Fundamentals of Acoustics 2 - Phase, sound sources

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

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.
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courses and material on https://alexisbaskind.net/teaching.
Except where otherwise noted, content of this course
material is licensed under a Creative Commons Attribution-
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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

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the Phase
• A sine wave is a periodical process, i.e. it describes a
cycle that repeats itself over time
time
period
= 1/frequency

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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
0° 90°
180°
270°
450°
= 90°...
360° = 0°

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phase (rad)
period = 2π rad
• ... Or in radians (full cycle = 2π)
the Phase
0 π/2
π
3π/2
5π/2
= π/2 ...
2π = 0

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• 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

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• 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

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• 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”)

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sine wave 1
sine wave 2
sine wave 1
+ sine wave 2
Case 1: signals are in phase
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

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Case 2: signals are in quadrature
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

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Case 3: signals are phase inverted (= in antiphase)
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

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Amplitude of the mix as a function of phase difference
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

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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

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• Practical example: a sinusoidal source is recorded with two
non coincident microphones which signals are mixed
together
=> What will be the resulting waveform ?
+ 1+2=?
d1
d2
d1 and d2 are the distances
from the source to
microphones 1 and 2,
respectively
1
2

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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!

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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

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d1
d2
1 2
2
1
source
frequency
1+2
=> Both signals are in phase
Time (ms)
=> delay = (d2-d1)/c ≈ 3ms
delay
low frequencies

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d1
d2
1 2
2
1
source
frequency
1+2
=> Both signals are in quadrature
Time (ms)
=> delay = (d2-d1)/c ≈ 3ms
delay
F=85Hz ( λ=4(d2-d1) )

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d1
d2
1 2
2
1
source
frequency
1+2
=> Both signals are in antiphase
Time (ms)
=> delay = (d2-d1)/c ≈ 3ms
delay
F=170Hz ( λ=2(d2-d1) )

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d1
d2
1 2
2
1
source
frequency
1+2
=> Both signals are again in phase
Time (ms)
=> delay = (d2-d1)/c ≈ 3ms
delay
F=340Hz ( λ=d2-d1 )

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d1
d2
1 2
2
1
source
frequency
1+2
=> again in quadrature, etc…
Time (ms)
=> delay = (d2-d1)/c ≈ 3ms
delay
F=425Hz ( λ=4/5(d2-d1) )

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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

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F0
2
F0 3F0...
3F0
2
=> This is called a comb filter
frequency (Hz)
(log scale)
Here F0=340 Hz
which is the
frequency for
which the
wavelength
equals the
distance d2-d1
gain
(dB)

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F0
2
F0 3F0...
3F0
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)

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 !

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• 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)

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• 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

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• 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
original signal
after phase inversion

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• 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
original signal
after phase inversion

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1. case: Both signals are mixed in mono => the
resulting signal is pure silence
Consequences of phase inversion
Signal 1
Signal 2
+
1+2=pure silence

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2. case: Stereo: signals are presented on separate
channels => the resulting signal cannot be
localized, and lacks low frequencies
Consequences of phase inversion
Signal 1
Signal 2
L
R
?
?
??
?

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• 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
But also:
. Kick drum
. Guitar amp
…

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• 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

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Omnidirectional sources
• Omnidirectional sources, or monopoles, are sources
which radiate the sound equally in all directions
• They create spherical waves
(this diagram is only 2-
dimensional, but should be
interpreted as 3D)
Image source: Daniel A. Russel

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Omnidirectional sources
• A closed loudspeaker can be approximated as an
omnidirectional source at low frequencies
• At higher frequencies it’s not true anymore
Image source: Daniel A. Russel

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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)
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)

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Omnidirectional sources – Distance Law (again)
• 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!

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Plane waves
• In plane waves, the
sound pressure and
velocity vary only
along one dimension
• Among others, the
sound pressure level is
independent of the
distance

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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
Big sensibility
to position

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• 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

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• 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)

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Bidirectional sources
• Bidirectional sources, or dipoles, are made of two
monopoles of opposite phase, separated by a small
distance compared to the wavelength
- +

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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:
• 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)

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• This is a simplified model of an unboxed speaker (or
earphones) at low frequencies (without enclosure)
Radiation pattern
+-
(from Daniel A. Russell)

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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

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- +
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)

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- +
1b – microphone in far-field / high frequencies
• The distance ratio from the source to each monopole is close to 1:
almost identical
• But the phase shift is not anymore negligible at high frequencies
The resulting pressure is not 0
green = sum
Time(s)

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- +
2a – microphone in near-field / low frequencies
• The distance ratio from the source to each monopole is not any more
close to 1:
not identical
• The phase shift is very small at low frequencies
The resulting pressure is not 0 and depends on
the distance ratio
green = sum
Time(s)

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- +
2b – microphone in near-field / high frequencies
• The distance ratio from the source to each monopole is not any more
close to 1:
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
green = sum
Time(s)

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F0
2
F0 3F0...
3F0
2
This phenomenon can be explained as a comb filter
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

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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

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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”

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The closer the
source, the lesser the
resulting level
fluctuations
frequency (Hz)
(log scale)
gain
(dB)
Hörbereich
near-field: source “very close”

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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 - Phase, sound sources

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