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# Fundamentals of Acoustics 2 - Phase, sound sources

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# Fundamentals of Acoustics 2 - Phase, sound sources

Visit https://alexisbaskind.net/teaching for a full interactive version of this course with sound and video material, as well as more courses and material.

Course series: Fundamentals of acoustics for sound engineers and music producers
Language: English
Revision: February 2020
To cite this course: Alexis Baskind, Fundamentals of Acoustics 2 - Phase, sound sources
course material, license: Creative Commons BY-NC-SA.

Course content:
1. The phase
sine wave, phase, angle, phase and complex signals, signals in phase, signals in quadrature, signals in antiphase, constructive and destructive interferences, comb filter, phase inversion, phase shift

2. Omnidirectional sources (monopoles)
Definition of omnidirectional sources, radiation pattern, spherical waves, Omnidirectional sources and distance law

3. Plane waves, near field, far field
definition of a plane wave, near field, far field

4. Bidirectional sources (dipoles)
Definition of bidirectional sources, radiation pattern

5. Dipoles in near-field and far-field
frequency-dependent behaviour of dipoles in near and far field

Visit https://alexisbaskind.net/teaching for a full interactive version of this course with sound and video material, as well as more courses and material.

Course series: Fundamentals of acoustics for sound engineers and music producers
Language: English
Revision: February 2020
To cite this course: Alexis Baskind, Fundamentals of Acoustics 2 - Phase, sound sources
course material, license: Creative Commons BY-NC-SA.

Course content:
1. The phase
sine wave, phase, angle, phase and complex signals, signals in phase, signals in quadrature, signals in antiphase, constructive and destructive interferences, comb filter, phase inversion, phase shift

2. Omnidirectional sources (monopoles)
Definition of omnidirectional sources, radiation pattern, spherical waves, Omnidirectional sources and distance law

3. Plane waves, near field, far field
definition of a plane wave, near field, far field

4. Bidirectional sources (dipoles)
Definition of bidirectional sources, radiation pattern

5. Dipoles in near-field and far-field
frequency-dependent behaviour of dipoles in near and far field

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### Fundamentals of Acoustics 2 - Phase, sound sources

