Fundamentals of Acoustics 1

Alexis Baskind
Fundamentals of Acoustics 1
The sound wave
Alexis Baskind, https://alexisbaskind.net

Alexis Baskind
The sound wave
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 1 - The sound wave, course material,
license: Creative Commons BY-NC-SA.
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Outline
1. What is acoustics?
2. What is a sound wave ?
3. Speed of sound and acoustic velocity
4. Superposition principle, interferences
5. Harmonic oscillators
6. Harmonic waves
7. Sound pressure level
8. Distance law

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What is acoustics?
• Acoustics is a branch of science that focuses on
sound and its propagation
• A Sound is a mechanical wave, i.e. the time and
space propagation of an oscillation in an elastic
medium, like:
– Gas (like air) => gas-borne (airborne) sound
– Liquids (like water) => liquid-borne sound
– Solids: wood, metal, concrete...
=> structure-borne sound

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What is acoustics
• Elasticity is the “ability of a deformed material
body to return to its original shape and size when
the forces causing the deformation are removed“
(Encyclopaedia Britannica)
• This explain why sounds can propagate: if some
molecules of the medium are moved with respect
to their position of rest, a tension/force will be
exerted to the neighboring molecules, that forces
them to move from their position of rest as well =>
Propagation

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What is acoustics?
• Sound can also be categorized with respect to its
oscillation frequency and the human hearing
range:
– Infrasound (under 20 Hz) (not audible)
– Audible sound (from 20 Hz up to 20 kHz)
– Ultrasound (from 20 kHz um to 1 GHz) (not audible)
– Hypersound (above 1 GHz) (not audible)
• This course deals only with audible sound, which is
a tiny fraction of all forms of sound!

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What is a sound wave ?
• A sound wave is the time and space evolution of
the sound characteristics, that is:
– The sound pressure (or “acoustic pressure”). Unit =
Pascal (Pa)
– The sound particle velocity (or “acoustic velocity”) (see
next part). Unit = m/s
• As already mentioned, sound can only be
transmitted in an elastic medium: no sound
propagation is possible in vacuum

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Longitudinal waves
• In gases and liquids, sound can only be propagated
through longitudinal waves: the particles oscillate
in the same direction than the wave)
Example: particles in a one-dimensional longitudinal wave
Direction of propagation
Image source:
Daniel A. Russel

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Longitudinal waves
=> The sound wave alternates between compression and
rarefaction of air
Example: particles in a one-dimensional longitudinal wave
compression rarefaction
Image source:
Daniel A. Russel

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Sound pressure
The sound wave alternates between compression
and rarefaction of air
But: the order of magnitude of those fluctuations is
very low!
Example:
– The average air pressure in the air is around 1 Bar, so
100000 Pa
– The range of pressure fluctuations due to sound is
around 20 Pa for very loud sounds (threshold of pain),
and around 0,00002 Pa for very soft sounds (threshold
of hearing), thus between 5000 and 5 billion times
less!

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Longitudinal waves
Spherical Wave (normally
tridimensional, hier shown
only in 2D)
In green: direction of propagation
Example: particles in a three-dimensional longitudinal wave
Image source: Daniel A. Russel

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Transverse Sound Waves
• In solids, sound can be propagated through
longitudinal waves, but also through transverse
waves (the particles oscillate in the other direction
than the wave)
Examples:
• Strings (mostly transversal, but also longitudinal waves)
• Plates (Cymbals, wooden plates...)
• Bars (Marimba, Vibraphone...)
Image source:
Daniel A. Russel

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Speed of Sound
• Speed of sound is not the velocity of sound
particles !
The sound particle velocity is the average speed of
the molecules around their stable position. It
depends on time and space and remains very low
(up to ca. 1 mm/s)
Direction of the wave travel
Image source: isvr

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Speed of Sound
Speed of sound is not the velocity of sound particles !
The speed of sound is the overall speed of
propagation of the wave. It’s constant under given
conditions (medium, temperature, humidity...) and
much bigger as the sound particle velocity
The speed of sound does not depend on frequency
Direction of the wave travel
Image source: isvr

