Fundamentals of Acoustics

1. Fundamentals of Acoustics

2. The Nature of a Sound Event • Sound consists of vibrations of air molecules • Air molecules are analogous to tiny superballs • Sound occurs when air molecules are disturbed and made to ricochet off of each other

3. The Nature of a Sound Event • The ricochets cause the density of the air molecules to oscillate Compressed Rarefied Normal

4. The Nature of a Sound Event • The ricochets cause the density of the air molecules to oscillate back and forth

5. Wave Types Sound consists of longitudinal waves The wave’s oscillation is in the same direction as its propagation propagation oscillation Water waves are transverse waves The wave’s oscillation is perpendicular to the direction of its propagation propagation oscillation

6. Sound Propagation Sound waves propagate in a sphere from the sound source (try to imagine a spherical slinky). Note that the molecules themselves are not travelling. What spreads is the energy of the wave.

7. Sound Perception • Speed of sound (in air): 1128 ft./sec (344 m/sec) • When sound waves reach the eardrum, they are transduced into mechanical energy in the middle ear • The mechanical motion is transduced into electrical current in the inner ear. The auditory nerves interpret the current as sound

8. Sound Wave Plots • Sound waves are typically represented with molecular density as a function of time compressed normal time rarefied molecular density

9. Music vs. Noise Musical sounds are typically periodic – the wave repeats regularly repeats Sine wave Though they don’t exist in nature, sine waves are often useful for demonstrating properties of sounds Noise is aperiodic – there is no repeating pattern Noise

10. Properties of a Musical Event A musical event can be described by four properties. Each can be described subjectively, or objectively (in terms of measured properties) Subjective Objective Pitch Frequency Volume Amplitude/Power/Intensity Timbre Overtone content Duration in beats Duration in time

11. c f Frequency/Pitch Frequency is measured in cycles per second, or Hertz (Hz) f = 2 Hz one second Wavelength (l), the distance between corresponding points on the wave, is the inverse of frequency. l 1000 ft./sec. = = = 500 ft./cyc. l 2 cyc./sec.

12. Frequency/Pitch Middle A = 440 Hz = 2.3 ft. l frequencies audible to humans 20 Hz < 20,000 Hz (20 kHz) < l = 0.05 ft. l = 50 ft. Sound wavelengths are significantly larger than light wavelengths

13. Waves reflect from a surface if its height/width is larger than the wavelength

14. Waves refract around surface if the surface dimensions are smaller than the wavelength This explains why we can hear sound from around corners, but cannot see around corners: Light wavelengths are far too small to refract around any visible surface

15. 0 1 2 3 4 5 6 x x x x x x x 2 2 2 2 2 2 2 55 55 55 55 55 55 55 Our Pitch Perception is Logarithmic Equivalent pitch intervals are perceived according to an equivalent change in exponent, not in absolute frequency For example, we hear an equivalent pitch class with every doubling of frequency (the interval of an octave) Frequencies of successive octaves of concert A 55 110 220 440 880 1760 3520

16. n/12 • multiply it by 2 for n = 0 to 11 3 12 0 1 2 4 5 6 7 8 9 10 11 12 12 12 12 12 12 12 12 12 12 12 2 2 2 2 2 2 2 2 2 2 2 2 220 220 220 220 220 220 220 220 220 220 220 220 x x x x x x x x x x x x Our Pitch System is Based on Equal Division of the Octave 12 Tone Equal Temperament – the octave is divided into twelve equal increments We can describe an octave by: • choosing a starting frequency A A# B C C# D D# E F F# G G# 220 233 247 261.6 277 293.6 311 329.6 349.2 370 392 415.3 Higher octaves may be created by doubling each frequency Lower octaves may be created by halving each frequency

17. Phase Phase = “the position of a wave at a certain time” If two waveforms at the same frequency do not have simultaneous zero-crossings, we say they are “out of phase” Wave 1 + Wave 2 Wave 1 Two waves at the same frequency but different phase Wave 2 In terms of sound perception, phase can be critical or imperceptible, as we’ll see...

18. Loudness Loudness is related to three measurements: • Power • Pressure • Intensity All three are related to changes in sound pressure level (molecular density)

19. Molecular Motion is Stationary • As sound travels, molecules are not traveling with the sound wave • What is traveling is an expanding sphere of energy that displaces molecules as it passes over them • How strong is the force behind this energy wave? • The more force is contained in a sound wave, the greater its perceived loudness.