1. 1. Alexis Baskind Fundamentals of Acoustics 2 Phase, sound sources Alexis Baskind, https://alexisbaskind.net
2. 2. Alexis Baskind Fundamentals of Acoustics 2 - Phase, sound sources Course series Fundamentals of acoustics for sound engineers and music producers Level undergraduate (Bachelor) Language English Revision January 2020 To cite this course Alexis Baskind, Fundamentals of Acoustics 2 - Phase, sound sources, course material, license: Creative Commons BY-NC-SA. Full interactive version of this course with sound and video material, as well as more courses and material on https://alexisbaskind.net/teaching. Except where otherwise noted, content of this course material is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License. Fundamentals of Acoustics 2
3. 3. Alexis Baskind Outline 1. The phase 2. Omnidirectional sources (monopoles) 3. Plane waves, near field, far field 4. Bidirectional sources (dipoles) 5. Dipoles in near-field and far-field Fundamentals of Acoustics 2
4. 4. Alexis Baskind the Phase • A sine wave is a periodical process, i.e. it describes a cycle that repeats itself over time Fundamentals of Acoustics 2 time period = 1/frequency
5. 5. Alexis Baskind phase (degrees) period = 360° • The phase describes the state of the sine wave cycle as a fraction of the period • The phase does not depend in frequency • It is measured as an angle, either in degrees (full cycle = 360°) the Phase Fundamentals of Acoustics 2 0° 90° 180° 270° 450° = 90°... 360° = 0°
6. 6. Alexis Baskind phase (rad) period = 2π rad • ... Or in radians (full cycle = 2π) the Phase Fundamentals of Acoustics 2 0 π/2 π 3π/2 5π/2 = π/2 ... 2π = 0
7. 7. Alexis Baskind • Phase is a notion that concerns all time-varying signals, and not only sine waves. • As a matter-of-fact, every signal (see previous lesson) can be considered as a (finite or infinite) sum of sine waves with various frequencies • This means, that for a complex signal, the phase can be calculated at any time for each frequency component • What means phase for music production? To what extent is it relevant for hearing? Phase and complex signals Fundamentals of Acoustics 2
8. 8. Alexis Baskind • The notion of phase is closely related to time: more precisely, a phase shift can be considered as a time shift with regard to the period • One of the most important important interpretations of phase in audio concerns the time synchronization between two signals: if two signals are perfectly synchronized, their phases are identical for all frequencies and vice-versa. Examples: – Time correction between tracks in a mix – Time alignment of a subwoofer Phase and interferences Fundamentals of Acoustics 2
9. 9. Alexis Baskind • Phase plays a major role in interferences, i.e. when two or several sinusoids with the same frequency are added • The result is also a sinusoid of the same frequency, but its amplitude depends on the relation between phases, i.e. the phase difference • Among others, there are three important cases: – Phase difference = 0° => both frequencies are in phase – Phase difference = ±90° => both frequencies are in quadrature – Phase difference = 180° => both frequencies are phase inverted (also called “in antiphase”) Phase and interferences Fundamentals of Acoustics 2
10. 10. Alexis Baskind sine wave 1 sine wave 2 sine wave 1 + sine wave 2 Case 1: signals are in phase Phase and interferences Fundamentals of Acoustics 2 The resulting sinusoid is doubled in amplitude (+6dB) The sine waves are synchronized Note: both sine waves are assumed here to have the same amplitude
11. 11. Alexis Baskind Case 2: signals are in quadrature Phase and interferences Fundamentals of Acoustics 2 When one sine wave is maximal or minimal, the other equals 0 The amplitude is increased of +3dB sine wave 1 sine wave 2 sine wave 1 + sine wave 2
12. 12. Alexis Baskind Case 3: signals are phase inverted (= in antiphase) Phase and interferences Fundamentals of Acoustics 2 The wave forms are opposite with respect to each other: the maxima of one correspond to the minima of the other one and vice-versa The amplitude is 0, the sine waves cancel each other sine wave 1 sine wave 2 sine wave 1 + sine wave 2
13. 13. Alexis Baskind Amplitude of the mix as a function of phase difference Phase and interferences Fundamentals of Acoustics 2 phase difference 0° 90° 180° 270° 360° gain (dB) CASE 1: in-phase (0°)=> +6 dB CASE 3: phase inverted (180°) => silence (-∞ dB) CASE 2: quadrature (+/-90°)=> +3 dB
14. 14. Alexis Baskind Phase and interferences Fundamentals of Acoustics 2 This explains why two acoustic sources create interferences patterns (see “Fundamentals of Acoustics 1”) • red: the pressure is greater than if there was only one source (constructive interferences) • green: the pressure is always almost zero (destructive interferences) • The interference pattern depends on frequency and distance between sources Test it yourself: http://www.falstad.com/ripple/ Image source: Oleg Alexandrov
15. 15. Alexis Baskind • Practical example: a sinusoidal source is recorded with two non coincident microphones which signals are mixed together => What will be the resulting waveform ? Phase and interferences Fundamentals of Acoustics 2 + 1+2=? d1 d2 d1 and d2 are the distances from the source to microphones 1 and 2, respectively 1 2
