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Lecture 1 Sound Hearing Sound Intensity Sound Level Assistant Prof. Matthias Möbius [email protected] Sound Waves Gas, liquid or solid is mechanically disturbed • Sound waves are produced Speed of sound in a substance depends on •physical properties •e.g. (density, temperature) When sound encounters a boundary between substances, some sound energy is transmitted and some reflected Reflection makes ultrasound imaging possible Sound Sound Waves (Longitudinal waves) A plucked string will vibrate at its natural frequency and alternately compresses and rarefies the air alongside it. direction compression Density of Air rarefaction Compressed air >>> increased pressure Rarefied air >>> reduced pressure organised vibrations of air molecules>> sound Sound Sound waves-(variation in air pressure) can cause objects to oscillate Example: ear drum is forced to vibrate in response to the air pressure variation Depending on: intensity of the sound frequency of vibration movement of the ear drum will stimulate nerve cells and the sound will be perceived. Sound Waves Speed of sound (v) in materials Depends on •Phase of the material •Characteristics of the material (elasticity, density & temperature) In general vsolids vliquids vgases •Greater in solids because molecules interact more strongly with each other •Greater in rigid materials Material Speed (ms-1) Air 344 Helium 965 Water 1450 Blood 1570 Body Tissue 1570 Copper 3750 Iron 5000 Glass 5000 Helium has a lower density than air. Resonant frequencies of vocal cavity increase. Spectral distribution of sounds shift to higher frequencies -timbre of sound changes Sound Waves Speed of sound (v) Depends on elasticity and density Solid bar v Liquid v Gas v kT m E B E Young’s Modulus density B bulk modulus cp cv Cp specific heat constant pressure Cv specific heat constant volume m molecular mass k Boltzmann’s constant T temperature (Kelvin) Calculate the speed of sound in air at 20 oC =1.4. Boltzmann’s constant =1.38x10-23J/K Avg. mass of “air molecule” = 47.97x10-27kg V kT m 23 1.4(1.38 10 J / K )[(20 273.15) K ] V 47.97 1027 kg V 343.6ms 1 Sound Speed of sound The speed of sound in water is 4.2 times the speed of sound in air. A whistle on land produces a sound wave with frequency f0. When this sound wave enters water, its frequency is: a) 4.2f0 b) f0 c) f0/4.2 d) Not enough information given • Frequency (f) of a wave is independent of the medium through which the wave travels. –It is determined by the frequency of the oscillator that is the source of the waves. Sound Diffraction Longer the wavelength compared to size of opening or object the greater the diffraction Light waves: •Wavelengths « dimensions of everyday objects •Little diffraction occurs •Relatively sharp shadows occur v 3 108 ms 1 500nm 14 f 6 10 Hz Sound waves: • Wavelength ≥ size of everyday objects v 344ms 1 •diffraction occurs 34.4cm f Example Sound source 1KHz Hearing Hearing Sound wave enters the ear. Forces exerted on eardrum due to air pressure variations cause it to vibrate. three small bones (hammer, anvil, and stirrup) in the middle ear amplify & transmit forces to fluid filled inner ear through the oval window (very small area compared with eardrum) result pressure x 30 Other amplification characteristics ?? The motion of the fluid disturbs hair cells within the Cochlea, which transmit nerve impulses to the brain corresponding to the sound heard. Outer ear hammer Middle anvil Inner ear Oval window Cochlea sound Ear canal stirrup ear drum Ear can detect very low intensity sounds Hearing All waves carry energy Audible sound waves carries very little energy Ear can detect extremely low intensity sounds Power output: Talk ≈10-5 W Talk 24 hours a day non stop for 114 years ≈106 hours Total energy output is ≈10-5 w x106 hrs =10 Wh Equivalent to quantity of energy consumed by a 100W bulb in 6 minutes Sound Waves Intensity Waves (energy) spread out from source Intensity (I) of a wave is defined as •Energy (E) carried per unit time per unit area (A) E /t I A therefore Power (P) E P t P I A Unit of intensity Watt per square metre (Wm-2) Sunlight intensity at Earth ≈103 Wm-2 Hearing Intensity Human ear can detect extremely low intensities ≈10-12 Wm-2 Maximum intensity without ear damage ≈1 Wm-2 Large range 1012 logarithmic units