20. Power Power = the amount of time it takes to do work (exert force, move something) Power is measured in watts, W There are two difficulties in measuring sound power levels. The range of human hearing encompasses many millions of watts. Sound power level is also relative, not absolute. Air molecules are never completely motionless. Given these two difficulties, sound power levels are measured on a scale that is comparative and logarithmic, the decibel scale.

21. Logarithmic Scale Logarithm = exponent (an exponent is typically an integer, a logarithm not necessarily) 102 = 100 log10100 = 2 103 = 1000 log101000 = 3 102.698 = 500 log10500 = 2.698 102.875 = 750 log10750 = 2.875 Logarithms allow us to use a small range of numbers to describe a large range of numbers

22. -12 W = 10 watts 0 L (dB) = 10*log (W/W ) W 0 10 The Decibel Scale • The decibel scale is a comparison of a sound’s power level with a threshold level (the lowest audible power level of a sine tone at 1 kHz). Threshold (W0): Power level of a given sound in watts, LW(dB):

23. Decibels Typical power levels: Soft rustling leaves 10 dB Normal conversation 60 dB Construction site 110 dB Threshold of pain 125 dB Halving or doubling sound power level results in a change of 3 dB. For example, a doubling of the threshold level may be calculated: LW(dB) = 3.01 dB Thus, a power level of 13 dB is twice that of 10 dB. A power level of 60 dB is half that of 63 dB, and so on.

24. Pressure changes The degree of fluctuation present in a vibrating object Peak pressure level: Maximum change in sound pressure level (more generally: in a vibrating system, the maximum displacement from equilibrium position) The amplitude level fluctuates with the wave’s oscillation. Thus, power is the cause, pressure change is the result

25. 2 Pressure level is measured in Newtons per square meter (N/m ) -5 2 2 x 10 N/m (p ) 0 Pressure changes Also may be described as changes in sound pressure level (molecular density). Threshold: There is a direct relationship between pressure and power levels: For any propagating wave (mechanical, electric, acoustic, etc.) the energy contained in the wave is proportional to the square of its pressure change. Pressure changes are also expressed in decibels, but in a way that describes an equivalent change in power level: L (dB) = 10*log10(W/W0) = 10*log10(p/p0)2 = 20*log10(p/p0) W This is how pressure is measured logmn = nlogm

26. Pressure changes In audio parlance, “amplitude” (the degree of pressure change) is often equated with “loudness.” The reason is that modifications to volume are made by adjusting the amplitude of electrical current sent to an amplifier. But perceived loudness is actually based on power level plus the distance of the listener from the source.

27. Power combined with distance is intensity, I, measured in watts per square meter (W/m ). 2 -12 2 L (dB) = 10*log (I/I ) I = 10 W/m I 0 0 10 Intensity Power corresponds to the sphere of energy expanding outward from the sound source The power remains constant, spread evenly over the surface of the sphere Perceived loudness depends primarily on the sound power level and the distance from the sound event Intensity is also measured in decibels:

28. Timbre The perceived difference in sound quality when two different instruments play at the same pitch and loudness Sine waves are useful as demonstrations because they are a wave with one frequency only, thus they are often termed pure tones Natural sounds are composed of multiple frequencies To understand how a wave can be composed of multiple frequencies, we can consider the behavior of a wave in a bounded medium, such as a string secured at both ends (or air vibrating within a pipe)

29. Timbre When we pluck a string, we initiate wave motion The wavelength is twice the length of the string The perceived pitch is the fundamental, the speed of sound divided by the wavelength

30. Timbre This curved shape represents the string’s maximum deviation It’s more accurate to think of it as a series of suspended masses (kind of like popcorn strung together to hang on a Christmas tree).

31. Timbre Each suspended mass can vibrate independently. Thus, many simultaneous vibrations/frequencies occur along a string. When a string is first plucked, it produces a potentially infinite number of frequencies.

32. Timbre Eventually, the bounded nature of the string confines wave propagation and the frequencies it can support Only frequencies that remain in phase after one propagation back and forth can be maintained; all other frequencies are cancelled out Only frequencies based on integer subdivisions of the string’s length, corresponding to integer multiples of the fundamental, can continue to propagate

33. Timbre These frequencies are called harmonics NOTE: These frequencies are equally spaced Therefore, they do not all produce the same pitch as the fundamental Therefore, other frequencies are introduced …etc.