16. 16. Alexis Baskind Here it is assumed that: – the wave is sinusoidal (pure tone) – the levels of the signals at both microphones are identical (attenuation due to distance is neglected) – both microphones are identical, and that they pick up the sound pressure exactly without filtering (perfect pressure microphones) In this case, both signals at the output of the microphones have identical frequencies and amplitudes But phases differ! Phase and interferences Fundamentals of Acoustics 2
17. 17. Alexis Baskind Phase and interferences d1 d2 1 2 2 1 source • The phase difference between both sinusoids depends on the difference between d1 and d2, on the speed of sound and on frequency 1+2 Time (ms) delay Example: The distance d2-d1 is 1 meter => delay = (d2-d1)/c ≈ 3ms Fundamentals of Acoustics 2 © Alexis Baskind
18. 18. Alexis Baskind Phase and interferences d1 d2 1 2 2 1 source • The phase difference between both sinusoids depends on the difference between d1 and d2, on the speed of sound and on frequency 1+2 => Both signals are in phase Time (ms) Example: The distance d2-d1 is 1 meter => delay = (d2-d1)/c ≈ 3ms delay Fundamentals of Acoustics 2 © Alexis Baskind low frequencies
19. 19. Alexis Baskind Phase and interferences d1 d2 1 2 2 1 source • The phase difference between both sinusoids depends on the difference between d1 and d2, on the speed of sound and on frequency 1+2 => Both signals are in quadrature Time (ms) Example: The distance d2-d1 is 1 meter => delay = (d2-d1)/c ≈ 3ms delay F=85Hz ( λ=4(d2-d1) ) Fundamentals of Acoustics 2 © Alexis Baskind
20. 20. Alexis Baskind Phase and interferences d1 d2 1 2 2 1 source • The phase difference between both sinusoids depends on the difference between d1 and d2, on the speed of sound and on frequency 1+2 => Both signals are in antiphase Time (ms) Example: The distance d2-d1 is 1 meter => delay = (d2-d1)/c ≈ 3ms delay Fundamentals of Acoustics 2 © Alexis Baskind F=170Hz ( λ=2(d2-d1) )
21. 21. Alexis Baskind Phase and interferences d1 d2 1 2 2 1 source • The phase difference between both sinusoids depends on the difference between d1 and d2, on the speed of sound and on frequency 1+2 => Both signals are again in phase Time (ms) Example: The distance d2-d1 is 1 meter => delay = (d2-d1)/c ≈ 3ms delay Fundamentals of Acoustics 2 © Alexis Baskind F=340Hz ( λ=d2-d1 )
22. 22. Alexis Baskind Phase and interferences d1 d2 1 2 2 1 source • The phase difference between both sinusoids depends on the difference between d1 and d2, on the speed of sound and on frequency 1+2 => again in quadrature, etc… Time (ms) Example: The distance d2-d1 is 1 meter => delay = (d2-d1)/c ≈ 3ms delay Fundamentals of Acoustics 2 © Alexis Baskind F=425Hz ( λ=4/5(d2-d1) )
23. 23. Alexis Baskind Phase and interferences Fundamentals of Acoustics 2 frequency (Hz) (linear scale)F0 2 F0 2F0 3F0... => This is called a comb filter gain (dB) 3F0 2 Here F0=340 Hz which is the frequency for which the wavelength equals the distance d2-d1
24. 24. Alexis Baskind F0 2 F0 3F0... 3F0 2 => This is called a comb filter Phase and interferences Fundamentals of Acoustics 2 frequency (Hz) (log scale) Here F0=340 Hz which is the frequency for which the wavelength equals the distance d2-d1 gain (dB)
25. 25. Alexis Baskind F0 2 F0 3F0... 3F0 2 Phase and interferences Fundamentals of Acoustics 2 frequency (Hz) (log scale) If the delayed signal is softer, there is still comb-filtering, but with lesser amplitude gain (dB) (here for example, the delayed signal is -6dB softer)
26. 26. Alexis Baskind • Comb-filters do not only concern pure tones, but all kind of sounds (since sounds can be decomposed in a sum of sinusoids) • Practical example (just try it yourself with a delay plugin in a sequencer): – pink noise – cymbal roll – Vocals … So be careful with delays ! Phase and interferences Fundamentals of Acoustics 2
27. 27. Alexis Baskind • To summarize: a delay corresponds to a phase shift that depends on frequency. • If two similar but not synchronous signals are superimposed in mono, a comb-filtering occurs • This should not be confused with the stereo presentation of a signal + delayed version. In the latter case, no (or little) comb-filtering occurs, but the precedence effect has to be considered (see course on spatial hearing) Phase and interferences Fundamentals of Acoustics 2
28. 28. Alexis Baskind • It is very important to be able to identify a comb filter quickly and, if needed, to correct it • Some typical causes for comb filters are: – Faulty (=double) signal paths (for instance Direct- Monitoring + DAW-Monitoring simultaneously) – Problem with latency compensation in a DAW – More as one microphone pro sound source (sometimes necessary, but then microphones should be positioned carefully) • However sometimes comb filters are desired: for instance, a Flanger is a time-modulated comb filter Phase and interferences Fundamentals of Acoustics 2