useful Human perception If we listen to two sounds (I1 and I2) and I2 seems twice as loud as I1 Measure intensities I2 is approximately 6 to 10 times I1 Convenient scale to measure loudness is the logarithm of the intensity Hearing Perceived loudness is roughly Logarithmic Ear response to sound • logarithmic • not linear Decibel scale for intensity Sound (Intensity) level in decibels (b) I b 10 log10 I0 12 2 I 10 Wm where (threshold of hearing 0 at 1000Hz) decibel (b) is a relative sound level measurement Threshold of discomfort = 1 Wm-2 Above this, pain is experienced, and there is potential for long term damage Logarithm Logarithm is the inverse of exponentiation: 10x =120 log10 (10x) = log10 (120) x log10 (10) = log10 (120) x=log10 (120) Note that logarithms can have different bases. The most common ones are: log10, log2, ln (natural logarithm with base e) log(a b) = log(a) + log(b) log(a/b) = log(a) - log(b) log(ab) = b log(a) log(1) = 0 for all bases Convert between different bases: logx(A) = logy(a) / logy(x) Hearing Sensitivity of ear Can detect sound intensity of ≈10-12Wm-2 Corresponds to pressure variation of ≈ 3x10-5 Pa (Atm. Pressure ≈ 101,325 Pa) Random fluctuation due to thermal motion of molecules ≈ 5x10-6 Pa Sensitivity: essentially due to mechanical layout •Area ratio: ear drum to oval window ≈ 30 •hammer, anvil and stirrup amplification ≈2 •canal resonance at 3kHz pressure increase ≈2 •Total pressure amplification ≈ 30x2x2 = 120 Intensity ( pressure)2 Intensity increases by factor of 1202=14,400 Brain: discriminatory role Filters unwanted noise Suppression: non-awareness of background noise ear is not equally sensitive at all frequencies Sound levels and Intensities Sound level (dB) Intensity (Wm-2) Sounds Vibration amplitude. air molecules 0 1x10-12 Threshold of hearing 1.1x10-11m 10 1x10-11 20 1x10-10 30 1x10-9 Quiet room 40 1x10-8 computer 50 1x10-7 60 1x10-6 Normal conversation 70 1x10-5 Busy traffic 80 1x10-4 Loud radio 90 1x10-3 100 1x10-2 110 1x10-1 120 1 Rock concert, Threshold of pain 140 1x102 Jet airplane at 30m 160 1x104 Bursting eardrums 1mm Computer 10 times louder than quiet room Does not seem so because of the logarithmic response of the ear Sound levels and Intensities Danger Hearing loss Damage Threshold 5 hours/week at > 89dB damage after 5 years > 100dB deemed hazardous 10 minutes at 120dB Temporarily changes your threshold of hearing from 0dB to 30dB Sound Waves (a) Calculate the sound level in dB of a sound intensity 10-8Wm-2 (b) Calculate the intensity in Wm-2 of a sound level of 80 dB I b 10 log10 (a) I0 108Wm2 b 10log10 12 2 10 Wm b 10log10 104 10 4 40d b (b) I 80 10 log10 I0 I 8 10 I0 I 8 log10 I0 I 108 1012Wm2 104Wm2 I 104Wm2 Hearing Hearing ability Loudness is a method of describing the acoustic pressure (or the intensity) of a given sound Intensity hearing range: 10-12Wm-2 →1Wm-2 Ability to hear is not only a function of intensity but also frequency Humans: Frequency range: 20 Hz → 20 kHz Infrasonic < 20 Hz Elephants: down to 1Hz Pigeons: down to 0.1 Hz 20 kHz < ultrasonic Dogs: up to 40 kHz Dolphins: up to 250 kHz. Bats: up to 120 kHz Hearing Human Hearing Ability Sound Intensity Level dB W/m2 100 120 10-2 100 10-4 80 10-6 60 10-8 40 10-10 20 10-12 0 20 Pain threshold Hearing threshold 100 1k 10k 20k Hz frequency Hearing ability as a function of intensity and frequency. The blue solid line is the pure tone threshold curve, below which the subject does not hear. Ear most sensitive at 3000 Hz Pain threshold almost frequency independent Hearing Why two ears Main advantage Sounds from different directions arrive at each of our ears at slightly different times and with slightly different intensities. Time difference of sound arriving at both ears used to locate the source of the sound Example: crossing a road direction of the car approximately how close it is Other advantages •easier to understand speech in noisy background • help judge loudness Sound intensity Distance Sound intensity is reduced by moving away from source By how much? Inverse Square Law Consider imaginary spheres Isotropic source power Intensity area P I1 4 r12 r1 r2 P I2 4 r22 I1 r22 2 I 2 r1 As the person gets further away, the sphere that intersects with them gets larger and larger Fraction = Area of person 4 π r12 Fraction = Area of person 4 π r22 Sound intensity