34. Timbre • Harmonics are well known to many instrumentalists • Strings • Brass

35. Timbre • The first six harmonics are often the strongest: 440 220 660 880 1100 1320 Octave Fundamental Perfect fifth Octave Major third Perfect fifth • People can learn to “hear out” harmonics

36. Timbre • Instruments and natural sounds usually contain many frequencies above the fundamental • These additional frequencies, as part of the total sound, are termed partials • The first partial is the fundamental

37. Timbre • The first partial is the fundamental • Other terms are also used • Overtones are partials above the fundamental (the first overtone is the second partial) • Harmonics are partials that are integer multiples of the fundamental

38. The Spectrum • Jean Baptiste Fourier (1768-1830) discovered a fundamental tenet of wave theory • All periodic waves are composed of a series of sinusoidal waves • These waves are harmonics of the fundamental • Each harmonic has its own amplitude and phase • The decomposition of a complex wave into its harmonic components, its spectrum, is known as a Fourier analysis

39. The Spectrum It is often more useful to represent complex waveforms with a spectral plot as opposed to a time domain plot = time domain spectral domain amplitude as a function of time amplitude as a function of frequency

40. Sound in Time • Our perception of sound and music events is determined by the behavior of frequency and loudness over time

41. Sound in Time • All instruments can be characterized by changes in amplitude over time (the envelope) loudness trumpet bowed violin harp time Changes in amplitude often correspond with changes in frequency content...

42. Sound in Time • Most instrument’s sound begins with an initial transient, or attack, portion • The transient is characterized by many high frequencies and noise • Example: the scraping of a bow or the chiff of breath • An instrument’s distinctiveness is determined primarily by the transient portion of its sound

43. Sound in Time • Following the transient, instruments usually produce a steady-state, or sustained, sound • The steady state is characterized by • Periodicity • Harmonic spectrum

44. The Spectrogram Most natural sounds (and musical instruments) do not have a stable spectrum. Rather, their frequency content changes with time. The spectrogram is a three-dimensional plot: Vibraphone note at 293 Hz (middle D) 2) frequency 3) power of a given frequency (darkness level) 1) time The instrument’s sound is characterized by the fundamental at 293 Hz and the fourth harmonic at 1173 Hz. The attack also contains noise below 2 kHz, the tenth harmonic at 2933 Hz and the seventeenth harmonic at 4986 Hz. Once the steady state portion sets in, the highest harmonic fades first, followed by a fading of the fundamental.

45. Localization • The auditory system localizes events through interaural time delay – the sound wave reaches the nearer ear a few milliseconds before it reaches the farther ear • For stereo systems, using delay for localization is impractical because it requires people to listen from a “sweet spot” • Localization effects are simulated through differences in loudness

46. Localization • In a multi-speaker system, a sound emanating from one speaker will be localized at that speaker • A sound produced at equal volume from two speakers will be perceived as a “phantom image” placed in space between them • Changing the volume balance between two speakers will cause the phantom image to “drift” towards the louder speaker

47. Measurement and Perception • Our perception of auditory events is based on all these measurements in combination • And more • An auditory event may be more than the sum of its parts

48. Measurement and Perception Phase • Changing the phase of components in a steady-state tone produces no perceptible change in sound, although the shape of the wave may change noticeably

49. Measurement and Perception Phase • The behavior of components in the attack segment is likely to be far more complex than in the steady state segment • Changing the phase of attack components can change the character of the attack • Solo performance sounds different from group performance because no two players can ever sound at exactly the same time; thus the attack is blurred • Since an instrument’s characteristics are defined primarily by the attack, the phase of attack components is critical

50. Measurement and Perception Timbre • We have discussed timbre as the result of overtone content • It is also judged by the sound’s envelope • Research in sound synthesis has shown the envelope shape to be more definitive than an exact match of overtone content • The attack portion is critical—a faster attack can be confused with “brightness” (more high frequency overtones) • Considerable research has gone into the creation of “timbre space,” a multi-dimensional plot in which timbres are classified according to overtone content, envelope and attack time