29. 29. Alexis Baskind • Another phenomenon, which is sometimes confused with the effects of a time delay, is phase inversion (also called phase reversal or polarity inversion) • This corresponds to a multiplication of the signal by -1 • A phase inversion occurs for instance in analog technology when the “+” and “-” conductors of a balanced connection are reversed Phase inversion Fundamentals of Acoustics 2 original signal after phase inversion
30. 30. Alexis Baskind • A Phase inversion corresponds to phase shift of 180° for all frequencies: all frequency components of the signal are in antiphase with the original • It can be corrected thanks to a polarity reversal button or plugin Phase inversion Fundamentals of Acoustics 2 original signal after phase inversion
31. 31. Alexis Baskind 1. case: Both signals are mixed in mono => the resulting signal is pure silence Consequences of phase inversion Fundamentals of Acoustics 2 Signal 1 Signal 2 + 1+2=pure silence
32. 32. Alexis Baskind 2. case: Stereo: signals are presented on separate channels => the resulting signal cannot be localized, and lacks low frequencies Consequences of phase inversion Fundamentals of Acoustics 2 Signal 1 Signal 2 L R ? ? ?? ?
33. 33. Alexis Baskind • In practice, a partial 180° phase shift is also possible (most of time at low frequencies) • Example: Snare-Drum recording: at low frequencies, a Snare-Drum behave like a dipole (see below) : the sound pressures above and below are opposite with respect to each other Partial Phase inversion Fundamentals of Acoustics 2 But also: . Kick drum . Guitar amp …
34. 34. Alexis Baskind • Remember: a time shift cannot be corrected by inverting the phase and vice-versa: they correspond to two different kinds of phase shift • Phase inversions and phase shifts may or may not be problems, depending on what you’re looking for: the best judge is your ears ! • But it’s anyway really important to be able to recognize and correct a phase issue Phase inversion and phase shift Fundamentals of Acoustics 2
35. 35. Alexis Baskind Outline 1. The phase 2. Omnidirectional sources (monopoles) 3. Plane waves, near field, far field 4. Bidirectional sources (dipoles) 5. Dipoles in near-field and far-field Fundamentals of Acoustics 2
36. 36. Alexis Baskind Omnidirectional sources • Omnidirectional sources, or monopoles, are sources which radiate the sound equally in all directions • They create spherical waves Fundamentals of Acoustics 2 (this diagram is only 2- dimensional, but should be interpreted as 3D) Image source: Daniel A. Russel
37. 37. Alexis Baskind Omnidirectional sources • A closed loudspeaker can be approximated as an omnidirectional source at low frequencies • At higher frequencies it’s not true anymore Fundamentals of Acoustics 2 Image source: Daniel A. Russel
38. 38. Alexis Baskind Omnidirectional sources – Distance Law (again) • An omnidirectional source has a limited sound power • This given power is spread out over all the surface of the wave (which is a sphere) Fundamentals of Acoustics 2 Caution: The previously mentionned distance law (attenuation of 6 dB of the sound pressure for a doubling of the distance) is only valid for monopoles!!! Image source: Borb (Wikipedia)
39. 39. Alexis Baskind Omnidirectional sources – Distance Law (again) Fundamentals of Acoustics 2 • The sound pressure in the center is infinite • In practice it’s impossible: exact monopoles (point sources) don’t exist in reality, it’s only a model!
40. 40. Alexis Baskind Outline 1. The phase 2. Omnidirectional sources (monopoles) 3. Plane waves, near field, far field 4. Bidirectional sources (dipoles) 5. Dipoles in near-field and far-field Fundamentals of Acoustics 2
41. 41. Alexis Baskind Plane waves Fundamentals of Acoustics 2 • In plane waves, the sound pressure and velocity vary only along one dimension • Among others, the sound pressure level is independent of the distance
42. 42. Alexis Baskind Near field – Far field • If the microphone is close to the source, there is a big difference as a function of the position => this is called the near field Fundamentals of Acoustics 2 Big sensibility to position
43. 43. Alexis Baskind Near field – Far field • If the microphone is far from the source, the wave behaves like a plane wave Small sensibility to position • The sound wave behave locally as a plane wave This is called far-field Fundamentals of Acoustics 2
44. 44. Alexis Baskind Near field – Far field • In the near field, the level, spectrum (see last part of this lesson) and phases are very sensitive to the position of the ears (or the microphone) • In the far field, the sound image is somehow more stable • In a mixing studio, there are usually near-field monitors and far-field monitors – Far-field monitors are meant for a group listening. They are usually bigger – Near-field monitors are designed for an individual listening (i.e. for the mixing engineer) Fundamentals of Acoustics 2
45. 45. Alexis Baskind Outline 1. The phase 2. Omnidirectional sources (monopoles) 3. Plane waves, near field, far field 4. Bidirectional sources (dipoles) 5. Dipoles in near-field and far-field Fundamentals of Acoustics 2
46. 46. Alexis Baskind Bidirectional sources • Bidirectional sources, or dipoles, are made of two monopoles of opposite phase, separated by a small distance compared to the wavelength Fundamentals of Acoustics 2 - +
