The Science of Sound
Module 1.1: Physics of sound .......................................................................................................................... 6
Chapter 1: What is sound?....................................................................................................................... 6
1.1 What is a sound wave? ........................................................................................................................... 6
1.2 Sources of sound ....................................................................................................................................... 6
1.3 Wanted and unwanted sound ............................................................................................................. 6
1.4 Distance, speed, and velocity ............................................................................................................... 6
1.6 Force and acceleration ........................................................................................................................... 6
1.7 Pressure........................................................................................................................................................ 7
1.9 Work and energy ...................................................................................................................................... 7
1.10 Power ............................................................................................................................................................ 8
Chapter 2: Vibrating systems ................................................................................................................. 9
2.1 Simple harmonic motion ....................................................................................................................... 9
2.2 Energy and damping ............................................................................................................................... 9
2.3 Simple vibrating systems ...................................................................................................................... 9
2.4 Systems with two or three masses ................................................................................................. 10
2.5 Systems with many modes of vibration ....................................................................................... 11
2.6 Vibrations in musical instruments ................................................................................................. 11
2.7 Complex vibrations: vibration spectra ......................................................................................... 11
Chapter 3: Waves ...................................................................................................................................... 12
3.1 What is a wave? ...................................................................................................................................... 12
3.2 Progressive waves ................................................................................................................................ 12
3.3 Impulsive waves; reflection .............................................................................................................. 12
3.4 Superposition and interference....................................................................................................... 13
3.5 Sound waves ............................................................................................................................................ 13
3.6 Wave propagation in two or three dimensions ........................................................................ 14
3.7 The Doppler effect................................................................................................................................. 14
3.8 Reflection .................................................................................................................................................. 15
3.9 Refraction ................................................................................................................................................. 15
3.10 Diffraction ................................................................................................................................................. 15
3.11 Interference ............................................................................................................................................. 15
Chapter 4: Resonance ............................................................................................................................. 15
4.1 Resonance of a mass-spring vibrator ............................................................................................ 15
, 4.2 Phase of driven vibrations ................................................................................................................. 16
4.3 Standing waves on a string ................................................................................................................ 16
4.4 Partials, harmonics and overtones................................................................................................. 17
4.5 Open and closed pipes ......................................................................................................................... 17
4.6 Acoustic impedance.............................................................................................................................. 17
4.7 Helmholtz resonator ............................................................................................................................ 18
4.8 Singing rods and wineglasses........................................................................................................... 18
4.9 Self-excitation ......................................................................................................................................... 18
4.10 Sympathetic vibrations: soundboards .......................................................................................... 18
Lecture Module 1.1 .................................................................................................................................. 19
Module 1.2: Vibroacoustics and musical instruments ......................................................................... 20
Chapter 10: String instruments........................................................................................................... 20
10.1 Summary ................................................................................................................................................... 20
10.2 Glossary ..................................................................................................................................................... 20
Chapter 13: Percussion instruments................................................................................................. 21
13.1 Summary ................................................................................................................................................... 21
13.2 Glossary ..................................................................................................................................................... 21
Lecture Module 1.2 .................................................................................................................................. 22
1.2.1 Violin strings....................................................................................................................................... 22
1.2.2 Mallets ................................................................................................................................................... 22
1.2.3 Transmission ...................................................................................................................................... 23
1.2.4 Sound radiation ................................................................................................................................. 25
Module 2.1: Sound perception ...................................................................................................................... 26
Chapter 5: Hearing ................................................................................................................................... 26
5.1 Range of hearing .................................................................................................................................... 26
5.2 Structure of the ear ............................................................................................................................... 26
5.5 Binaural hearing and localization ................................................................................................... 27
5.7 Logarithms in sound and music ...................................................................................................... 28
Chapter 6: Sound pressure, power and loudness ......................................................................... 28
6.1 Decibels ..................................................................................................................................................... 28
6.2 Sound intensity level ............................................................................................................................ 28
6.3 Sound pressure level ............................................................................................................................ 29
6.5 Loudness level ........................................................................................................................................ 29
Chapter 7: Pitch and timbre ................................................................................................................. 29
7.1 Summary ................................................................................................................................................... 29
, 7.2 Glossary ..................................................................................................................................................... 30
Lecture Module 2.1 .................................................................................................................................. 31
2.1.1 Anatomy of the ear ........................................................................................................................... 31
2.1.2 Spatial hearing ................................................................................................................................... 32
2.1.3 Pitch ....................................................................................................................................................... 32
2.1.4 Timbre ................................................................................................................................................... 33
2.1.5 Loudness .............................................................................................................................................. 33
2.1.6 Masking and critical bands ........................................................................................................... 33