Variation of Sound Intensity with distance from a point source Inverse square law Intensity I1 at a distance r1 from source Intensity I2 at a distance r2 from source 2 2 2 1 I1 r I2 r Intensity is inversely proportional to the square of the distance from the source. NOTE: Sound level (dB) is not inversely proportional to distance squared ! Sound intensity Examples The intensity falls off as 1/r2 (where r is the distance) so moving 4 times as far away will decrease the exposure by a factor of 16. A person near a source of loud noise wants to decrease their exposure to it by a factor of 10. How far away do they have to move? 2 2 2 1 I1 r I2 r r22 10 2 r1 I1 r22 2 I1 r1 10 r2 10 3.16 r1 They have to move 3.16 times further away Waves Example A bat can detect sound frequencies up to 120,000 Hz. What is the wavelength of sound in the air at this frequency? v f v f v 344ms 1 2.87 103 metres f 120, 000 Hz =.287cms High frequency—short wavelength Wave only disturbed by objects with dimensions similar to or greater than the wavelength Smaller objects have little effect Bats use ultrasound for navigation Can distinguish between insect and falling leaf Waves Resonance Most objects have a natural frequency: Determined by • size • shape •composition 1 l Simple pendulum T 2 Only one natural frequency f g Resonance occurs if frequency of the driving force equals natural frequency of the system Example: child being pushed on a swing. Swing is kept in motion at its natural frequency by a series of appropriately timed pushes. Difficult to get it to swing at any other frequency If an object is subjected to an intense wave oscillating at object’s natural frequency a large response (Resonance) occurs Waves Resonance: examples Opera singers with powerful voices can set glasses into audible vibration If frequency of note is the same as the natural frequency of the glass, the glass may vibrate with a large amplitude and may break Roman foot soldiers were instructed to break step when marching over a bridge •Prevented possible resonance response and bridge damage Air passages of the mouth, larynx Nasal cavity together form an acoustic resonator. Voiced sound depend on •resonant frequencies of the total system ------depends on system’s volume and shape Resonance: examples Half-closed pipe Resonance (e.g. ear canal): f Re sonance sound / Fundamental mode: f1 sound /( 4L) Electrical Resonance: Example: Tuning in radio station Adjust resonant frequency of the electrical circuit to the broadcast frequency of the radio station To “pick up” signal Sound Waves Travel distance is a function of frequency Traveling waves transfer energy from one place to another Sound energy dissipates to thermal energy when sound travels in air. Higher frequency sounds dissipate more quickly, because more energy transferred to the medium; so lower frequency sounds travel further. Examples • foghorns have a low frequency •Elephants communicate over long distances (up to 4 km), frequencies as low as 14 Hz Lecture 2 Sound Beats Doppler Effect Ultrasound Applications Waves Superposition Simple case: Addition of two waves with same frequency and amplitude Wave 1 Wave 2 resultant Beats If the two waves interfering have slightly different frequencies (wavelengths), beats occur. In step (in phase) In step (in phase) Out of step (out of phase) Waves Beats If the two waves interfering have slightly different frequencies (wavelengths), beats occur. Wave 1 Wave 2 Resultant envelope Waves get in and out of step as time progresses Result• constructive and destructive interference occurs alternately •Amplitude changes periodically at the beat frequency Beat frequency fb = f1-f2 Absolute value: beat frequency always positive Waves Beats fb = f1-f2 If frequency difference = zero No beats occur Wave 1 Wave 2 resultant Waves Beats Beats can happen with any type of waves Sound waves Beats perceived as a modulated sound: loudness varies periodically at the beat frequency Application Accurate determination of frequency Example Piano tuning Adjust tension in wire and listen for beats between it and a tuning fork of known frequency The two frequencies are equal when the beats cease. Easier to determine than when listening to individual sounds of nearly equal frequencies f1 = 264Hz f2 = 266 Hz Beat frequency 2Hz Sound Waves Doppler Effect Change in perceived