Acoustics 170
Class Progress

Week 0 (Sep. 25): Class Orientation
PTEs will be distributed on Tuesday.

Week 1 (Sept 30): Lecture: Introduction; Philosophy of Science
What is Science?
19th C. Science: Cartesian view.
Process of empirical investigation.
Aristotle: 1. Law of identity; 2. Law of non-contradiction

(Oct 2): No lecture by professor (substituted by the TA). Frequency estimation from a 2-dimensional signal graph.
Step 1: Find the number of grids (divisions) used by a complete cycle of signal.
Step 2: Find the time length that each grid (division) represents.
Step 3: Convert it into the time in seconds, if necessary.
Step 4: Apply the following formula:

f = 1/T, where f = frequency (Hz, or cps), T = time to complete one cycle (sec).

*Note P (period), instead of T is used in the text.

The textbook will be available in the bookstore in a few days. It may cost $80+. It will also be on reserve at the Music library, hopefully by October 19th, but no guarantee. ----(Back to the Top)

Week 2 (Oct. 7)

Demonstration of Frequency Range using Audio Test CD: Frequency Sweeps (#57-58). 20~2,000Hz. Loudness changes as frequency changes. Ear is sensitive for frequencies around 2,000Hz. (See 6-6: Equal loudness contour by Fletcher & Munson.) Check the frequency range of most musical instruments (Text backcover). Frequency and pitch are closely related: American standard pitch notation. A4 = 440 Hz by American Standard.
Demonstration of low frequency using Laser Disks. "Sonic Boom". Space shuttle launch (low 2~10Hz). "Jurassic Park"

Sound Propagation

Wave Directionality. Lower frequency-wave = longer wavelength = less directional (~omnidirectional). Long waves bend. Higher frequency = shorter wavelength = more directional (~unidirectional).

Speed of Sound (Text 1-4).

The wave transmitting in medium like air is spherical (3-dimensional). Air is liner medium, i.e., linear (predictable) and non-dispersive (regardless the frequency, sound wave travels at the same speed). Water surface is dispersive on which long waves travel faster than shorter ones.

The speed of sound in the air formula:

Vt = 344 + 0.6 (t-20); where V = velocity (m/s), and t = temperature in Celsius (and Vt is the velocity at t)

Oscilloscope (Box 2-1): Estimation of frequency (See 2-1)

• Changing the sweep speed changes how a signal is displayed, as the time represented by a grid alters.
• In exam, show the process of calculation. The students can bring in calculator.
• Remember to use time in second at the end. ms: Millisecond = One thousandth second 10^-3; µs: Microsecond = One millionth second. 10^-6

3 fundamental physical quantities of any repetitive wave: Speed = v (m/s); Wavelength = l (m), Frequency = f (Hz). These variables are in the following relation: v= fl.

Simple harmonic motion: The restoring force is proportional to the displacement force. Move around the point of equilibrium. Non-simple harmonic motion can be produced by metal tong thin at one end thick at the other.
Amplitude Envelope: The pattern of amplitude changing through time.

Video "Demonstration of Acoustics" (by Berg) ----(Back to the Top)

(Oct. 9)

Fourier theorem: Any repetitive complex wave is the sum of sine (sinusoidal) waves.
Phase. Wave Cancellation: When two waves have the phases that differ in 180 degree (=1p ).

Notation: l = wavelength (m); F = frequency (Hz).

Resonance, Damping

Week 3 (Oct 14)

Quiz on Thursday, 16th will include the material covered up to Tuesday, 14th.

2 parts 1-Definitions (8); 2-Problem (1): Speed of sound

(1-5) Energy of wave can be measured as

• Amplitude displacement: Measured in distance.
• Intensity amplitude: Measured in watt (electricity).
• Pressure (p): Force (F: N or Newton) par area (S: m²), or p= F/S (N/m²) in medium. The faintest audible sound has p = 2*10^-5 (N/m²). Ambient (normal) air pressure is 10⁵ (N/m²) = one atmosphere = 50 atm.

