why are instruments warmed up? tension How do bugles work? v= mass per unit length FBB102  Standing Waves – Worked Example GIVEN : A pipe is 2.46 m long. Determine the frequencies of the first three harmonics if the pipe is open at both ends. Take 343 m/s as the speed of sound in air. How many harmonic frequencies of this pipe lie in the audible range, from 20 Hz to 20 000 Hz? What are the three lowest possible frequencies if the pipe is closed at one end and open at the other? 44 FBB102  Standing Waves – Worked Example Suppose the smallest value of L, for which a With this measurement, determine peak occurs in the sound intensity, is 9.00 cm. the frequency of the tuning fork. OPEN AT ONE END Find the wavelength and the next two air-column lengths giving resonance. 45 FBB102  Standing Waves - Music Nose and throat = air column closed at one end – resonates to vibrations produced in voice box. Figure 16.18 Helium voice ? 46 FBB102 Beats Beats are formed by the combination of two waves of slightly different frequencies. f b = f 2 − f1 FBB102 Beats tension v= mass per unit length 48 FBB102  Beats - Example A certain piano string is supposed to vibrate at a frequency of 440 Hz. To check its frequency, a tuning fork known to vibrate at a frequency of 440 Hz is sounded at the same time the piano key is struck, and a beat frequency of 4 beats per second is heard. Find the two possible frequencies at which the string could be vibrating. FBB102  Beats - Example tuning fork frequency = 440 Hz Suppose the piano tuner runs toward the piano, holding the vibrating tuning fork while his assistant plays the note, which is at 436 Hz. At his maximum speed, the piano tuner notices the beat frequency drops from 4 Hz to 2 Hz (without going through a beat frequency of zero). How fast is he moving? Use a sound speed of 343 m/s. Frequency of piano note observed by piano tuner : While the piano tuner is running, what beat frequency is observed by the assistant? Frequency of tuning fork observed by assistant : FBB102