frequency depending on the relative motion of the source and listener. Occurs with all types of waves – most notable •sound waves, •light waves. Christian Doppler 1803-1853 Austrian Physicist, Mathematician Example: Perceived pitch (or frequency) of a moving source such as a fire engine siren changes as it goes past Frequency of sound emitted does not change Longer Lower f stationary moving→ Shorter higher f Waves Doppler effect is observed because the distance between the source of sound and the observer is changing. source always emits the same frequency. Source moving towards the observer •sound waves reaching observer perceived to be at a more frequent rate (higher frequency) sound waves compressed into shorter distance Source moving away from the observer, •sound waves reaching observer perceived to be at a less frequent rate (lower frequency) Sound waves expanded into longer distance Waves Observed frequency for a moving source f observer vwave f source vwave vsource + sign: source moving away from observer - sign: source moving towards observer Stationary source, moving observer f observer vwave vobserver f source vwave +sign: observer moving towards source - sign: observer moving away from source f = Frequency v = Speed Waves Example moving observer A stationary siren has a frequency of 1000 Hz. What frequency will be heard by drivers of cars moving at 15ms1? a) away from the siren? b) toward the siren? f observer (a) vwave vobserver f source vwave vw vo fo fs vw 344ms 1 15ms 1 fo 1000 Hz 956 Hz 1 344ms (b) vw vo fo fs vw 344ms 1 15ms 1 fo 1000 Hz 1044 Hz 1 344ms Example: Moving Source A Garda car with a 1000 Hz siren is moving at 20 ms-1. What frequency is heard by a stationary listener when the police car is: a) Moving away from b) approaching the listener If you were to replace the Garda car with 2 stationary sirens emitting at the two frequencies as perceived in (a) and (b), what would be the beat frequency between them? (a) f observer (b) f observer f observer vwave f source vwave vsource 344ms 1 1000 Hz 945Hz 1 1 344ms 20ms f observer vwave f source vwave vsource 344ms 1 1000 Hz 1062 Hz 1 1 344ms 20ms Beat frequency fbeat f a fb fbeat 945Hz 1062Hz fbeat 117 Hz Waves Doppler effect can be used to measure speed of the source Radar: RAdio Detecting And Ranging Police radar uses radio waves: measures Doppler shift to determine speed of car •compares frequency of reflected wave from car with that emitted from radar Doppler RADAR •Weather •Rainstorms, tornadoes •Wind sheer at airports Swirling air & water droplets RADAR Wave source Sound Waves Reflection of waves (echoes) •Caused by solid object •Change in nature of medium Sound waves applications SONAR (sound navigation and ranging) - Underwater navigation and observation • Measuring the travel time of sound waves in the ocean can help monitor sea temperatures and global changes Ultrasound Frequency greater than range of human hearing Sound with frequencies above 20 kHz Normally 1 →20MHz Applications •Navigation •Diagnostics •Surgery •Therapeutic •Cleaning Ultrasound Bats can determine distance, speed and direction of their prey (using reflection time and Doppler effect) Typical prey: moths (dimensions cms) Bats use ultrasonic echolocation methods to detect their presence. why do bats use ultrasound? v 344ms 1 Audible 344 103 m 34.4cm f 1kHz v 344ms 1 Ultrasound 6.88 103 m 0.7cm f 50kHz Ultrasound- Shorter wavelength •Reflection, not diffraction occurs at moth. Submarines, dolphins and bats use ultrasound for navigation 30-100kHz Ultrasound Medical applications Ultrasound Imaging Ultrasound probe passed over region of interest Reflections of ultrasound pulses from patients occur at interfaces between different tissues of different density Good contrast: reflection from boundaries between materials of nearly the same density Reflection time provides depth information Image constructed from echo and position information Ultrasound Medical applications Ultrasound & Doppler effect can be used to measure • Blood flow speed in arteries and veins, measure arterial occlusion •Echocardiogram , examination of the heart • measure blood flow in and out •fetal heart beats • pulsation of artery walls Stroke: early warning Monitor blood speed in carotid artery in neck Ultrasonic Doppler flow meter Transmitter Receiver Red blood cell