Quiz: S = 10 m², With p normal, F = 10⁶ N. If the atmosphere pressure drops 2%, then what is F?

Vector, Acceleration (m/s²), and gravity (on the Earth surface): g = 9.8 (m/s²), Q

Frequency of Pendulum: F (Hz) = (1/2p )(g/l)^1/2, where g: gravity and l the length of arm.
Frequency is constant regardless the weight. Lengthening the string lower its natural frequency.

Natural vibrating frequency (resonance frequency). Helmholtz resonators is a mechanical method of measuring resonance frequency.

Sound Propagation (Chap. 4)

• Reflection: If a vibration hits an impermeable surface, reflection occurs. The angle of reflection is equal to that of incidence (approach).
• Refraction: Occurs when there is density of medium change (or to different medium).
• Diffraction: Property of wave, bending around or pass through an object.

Video (Berg) Demo 5: A bell in a jar. As the air in the jar is vacuumed, ringing sound of the bell becomes inaudible. Demo 6: Speed of Sound, measured by the delay. Demo 7: Reflection, Refraction and Diffraction. ----(Back to the Top)

(Oct. 16) Quiz 1
Sound Intensity and Its Measurement (Chap. 5)
Intensity is measured in watt/m². Psychological loudness is measured in dB (deci-bell, <A. G. Bell). There are several kinds of loudness (e.g. dB_IL, dB_SPL).

dB_Io = 10log₁₀(I_o/I_ref), I_o: Intensity observed, I_ref Reference intensity.
Or, when pressure (N/ m²) is used instead of intensity, dB_Po = 20 log₁₀(P_o/P_ref).

• I_ref = Threshold of hearing = 10^-12 watt/m². The loudness corresponding this intensity is set as 0 dB.
• Threshold of pain = 10⁰ watt/m² = 1 watt/m² = 120 dB.

Loudness in dB cannot be added or subtracted, because it is in log-scale. Loudness requires conversion first into intensity (or pressure) for addition/subtraction. The obtained intensity (or pressure) is then converted back to loudness (dB_IL.or dB_SPL). (See Ex. 1 and 2)

Week 4

(Oct 21) No Class

(Oct. 23)Quiz returned.

Inverse-square Law (5-3). Sound intensity decreases in proportion to the square of distance.
I = P/(4p r²), where P: the acoustic power of the source (watt); r: the radius of a spherical wave, i.e., the distance from the source. Or, I₂/I₁ = (r₁/r₂)² (Quiz: The loudness of bomb exploded at some distance.)

Absorption: I_A = I – I_R where I: The intensity of sound projected; I_A: The intensity absorbed; I_R: The intensity reflected.
Absorption Coefficient: a = I_A/I. It ranges from 0 (= no absorption) to 1 (= total absorption).

A string can be conceived as a series of infinite # of evenly distributed masses.

Modes of vibration:

1. F₁ = Fundamental (usually perceived Pitch) = 1st harmonic = 1st partial
2. F₂ = 2F₁ = Octave above = 2nd harmonic = 1st overtone
3. F₃ = 3F₁ = Oct + 5th above = 3rd harmonic = 2nd overtone
4. Etc.

• A harmonic is an integer multiple of the fundamental (= the 1st harmonic), and the frequency of a harmonic is expresses as nF₁, where n is a natural number.
• A partial is not necessarily an integer multiple of the fundamental.

Spectrum: The number of partials (harmonics) and the amount of energy in each partial.

The formula for some signals. (An: the amplitude of the n-th harmonic.)

• Sawtooth-wave: An = 1/n
• Square-wave: An = 1/(2n-1), n = natural number. contains only odd-numbered harmonics. Energy fall off with 6dB per octave.*
• Triangle-wave: An = 1/(2n-1)² where n = natural number.*

*All symmetric wave has energy only in odd-numbered harmonic.

Timbre: Sound quality. E.g., vowel sounds can be differentiated for spectrum difference. -(Back to the Top)

Week 5 (Oct. 28)

Fourier Theorem: All periodic complex signal is the sum of sines. (Inharmonic signal can be represented if it is periodic.)