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• Speed of sound
Speed of Sound
Green: Direction of
the wave travel
Red: Fluctuation of
the position of single
molecules

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Speed of Sound
The speed of sound depends on the medium (air,
water, wood…)
The speed of sound in the air is quite low:
(this has to be considered as an order of magnitude, since it depends on several
factors)
In more dense media (liquids and solids), sound is
normally much faster than in the air
cair,T=15°C ≈ 340 m/s = 1225 km/h

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Speed of Sound
• Orders of magnitude of speed of sound in other
materials
Medium Speed of sound
(m/s)
Air 340
Brass 3475
Steel 6100
Concrete 3200-3600
Wood 3300-4000
Water 1433

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Speed of Sound
The speed of sound in the air depends on humidity
and temperature (thus on the altitude as well), but not
on the average air pressure:
– speed of sound increases with temperature
– speed of sound increases with humidity
In liquids and solids, there are much more variables that
have an influence on the speed of sound

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Superposition principle
• What happens when two sound waves are
superposed ?
=> Their contributions just add, since air is a linear
medium (except at very high levels)
+
=

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Interferences
Interferences emerge because
of this superposition:
depending on the position-
dependent phase difference
(see next lesson), the sound
waves locally:
• partially or totally cancel
out each other (destructive
interferences)
• reinforce each other
(constructive
interferences)
Image source:
Oleg Alexandrov

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Interferences
The resulting interference
pattern depends on frequency
and on distance between
sources
Image source:
Oleg Alexandrov

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Image source: Debenben (Wikipedia)
Displacement y
Period T
Time t
Harmonic oscillations are (undamped) oscillations with
a sinusoidal motion
• Examples of harmonic oscillators:
1. Spring/Mass-System
The Mass oscillates endlessly
without damping (friction) (see
later). The frequency of
oscillation (=eigenfrequency):
• grows with increasing spring
constant (stiffness)
• decreases with increasing
mass
Harmonic oscillators

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Harmonic oscillators
Harmonic oscillations are (undamped) oscillations with
a sinusoidal motion
• Examples of harmonic oscillators:
2. Pendulum 3. LC electric Circuit
Rest position
Image source: Stündle (Wikipedia) Image source: Chetvorno (Wikipedia)

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Damped Harmonic Oscillators
• All real physical systems present damping
• Oscillations with damping are not any more purely
harmonic: they are called damped harmonic oscillations
1. Spring/Mass-System with friction
The eigenfrequency:
• increases with increasing spring
constant
• decreases with increasing mass
• decreases with increasing friction
The decay time:
• increases with increasing mass
• decreases with increasing friction
Image sources: Jahobr and Oleg Alexandrov (Wikipedia)

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• All real physical systems present damping
• Oscillations with damping are not any more purely
harmonic: they are called damped harmonic oscillations
2. RLC electric circuits
The Energy is progressively transformed in heat in the resistance

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Damped oscillators are everywhere in acoustics and
sound engineering. Examples:
• Loudspeakers: in the drivers (mechanic)
• Loudspeakers: Bass reflex systems (acoustic)
• Microphones (mechanic/acoustic)
• Room acoustics absorbing modules (membrane
absorbers, Helmholtz resonators)
• Sound holes in acoustic instruments
• ...

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Wavelength (λ)
= distance between
two pressure maxima
(or minima) (in meters)
Harmonic Waves
• In elastic media, harmonic oscillations propagate as
harmonic waves, i.e. sinusoidal oscillation that
depend on space...