47. 47. Alexis Baskind Bidirectional sources The radiation pattern result from interfences between both poles. However, because of their reversed polarity, the interference pattern looks different as in part 1 of this lesson: Fundamentals of Acoustics 2 • On the sides (90°), the resulting pressure is always zero (particles don’t move) • On axis, the sound pressure level is maximum (from Daniel A. Russell)
48. 48. Alexis Baskind Bidirectional sources • This is a simplified model of an unboxed speaker (or earphones) at low frequencies (without enclosure) Fundamentals of Acoustics 2 Radiation pattern +- (from Daniel A. Russell)
49. 49. Alexis Baskind Outline 1. The phase 2. Omnidirectional sources (monopoles) 3. Plane waves, near field, far field 4. Bidirectional sources (dipoles) 5. Dipoles in near-field and far-field Fundamentals of Acoustics 2
50. 50. Alexis Baskind Dipoles in near-field and far-field • Contrary to monopoles, dipoles have different frequency behaviors in near- and far-field: – In far-field, low frequencies cancel out each other – In near-field, low frequencies are increased • This phenomenon, that concerns all directional sources (and not only dipoles), is conceptually very similar to the so-called “proximity effect” for directional microphones (see lesson about microphones) • The explanation of this phenomenon requires understanding the distance law as well as the frequency- and distant-dependent phase shift that occur Fundamentals of Acoustics 2
51. 51. Alexis Baskind Dipoles in near-field and far-field - + 1a – microphone in far-field / low frequencies • The distance ratio from the source to each monopole is close to 1:  The sound pressure levels corresponding to each monopoles are almost identical • The time shift compared to the period (= phase shift) is very small The resulting pressure is close to 0 red= “+” pressure blue= “-” pressure green = sum Time(s) Fundamentals of Acoustics 2
52. 52. Alexis Baskind Dipoles in near-field and far-field - + 1b – microphone in far-field / high frequencies • The distance ratio from the source to each monopole is close to 1:  The sound pressure levels corresponding to each monopoles are almost identical • But the phase shift is not anymore negligible at high frequencies The resulting pressure is not 0 Fundamentals of Acoustics 2 red= “+” pressure blue= “-” pressure green = sum Time(s)
53. 53. Alexis Baskind Dipoles in near-field and far-field - + 2a – microphone in near-field / low frequencies • The distance ratio from the source to each monopole is not any more close to 1:  The sound pressure levels corresponding to each monopoles are not identical • The phase shift is very small at low frequencies The resulting pressure is not 0 and depends on the distance ratio Fundamentals of Acoustics 2 red= “+” pressure blue= “-” pressure green = sum Time(s)
54. 54. Alexis Baskind Dipoles in near-field and far-field - + 2b – microphone in near-field / high frequencies • The distance ratio from the source to each monopole is not any more close to 1:  The sound pressure levels corresponding to each monopoles are not identical • the phase shift is not any more negligible at high frequencies The resulting pressure is not 0 and depends on the distance ratio and phase shift Fundamentals of Acoustics 2 red= “+” pressure blue= “-” pressure green = sum Time(s)
55. 55. Alexis Baskind F0 2 F0 3F0... 3F0 2 This phenomenon can be explained as a comb filter Dipoles in near-field and far-field Contrary to a “classical” comb- filter, destructive interferences occur at low frequencies (because of polarity reversal of one of the poles) frequency (Hz) (log scale) gain (dB) far-field Fundamentals of Acoustics 2
56. 56. Alexis Baskind This phenomenon can be explained as a comb filter Dipoles in near-field and far-field If the monopoles are very close to each other,, the next cancellation occurs in ultrasonic range => In the hearing range, this can be modeled as a first- order high pass filter frequency (Hz) (log scale) gain (dB) Hearing range slope: 6 dB / Octave far-field Fundamentals of Acoustics 2
57. 57. Alexis Baskind This phenomenon can be explained as a comb filter Dipoles in near-field and far-field In near-field, the cancellation is not total because of the level difference frequency (Hz) (log scale) gain (dB) Hearing range near-field: source “close” Fundamentals of Acoustics 2
58. 58. Alexis Baskind This phenomenon can be explained as a comb filter Dipoles in near-field and far-field The closer the source, the lesser the resulting level fluctuations frequency (Hz) (log scale) gain (dB) Hörbereich near-field: source “very close” Fundamentals of Acoustics 2
59. 59. Alexis Baskind Dipoles in near-field and far-field To summarize: • For dipoles (and actually for all directional sources), high frequencies are radiated farther than low frequencies • Low frequencies can be only be captured and perceived close to the source • As previously mentioned, this effect is in principle exactly symmetrical to the so-called proximity effect for directional microphones (see lesson about microphones) This why headphones and especially earphones sound thin when they are not close to the ears Fundamentals of Acoustics 2