Inharmonic spectrum (aperiodic) produced by non-pitched instruments, e.g., snare drums, cymbal. These instruments have mechanism that disturb normal modes of vibration.

Missing Fundamental Phenomenon: People assign the pitch corresponding to the fundamental of a harmonic series even if no energy exists physically for the fundamental.

CD Demonstration on "Timbre" (1) Effect of Spectrum on Timbre (#53). (2) Effect of Tone-Envelope on Timbre (#54~56 Bach Chorale). (3) Change in Timbre in Transposition (#57 Bassoon playing scales).

Envelope is a profile of amplitude change through time. It consists of attack, steady-state, and decay. It is an important signature of an instrument.

Two types of signal envelope: (1) Impulse signal: With attack and decay only; no steady state, e.g., plucked, struck instruments. (2) Continua signal: With attack, steady state, decay, e.g., wind, bowed instruments.

(True steady-state can be produced in a static vibrational system, by electronic signal generator. The ‘steady-state’ of sound produced by real instruments is quasi-steady.)

Time-variant (3D representation with x-, y-, z-axis for frequency, amplitude, time, respectively).

Thursday 5:30 PM Review session by Hi'ro

Mid-term on Tuesday (20 definitions ~ 2.5 * 20 = 50; 3 short-answer questions ~ 10 * 3 = 30, e.g., digital, Fourier, concept of dB; 1 problem ~ 20, dB, oscilloscope, speed, frequency.) -------(Back to the Top)

(Oct. 30)

Missing Fundamental: One theory explains that we may assign the pitch of fundamental, by comparing two succewssive harmonics which have physical energy.
Demonstration of missing fundamental (virtual pitch) phenomena (CD #43-45, "Westminster Chime")

Beating: Periodic amplitude modulation (i.e., fluctuation of intensity) caused by constructive and destructive interferences of two or more simultaneous periodic signals (waves). Beat frequency or beating rate (i.e., the number of beats per second) = F2 –F1.

Chorus Effect: Close frequency complex spectra sound full like a group, because of beating component.

Theramin uses the mechanism of beating for sound production.

Sound Reproduction (Chap. 16)

Transduction (16-1, 16-2): Transformation of energy from one form to another. A devise that does transformation is called transducer. M. Faraday

Analog Sound Re/Production. Electric-Magnetic transformation: As the pressure of wave moves the cone of microphone, magnet attached to it induces electricity (voltage) in wire. The inversion of this process results in pressure from speaker. In mechanical recording, as electricity is transformed to vibration, a needle etches signal onto a vinyl (record).

Digital Sound:

Sampling: The measurement of amplitude at a given moment in time.

1. The speed of sampling. Nyquist Frequency (Rate): The number of samples needed to resolve and represent a given frequency accurately is twice of that frequency. I.e., 40 kHz (cps) samples for 20 kHz signal.
2. The amount of energy per sample. S/N (Signal to Noise) Ratio: dB = #bits * 6. E.g., 16-bit = 96 dB S/N. (bit = binary digit. Binary system indicates one of the two states of condition: ON/OFF = 1/0. 8 bits = 2⁸ = 256 states)

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Week 6 (Nov. 4)

Class Canceled

(Nov. 6)

(Chap. 5, 6)

Psychoacoustics: The study of relationships between physical and psychological measurements.

Equal-loudness curves (loudness-level) contour. (Fletcher & Munson, 1934) obtained through perceptual experiments.

Phon: Perceptual loudness level, dB loudness level = dB_LL. dB_IL = dB_LL at 1,000Hz

Son loudness. (S. S. Stevens, 1950): Frequency-independent perceptual loudness curve, can be converted between dB_LL. It is on ratio-scale, i.e., arithmetic can be performed. Magnitude estimation method.

Note: All these measurements are done using pure tone (sine signal).

Ear Physiology (Remember the diagram and names of parts in the text):

Ear drum, ossicles, Cochlea, basilar membrane (ca. 3~4cm), hair cells, perilymph, oval window, round window, helicotrema,

Basilar membrane acts a series of tuned resonators. It is flexible, when it becomes stiff due to aging, loss of hearing called presbycusis occurs. (Other hearing problems: Tenatis: ringing in the ear. Temporary threshold shifting, TTS.)

