Waves and Sound

1. Waves and Sound Chapter 11

2. Ways to Transport Image right: Recent Cassini images of Saturn's moon Enceladus backlit by the sun show the fountain-like sources of the fine spray of material that towers over the south polar region. Image credit: NASA/JPL/Space Science Institute+ Full image and caption+ Movie: Enceladus plumes+ Browse version of image • Two ways to transport energy and momentum • Streaming particles • Flowing waves

3. Sound • Two ways to study • Psychological (mind) and physiological (body) • What we hear • Physical • What sound is – compression wave

4. Waves • Moving self-sustained disturbance of a medium • Medium • Field • Substance • Mechanical wave – in material media

5. Wave Characteristics • Atoms • Push together – repel • Pull apart – attract • Objects are made of atoms • When atoms are distorted they act like attached by springs • Displacement causes a wave

6. Progressive or Traveling wave • Self-sustaining disturbance • Examples • String • Liquid waves • Sound waves • Compression waves • The main difference between particle stream and wave is: • Medium stays in place as the wave progresses

7. Wave Forms • Longitudinal • Sustaining medium is displaced parallel to the direction of propagation • Ex – Sound waves • Transverse • When the sustaining medium is displaced perpendicular to the direction of propagation • Ex – Guitar string • Torsion • Variation of transverse waves • Water waves • Combination of Longitudinal and Transverse waves

9. Types of Waves • Longitudinal – move back and forth • Transverse – move up and down • Water – move in circle

10. WavePulse • One cycle of a wave • Profile – outline or shape of the wavepulse • Determined by the driver of the wave • Speed – Determined by the medium • Examples • Gunshot • Grunt • Tsunami

11. WaveTrain • Disturbance of waves with a beginning and end • Amplitude varies • Carrier wavelength – Steady sinusoidal oscillation

12. Periodic • Ideal disturbance composed of endless repeats of the same profile wave

13. Labeling a Wave • Period – how long it takes one profile to pass a point in space • Frequency – number of profile waves passing per second • Wavelength – λ (lambda) - distance in space over which the wavetrain executes one cycle • Amplitude – Height of the waves

14. Velocity of Wave • v = fλ • V – velocity (m/s) • f – frequency (cycles/sec or Hz) • λ – wavelength (m)

15. Problem • Waves pass the length of a 4.5 m boat. It takes 1.5 seconds for the wave to go from end to end. If the waves are 0.5 seconds apart, what is the period, frequency and wavelength? • T = 0.5 seconds • f = 1/T = 1/0.5 = 2.0 Hz • v = L/t = 4.5 m/1.5 sec = 3.0 m/s • λ = v/f = 3.0 m/s / 2.0 Hz = 1.5 m

16. Transverse Waves : Strings • Speed of the waves is determined by the properties of the medium, not in any way the motion of the source • Velocity of wave in string • v = √FT/m/L • v - velocity (m/s) • FT = Tension (N) • m /L – mass/unit length

17. Problem • What is the speed of a wave pulse in a 20 cm, 40 g guitar string with the tension of 19.6 N? • v = √FT/m/L • v - ? • FT = 19.6 N • m /L – .040 kg / 0.20 m = 0.020 kg/m • v = √FT/m/L =√19.6 N / 0.020 kg/m = 31 m/s

18. Reflection, Absorption, Transmission • Reflected – carries all the original energy • Absorbed – Friction stops wave • Transmission – moving from one media to another • Velocity may change when moving between medias

19. Compression Waves • Solids – longitudinal elastic wave • Ex – Earth quake • Fluids – acoustic waves • Ex – sound waves • Parts • Rarefaction – distance between atoms is elongated • Compression – distance between atoms is squeezed • Direction of movement – in the direction of oscillation • Each atom is in SHM

20. Speed of Waves in Media • Can be determined by the restoring force and its density • Use • Bulk Modulus • Bernoulli’s equation • Young’s Modulus

21. Ultrasound • Dolphins use chirps to locate items underwater • Size of wave – 1.4 cm • Can “see” fish and other small items • Above our hearing range - 10⁵ Hz

22. Other uses of Ultrasound • Autofocus cameras • Bats • Medicine • Tumor and Kidney stone destruction • Probe body • Joints • Baby

23. Infrasound • Wave lengths below our hearing range (less than 20 Hz) • Examples • Elephants • Submarines • Subwoofers in Rock Bands • Vibrate our internal organs • http://www.pbs.org/wnet/nature/animalspredict/video2.html

24. SOUND • Human hearing range – 20 Hz to 20 khz • Usually can not hear through entire range • Diminishes with age (above 20 years) and loud noises

25. Acoustics • First considered in Rome • Marco Vitruvius Pollio – designed amphitheaters • Though sound travel through air like water waves • Sound needs a media to travel through • No sound in a vacuum • No sound in explosions in space

26. Making Waves • Speaker vibrates • Creates pressure variations • Quiet – less than 0.002 Pa • Loud – about 10 Pa • Loudness – depends on how far the air molecules move • Period and Frequency – depends on time for speaker to move through a cycle • Wavelength – distance between rarefactions