Ultrasound Surgery Ultrasonic scalpel (55kHz) Precise cutting and coagulation •Tumour removal •Tonsillectomies Medical ultrasound without harmful effects •intensity kept low (≈10-2 Wm-2) to avoid tissue damage Ultrasound scanning during pregnancy Ultrasound Imaging Why use ultrasound---not audible sound Smallest detail observable ≈ one wavelength Audible sound wavelength in tissue v 1570ms 1 0.5m f 3000 Hz Ultrasound wavelength in tissue 1570ms 1 1cm 150kHz In tissue, higher frequencies are attenuated more Compromise between spatial resolution of image and penetration depth 1MHz: penetration depth ≈ 6cm 3MHz: superficial conditions (eg. Tennis elbow etc) •Frequency is selected based on the depth of the tissue to be treated. Example: deep heat therapy (low frequency) Ultrasound Example Ultrasound speed =1500m/s in tissue. Using an ultrasound frequency of 2MHz, calculate (a) smallest detail visible (b) time for reflected wave to return to probe from a depth of 5cm v f (a) v 1500m / s 6 f 2 10 Hz l = 0.75mm (b) time for reflected wave to return to probe s 2 0.05m 5 t 6.6 10 sec 1 v 1500ms Ultrasound Other uses in medicine •Destructive effects •Intense ultrasound produces large density and pressure changes • Results − Large stresses −Heat is produced in most materials − microscopic vapour bubbles formed and implode releasing energy (cavitation) Non-invasive removal of kidney stones Dental applications ultrasonic scalar Consists of a ultrasound probe with a small tip. The ultrasound in combination with water flow effective in plaque and tartar removal Ultrasound Component surface cleaning Component placed in fluid in ultrasonic bath Ultrasound creates a periodic compression and expansion in the fluid. Results in Acoustic cavitation Bubbles formed, grow, and implosively collapse localised heating (>1000K) and high pressures (>100 atmospheres) Result: effective surface cleaning Auto-focus cameras computes time taken (and hence distance of subject) for the reflected ultrasonic sound wave to reach the camera lens position and then sets focus accordingly. Sound Supersonic speed Moving source, approaching listener When speed of source approaches the speed of sound, waves ahead of source come close together. f observer vwave f source vwave vsource f observer approaches infinity Nearly infinite number of wave crests reach observer in very short time At supersonic speeds the waves overlap and there are many points of constructive interference, shock wave results Wave front produced when vsource vwave is known as a shock wave sonic boom Sound Supersonic speed vs 0 Circles represent wave fronts emitted from sound source Stationary source v = 0 Speed of sound in air =vs v vs Waves ahead of source come closer together subsonic v vs Waves pile up at front Mach 1 v vs supersonic Waves overlap: Shock wave, Sonic boom. Sound Supersonic speed Circles represent wave fronts emitted from sound source vs t In time interval t vt Sound wave travels a distance vst Source travels distance vt Tangent lines lie on surface of cone vs t vt v Ratio is called Mach number M vs object speed v 1 M M speed of sound vs sin sin since sin 1 No shock unless M 1 v vs vwwt Bow sin vt Waves Speed of boat v > Water wave speed Vww Question What is the speed of ultrasound with a wavelength of 0.25 mm and a frequency of 6 MHz? How does this compare with the speed of sound in air? v f v 6 106 Hz 0.25 103 m 1.5 103 ms 1 Compare with speed of sound in air 1.5 103 ms 1 4.4 1 344ms Question Lightening strikes 10 km away. (a) How long after the strike will you see the light? (b) How long after the strike will you hear the sound? (a) c = 3*108 m/s, s = 10 km, t = ? s = vt t = s/v t = (10,000 m)/(3*108 m/s) = 3.3*10-5 s (b) v = 344 m/s, s = 10 km, t = ? s = vt t = s/v = (10,000 m)/(344 m/s) = 29 s If you hear the sound 3 seconds after you see the lightening how far away is the strike? s = vt =(344 m/s)(3 s) = 1002 m Question (a) What is the sound level in decibels of a sound with an intensity of 0.0200W/m2? (b) If you had 3 such sounds what would the sound level be? 2 102Wm2 b 10log10 12 2 10 Wm (a) I b 10 log10 I0 b 10log10 2 1010 b 10 log10 2 10 log10 10 b 10 0.3 10 (b) b 103dB 3 2 102Wm2 b 10log10 12 2 10 Wm b 10log10 6 1010 b 10 log10 6 10 log10 10 b 10 0.78 10 b 107.8dB Not equal to 3x103 dB ! Sound levels are logarithms of intensity