"Roughness" is due to overlapping of excited regions on the basilar membrane. Critical Band width (Bark) is about 1/3 octave. There are about 20 barks per basilar membrane.

Loudness of complex tones is approximated by summing loudness of each excited critical band in sons.

Non-isomorphic relationships (cross mapping) among physical and perceptual variables. e.g., Frequency and Pitch

Pitch circularity. Drobish, Shepard double helix model in which tritones locate at the opposite of the circle sphere.

CD Demo (auditory barber-pole effect): Shepard, Risset.

Octave: Perceptual similarity for sounds having 2:1 frequency ratio. (Octave stretches--i.e., the ratio is about 2.009--in reality, but not to be discussed in this class.)

Chunking, short-term memory, Millar’s magic number 7+/-2.

Tuning, Intonation

Pythagorean Tuning: Derived from using perfect 5th as 2:3 frequency ratio. Pythagorean coma = 23.5 cents

Just Intonation: Most simple frequency ratios for 4, 5, 6th harmonics.

Tempered Tuning: Tempered to resolve commas.

Equal-temperament: 12 semitones per octave. Equal distance between any two semitones. 12th root of 2 = 1.05946.

Cent (<A. J. Ellis): 100 cents per one semitone in the Equal Tempered Tuning.

Quiz: Obtain the frequency of the tone, given the number of semitones from a particular tone with specified frequency. Let Fo to be the frequency to be obtained, Fg to be the frequency of the given tone. Fo = Fg*(2^1/12 raised to the number of semitones) = Fg * 1.059^{#of semitones} .

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Week 7 (Nov. 11)

(Nov. 13)

Organology: The study of instrument.

Instrument classification (taxonomy):

Four main classes--(1) Chordphone, (2) Aerophone, (3) Idiophone, (4) Membranophone--in Holmbostel-Sachs system is based mostly on the kind of driver involved.

Classification can be based upon the types of Driver--Resonator Coupling (Kendall)

Most resonator has fixed body but can have more than one resonance frequency. Sympathetic strings in Sitar are a kind of resonator.

Resonator: Device that increases the amplitude provided by the driver.

Standing wave: Wave that is in stasis, stable, equilibrium. Consisting compression and rarefaction.

Node: Point where molecules do not move, e.g., point of compression, the end of a vibrating string.

Anti-node: Point of maximum displacement, e.g., point of rarefaction.

Modes of Vibration

Node needs to be at the reflecting points.

Modes of vibration in various instruments.

Freq. of string (No calculation in exam): F = (1/2L)(T/P)^1/2, where L is the string length; T is tension; P

In open-ended cylindrical aerophone (e.g., clarinet), only odd harmonics exist. Keys change the length of Driver, as well as the resonator.

Modes of vibration produced depend on the geometry (construction of instrument).

String which is conceived as one-dimensional with nodes at both ends, therefore, modes of vibration as integer multiples of the fundamental. Air-column cylinder does not have node, thus, no even-numbered harmonics. Aerophone with cone

Bars of xylophone are held at nodal points, thus, the supporting mechanism does not affect modes of vibration.

Video: Demonstration of Acoustics (Berg)

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Week 8 (Nov. 18)

Reminder:

• No guest lecturer on Thursday.
• Quiz 2 on Nov. 25. Only the material covered after the Mid-term.
• Begin working on individual report next week. Review on 10th week.

Acoustical Organology

Coupling by air, mechanical (e.g., bridge). How does the energy from the generator transmitted to the resonator.

"Musical Instruments of the World (Microsoft Multimedia CD-ROM)" --Music library has this CD-ROM.