27. Problem • What is the wavelength of a tuning note (A440) which is 440 Hz. The speed of sound at room temperature is 343.9 m/s? • λ = v/f = 343.9 m/s / 440 Hz = 0.782 m

28. Superposition of Waves • Waves can move through the same area of space and have a combined effect • Are not changed or scattered • Superposition Principle -When two waves overlap, the resultant is the algebraic sum of various contributions at each point

30. Fourier Analysis • Jean Baptiste Joseph, Baron de Fourier • Proved that a periodic wave having a wavelength can be synthesized by a sum of harmonic waves • A wave profile is a result of overlapping sines and cosines

31. Wavefront and Intensity • Waves move out in a circle or sphere • In-phase at different distances • As the wave moves out it becomes diffused

32. Acoustic Power • Power – Joules/sec – Watts P = Work/sec • Joules – Newton-meters Work = Force x Distance • Measuring – • Depends on area the detector • Depends on the amount of time

33. Intensity • The average power divided by the perpendicular area across which it is transported • I = Pav/A (Watt/meter²) • Area of spherical wave = 4ΠR² • The farther from the source, the greater the area, therefore the less the intensity

34. Inverse Square Law

35. Problem • An underwater explosion is detected 100 m away, where the intensity is 1.00 GW/m². About 1 second later the sound wave is recorded 1.5 km away from the explosion. What will its intensity be? R1 = 100 m R2 = 1.5 km I1 = 1.00 GW/m² Δt = 1 sec Power in first square = power in second square I1 4ΠR² = I2 4ΠR² I2 = (1 x 10⁹ W/m²) (100 m)² / (1500 m)² = 4.4 x 10⁶ W/m²

36. Speed of Sound in Air • In 1636, Father Mersen used echoes to measure speed of sound • Speed of sound increases with temperature of air • Air temperatures aren’t constant • Velocity varies depending on the gas • Speed of sound does not depend on frequency • All waves get there simultaneously

37. Problem • During a thunder storm, you hear thunder 3.50 seconds after you see a bolt of lightening. How far away, in meters and miles, did the lightening strike?

38. Hearing Sound • Three parts of ear • Outer – From outer ear to ear drum • Sound resonates in canal • Amplifies waves from 3 kHz to 4 kHz • Middle – links eardrum to 3 bones to oval window • Increases sound pressure • Inner – Transducer that converts pressure to electrical impulses • Hairs in the cochlear vibrate at different frequencies and amplitude

40. Pitch • Human response to frequency • Pure tone – sine wave • Higher the frequency, the higher the pitch • Varies in people • Increasing intensity makes you think you also increased pitch • Human voices • Men 80 Hz – 240 Hz (700 Hz in song) • Woman 140 Hz – 500 Hz (1100 Hz in song)

41. Timbre • Waveform blend of: • Harmonic – fundamental tone (f) • Overtones – tones that are over the harmonic • May or may not be harmonics (2f, 3f, etc) • Combination of harmonic and overtones makes the timbre

42. Intensity - Level • Intensity-level • Number of factors of 10 that is its intensity is above the threshold of sound • measured in bel (In honor of Alexander Graham Bell) • Io(hearing) = 1.0 x 10­¹² W/m² • Decibel (dB) – 1/10th of a bel • Unitless • β = 10 log10 I / Io • Condenses the range from 1.0 to a million millionth to 0dB to 120 dB

43. Logarithm identities • Log A/B = log A – log B • Log AB = log A + log B • β = 10 log I / Io • Δβ = 10 log I1 / I2 • This means that if you have a 12-W system and want to make it 2X louder, you have to increase the power to 120-W

44. Noise • Noise – Unrelated jumble of disturbances • Non-periodic • Continuous frequency • White noise – broad bandwidth of sounds out equal intensities • Ex – wind, pouring water, radio static • We can distinguish between wavepulses up to about 20 beats per second – then it becomes a hum

45. Beats • Interference caused in sound waves of different frequency • Used to tune guitars and pianos • Carrier wave = f1 + f2 / 2 • Beat frequency = f1 - f2 • f1 = higher f

46. Standing Waves • Waves reflected back and forth in a finite medium • Very common • All instruments • Our speaking and singing voice • Ringing bells • Lasers

47. Nodes and Antinodes • Nodes – when the resultant is zero • Antinodes – midway between nodes • Wavelength – twice the node-to-nodes distance

48. Standing Waves on Strings • First harmonic • Fundamental • 2nd harmonic • 1st overtone • 3rd harmonic • 2nd overtone • 4th harmonic • 3rd overtone • 5thharmonic • 4th overtone

49. String Standing Wave systems • Resonance in the system • Amplifies the input • Guitar • Each string has a different tension and linear mass-density • Fingering – Changes the length of the string – increases the fundamental frequency • L = ½ Nλ(N - whole number of nodes) • fN = N/2L FT/m/L • Falsetto voice – increase tension to increase frequency

50. Problem • What must the tension on a 300 mm fiddle string be to be tuned to 660 Hz? The mass-length is 0.38 g/m. • FT = (m/L)(2Lf)² = =0.38 g/m (2 x 0.300 m x 660 Hz)² =72 N (about 16 lbs)