Fundamentals of Acoustics 2 - Phase, sound sources

1. Alexis Baskind Fundamentals of Acoustics 2 Phase, sound sources Alexis Baskind, https://alexisbaskind.net

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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. 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. 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. 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!

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. 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. 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. 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

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. 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. 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)

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. 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. 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. 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. 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. 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. 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. 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. 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. 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

Fundamentals of Acoustics 2 - Phase, sound sources

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