• Hammered Dulcimer, string as generator = chordphone. mechanical coupling through bridge.
• Impedance: Acoustical resistance.
• Gourd resonators receive the energy through the supporting point (e.g., Vina).
• Zither’s soundboard usually has broad Q resonance bandwidth (e.g., Qin).
• Appalachian Dulcimer:
• Harp: Resonator is set perpendicular to the strings.
• Guitar: Electric guitar pickups are placed at points for different modes of vibration.
• The material used to make string influences the timbre. mass, behavior
• Brass instruments: lip-reed generator = lips do SHM
• Trumpet: conical tube.
• Accordion: metal reeds generator. resonator
• Alpenhorn: Playing only using upper harmonics
• Dijiridji: vocal chords, lips as generator, the whole tube as resonator
• Jews Harp: Metal tong as driver, mouth cavity as the resonator.
• Mbira: lamelophone: metal tong as the generator.
• Steel drum
• Clave: the room created between the holding hand and the wooden block becomes the resonator.
• Chimes: pitched but has aperiodic waves, both harmonic and non-harmonic (<torsional motion) components.
• Xylophone:
• Marimba: the bar is curved concave to damp modes of vibration on surface.
• Celesta: A series of Glockenspiel bars underneath keys.
• Glockenspiel: Loud because of its frequency range
• Electric Piano: metal bars struck

(Nov. 20)

No lecture due to the incapacitated professor. (Quiz 2 will be on 11/25, Tuesday as scheduled.)

Review by TA. Hint for Quiz 2: (1) Study Handout; (2) Be comfortable with Equal-Loudness Contours and some new measurements of loudness.

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Week 9

(Nov. 25) Quiz 2

Microsoft Musical instruments of the world (continued from last week).

Modes of vibration, wavelength (l ), and frequency (F). (Notation: L: the length of generator, n: natural number indicating n-th mode of vibration.)

• Chordphone. Nodes form at both ends of a string. l _n = 2L/n, F_n = nF₁.
• Aerophone with a conical bore. Both ends become nodes, and modes of vibration are similar to those of a string. l _n = 2L/n, F_n = nF₁.
• Aerophone with a cylindrical tube. When an instrument has one open and one closed ends, node is at the closed end and anti-node at the open end. l _n = 4L/(2n-1), F_n = (2n-1)F₁. For example, by overblowing, you get the 2nd mode of vibration, whose frequency is expressed as F₂ = 3F₁. This results a pitch that is a perfect 12th (= an octave and a perfect 5th) higher than the fundamental.

With aerophones, the active length is modified commonly by finger holes, at which nodes form. A register key (e.g., Clarinet), and octave key (e.g., Saxophone) can also change nodal points.

Room Acoustics (Chap. 15)

• Concert halls are designed to eliminate standing waves. To "mix" frequencies by random reflection, various devices are used.
• Sound arrives at audience directly and as reflection.
• Reverberation Time: Time to reduce intensity by 10⁶ w/m² (60dB) in a room.
• Desired reverberation time varies depending on the context. (e.g., Speech: 0.5~1.0 sec; Baroque music: 1~1.5 sec; 19th C Orchestra: 1.5~2.5 sec)
• The bigger concert halls, more reverberation time.

(Nov. 27) No class. Thanks Giving Holiday (Back to the Top).

Week 10 (Dec. 2)

Room Acoustics (continued)

W. C. Sabine (ca. 1900); L. L. Beranek (1962). Music, acoustics & architecture.; Knudsen

Calculation of Reverberation Time (RT)--Sabin's formula:

RT (sec) = .16 (V/A), V = Total volume of the system, A = Total absorption of the system, or the effective absorption area. (The text uses Tr for RT, Se for A). A = Se = the sum of absorption.

Student Project Presentation: Moog synthesizer.

(Dec. 4)

Student Project Presentation (continued):(Back to the Top)

Week 11

Final Exam: 12/10 Wednesday 15:00-18:00.

Probable Part 3 exam question: Calculation using (1) the equal-loudness contour; (2) RT formula.

NOTE:

• There will be NO make-up exam for this one, unless, of course, with a valid reason.
• Please indicate if you want the final exam score to account for 70% toward the total and ignore the mid-term score.

PS1: Good Luck on your finals!

PS2: Have a nice winter break! : )

This page is prepared by me.

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