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Harmonic Waves
... and time.
wavelength λ (meters)
period T (seconds)
Period (T):
= time between two
pressure maxima on a
given point (in seconds)
Wavelength (λ)
= distance between
two pressure maxima
(or minima) (in meters)

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Harmonic Waves
Relation between wavelength, period and speed of sound:
frequency (f): inverse of
the period (in Hertz)
Relation between
wavelength, frequency
and period:
l =
c
f
= cT
f =
1
T
wavelength λ (meters)
period T (seconds)

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Harmonic Waves: sound waves
• Some orders of magnitudes about frequencies
and wavelengths (with c = 340 m/s)
Pitch (German/English notation) Frequency (Hz) Wavelength
Infrasound < 20 > 17 m
Lowest note of the piano (,,A / A0) 27.5 12.36 m
Middle C (c‘ / C4) 261 1.3 m
Middle A (a‘ / A4) 440 77 cm
Highest note of the piano (c‘‘‘‘‘ / C8) 4186 81 mm
Ultrasounds > 20000 < 17 mm

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Harmonic Waves: sound waves
• Most of sound waves are not harmonic
(=sinusoidal)
• However, every complex sound can be considered
as the combination of many harmonic elements
(see lesson „the overtone spectrum“)
• This means, that the model of harmonic waves is
still valid for complex waves, if they each
frequency is considered separately

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Instantaneous
pressure (with
peak values)
Sound Pressure Level
• The pressure is measured in pascal (Pa)
• We often refer to the effective pressure, i.e. the
root mean square (RMS) of the pressure over time
pressure
time (seconds)
RMS pressure
pRMS = p2
...With meaning
“time average”

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• According to the Weber-Fechner Law (see
appendix “dB or not dB”), human perception
(brightness - vision, loudness - hearing, etc…) is in
general roughly proportional to the logarithm of
the corresponding physical quantity (light
intensity, sound intensity, etc…)
• This logarithm is generally called level
• A level can be:
– either absolute (then an absolute reference value is
needed) => dB SPL, dB FS, dBu, dBV...
– or relative (between two values) => dB (without
anything after “dB”)

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• The Sound Pressure Level (SPL) is the logarithmic
ratio of the RMS pressure to a reference value :
𝐿 𝑆𝑃𝐿 = 20. 𝑙𝑜𝑔10
𝑝 𝑅𝑀𝑆
𝑝0
(𝑑𝐵)
… with the reference value
LSPL = 0dBSPL
p0
p0 = 20 mPa = 0,00002 Pa
• Why this value for ? Because it corresponds to
the absolute minimal threshold of hearing at 1kHz
(see next course on psychoacoustics)
p0
• Then if ,pRMS = p0

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Sound Pressure Level - references
Bedroom
Library
Conversational
Speech
Heavy Truck
Pop Group
Jet Engine at 100m
Forest
Living Room
Business Office
Average
Street Traffic
Jack hammer at 1m
dB SPL
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Vuvuzela at 1m
Threshold of
hearing at 1kHz
200 Pa
pRMS
20 Pa
2 Pa
0.2 Pa
0.02 Pa
0.002 Pa
0.0002 Pa
0.00002 Pa

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Distance Law
• The sound intensity emitted from a source is
distributed over an increasing surface with increasing
distance to source.
If the distance to the
source is doubled, the
surface is multiplied by 4,
and the sound intensity
divided by 4
This corresponds to
halving the sound
pressure level
 -6 dB/doubling of
distance to sound source
Image source: Borb (Wikipedia)

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Distance Law
• Distance law: in free-field conditions (i.e. without
room acoustics), if we do not consider air
absorption, each time the distance to the source
is doubled, the SPL drops by 6dB
Note: strictly speaking, this rule is only true for
monopoles (see “Fundamentals of Acoustics 2”)
Distance x 2 ⇒ LSPL- 6dB

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Conclusion
• Sound can propagates in gases, liquids and solids.
• Sound is a pressure wave, which travels (but not particles) at
a speed that does only depend on the transmission medium
• This wave is characterized by a time- and position-dependent
pressure and velocity
• The pressure is measured in Pascal (Pa)
• The Sound pressure level is measured in dB SPL
• Sound particle velocity ≠ speed of sound
• Among all sonic waves, only a limited amount (between 20 Hz
and 20 kHz) is hearable
• Simple relation between f, T, c und λ
• Sound obeys to the superposition principle => interferences

Fundamentals of Acoustics 1

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