Module 2.2: Acoustic communication ........................................................................................................ 34
Chapter 15: Speech production ........................................................................................................... 34
15.1 The vocal organs .................................................................................................................................... 34
15.2 The larynx and the vocal folds ......................................................................................................... 34
15.3 The vocal tract ........................................................................................................................................ 35
15.4 Articulation of speech .......................................................................................................................... 36
15.5 Resonances of the vocal tract: formants ...................................................................................... 37
15.6 Models of the vocal tract .................................................................................................................... 37
Chapter 16: Speech recognition, analysis and synthesis ........................................................... 38
16.1 Summary ................................................................................................................................................... 38
16.2 Glossary ..................................................................................................................................................... 38
AR Articles .................................................................................................................................................. 39
Animal communication: Sound and sound signal production .......................................................... 39
Animal communication: Sound signal propagation & reception ..................................................... 42
Lecture Module 2.2 .................................................................................................................................. 45
2.2.1 Human vocal organs ........................................................................................................................ 45
2.2.2 Vowels ................................................................................................................................................... 46
2.2.3 Consonants .......................................................................................................................................... 46
2.2.4 Speech recognition, analysis and synthesis ........................................................................... 46
2.2.5 Speech intelligibility and room acoustics ............................................................................... 46
2.2.6 Animal acoustic communication ................................................................................................ 47
Module 3: Room and electro-acoustics...................................................................................................... 49
Chapter 23: Auditorium acoustics ..................................................................................................... 49
23.1 Sound propagation outdoors and indoors .................................................................................. 49
23.2 Direct, early and reverberant sound ............................................................................................. 49
23.3 Direct and early sound: the precedence effect .......................................................................... 49
23.4 Reverberant sound ............................................................................................................................... 50
, 23.5 Calculation of the reverberation time ........................................................................................... 50
23.6 Air absorption ......................................................................................................................................... 50
23.7 Criteria for good acoustics ................................................................................................................. 51
23.8 Concert halls ............................................................................................................................................ 51
23.9 Background noise .................................................................................................................................. 52
23.10 Avery Fisher hall: a case study .................................................................................................... 52
23.11 Variable acoustics ............................................................................................................................. 52
23.12 Churches ............................................................................................................................................... 52
23.13 Classrooms .......................................................................................................................................... 53
Chapter 24: Electronic reinforcement of sound ............................................................................ 53
24.1 Sound sources in a room .................................................................................................................... 53
24.2 Sound fields.............................................................................................................................................. 53
24.3 Power considerations .......................................................................................................................... 54
24.4 Loudspeaker placement ..................................................................................................................... 55
24.5 Loudspeaker directivity...................................................................................................................... 55
24.6 Acoustic feedback .................................................................................................................................. 56
24.7 Equalization ............................................................................................................................................. 56
24.8 Time delay ................................................................................................................................................ 56
24.9 Enhancement of reverberation........................................................................................................ 56
24.10 Microphone placement and mixing........................................................................................... 57
24.11 Reinforcement for the hearing impaired ................................................................................ 57
24.12 Outdoor sound systems ................................................................................................................. 58
Chapter 25: Small room home listening rooms and recording studios ................................ 58
25.1 Acoustics of small rooms .................................................................................................................... 58
Lecture Module 3 ...................................................................................................................................... 59
3.1 Reverberation time ............................................................................................................................... 59
3.2 Sound absorption .................................................................................................................................. 60
3.3 Direct and diffuse sound fields ........................................................................................................ 61
3.4 Loudspeakers .......................................................................................................................................... 61
3.5 Small rooms ............................................................................................................................................. 62
3.6 Room design ............................................................................................................................................ 62
Module 4: Environmental acoustics ........................................................................................................... 64
Chapter 30: Noise in the environment.............................................................................................. 64
30.1 Summary ................................................................................................................................................... 64
30.2 Glossary ..................................................................................................................................................... 64
,Chapter 32: The control of noise ........................................................................................................ 64
32.1 Summary ................................................................................................................................................... 64
32.2 Glossary ..................................................................................................................................................... 65
AR Articles .................................................................................................................................................. 65
Computational atmospheric acoustics ........................................................................................................ 65
Experimental study of sound propagation in a street – Picaut et al. .............................................. 65
Predicting outdoor sounds............................................................................................................................... 66
Lecture Module 4 ...................................................................................................................................... 66
4.1 Ground effect ........................................................................................................................................... 67
4.2 Meteorological effects.......................................................................................................................... 69
4.3 Screening .................................................................................................................................................. 71
4.4 Urban effects ........................................................................................................................................... 72
, Module 1.1: Physics of sound
Chapter 1: What is sound?
Sound is described as 2 different things:
An auditory sensation in the ear
The disturbance in a medium that can cause this sensation
1.1 What is a sound wave?
All waves carry information from one point to another, and transport energy. Sound waves
travel in a solid, liquid, or gas.
Longitudinal waves: back-and-forth motion of air is in
the direction of travel of the sound. As the wave
travels the air pressure changes by a slight amount,
and this allows our ears to detect sound. Figure 1. Longitudinal wave.
Transverse waves: back-and-forth motion is perpendicular
to the direction of wave travel.
1.2 Sources of sound Figure 2. Transverse wave.
Sound can be produced by a number of different processes, such as:
Vibrating bodies: displaces air and causes the local air pressure to increase and decrease
slightly, these fluctuations travel outwards as a sound wave.
Changing airflow: air flow increases or decreases resulting in a sound wave (vocal folds
open and close when speaking).
Time-dependent heat sources: explosion produces a bang due to the expansion of the air
caused by its rapid heating (same for thunder).
Supersonic flow: shock waves result when a supersonic airplane or a speeding rifle bullet
forces air to flow faster than the speed of sound.
1.3 Wanted and unwanted sound
Unwanted sounds are often referred to as noise. Control of environmental noise is important but
only got important recently. The problem of this is that noise differs to different person (person
at a party and their neighbors). Unwanted noises can be stopped by soundproofing a room.
1.4 Distance, speed, and velocity
The difference between speed and velocity is that the velocity contains a direction. We calculate
the velocity with:
𝒔𝒔
𝒗𝒗 =
𝒕𝒕
Where v = velocity [m], s = distance [m] and t = time [s]. For average speed(vav) over a complete
journey you add all distances and all times. In sound waves the velocity often has only one
direction. Speed is often measured with a stroboscopic lamp.
1.6 Force and acceleration
The force applied to an object can be calculated with Newton’s second law of motion:
𝑭𝑭 = 𝒎𝒎 ∙ 𝒂𝒂
Where F = force [N], m = mass [kg] and a = acceleration [m/s2].
6|The Science of Sound
,Acceleration refers to a change in speed. It is defined as the rate of change of speed. Average
acceleration can be calculated with:
∆𝒗𝒗
𝒂𝒂𝒂𝒂𝒂𝒂 =
∆𝒕𝒕
Where aav = average acceleration [m/s2], ∆𝑣𝑣 = change of speed [m/s] and ∆𝑡𝑡 = time interval [s].
This means that a = 0 when the speed is constant since there is no change in speed nor is there a
time interval. If an object is in a free fall a = 9.81 m/s2. Acceleration can also decrease speed (e.g.
throw object upward).
1.7 Pressure
When we weigh an object we can consider the entire force of gravity acting at one point, or in
other words; the center of gravity or center of mass, even though the forces are applied to
different places. Other times the distribution of forces is important, otherwise called; pressure:
𝑭𝑭⊥
𝒑𝒑 =
𝑨𝑨
Where p = pressure [Pa] or [N/m ], 𝐹𝐹⊥ = force perpendicular to the surface [N] and A = area
2
surface [m2].
Fluids exert forces on the walls of their containers and anything immersed in them, the pressure
acts perpendicular to all surfaces. The pressure at any point in an open container of fluid is
determined by the weight of the fluid above that point. The buoyancy (drijfvermogen) of an
object is due to the excess upward pressure on its bottom surface.
Slow variations in atmospheric pressure are indicative of changing weather. Sound waves
consist of very small but rapid variations in pressure.
1.9 Work and energy
Work is done when a force is applied to an object that moves, and is calculated with:
𝑾𝑾 = 𝑭𝑭𝒈𝒈 ∙ 𝒅𝒅
Where W = work [J], Fg = force of gravity calculated [N], d = distance moved parallel to the force
[m].
𝑭𝑭𝒈𝒈 = 𝒎𝒎 ⋅ 𝒈𝒈
Where Fg = force of gravity calculated [N], m = mass [kg] and g = 9.81 m/s2.
If an object falls a vertical distance the work done by gravity, or raising an object to an height
you calculate the work with:
𝑾𝑾 = 𝒎𝒎𝒎𝒎𝒎𝒎
Where W = work [J], m = mass [kg], g = 9.81 m/s2 and h = height [m].
Energy is a central idea in all sciences. There are many forms; mechanical, electrical, thermal,
chemical, radiant and nuclear. Acoustics mostly focusses on mechanical energy which is closely
related to work. Work is the transfer of energy, systems with mechanical energy have the
potential to do work. Vibrating systems have mechanical energy, it is carried by the moving
molecules in a sound wave. Energy of motion is referred to as kinetic energy, stored energy is
referred to as potential energy which is the capacity to do work.
7|The Science of Sound
,Kinetic energy, e.g. a baseball flying through the air, is calculated with:
𝟏𝟏
𝑲𝑲𝑲𝑲 = 𝒎𝒎𝒗𝒗𝟐𝟐
𝟐𝟐
Where KE = kinetic energy [J], m = mass [kg] and v = speed [m/s].
Potential energy, e.g. a block of wood lifted to a height, is calculated with:
𝑷𝑷𝑷𝑷 = 𝒎𝒎𝒎𝒎𝒎𝒎
Where PE = potential energy [J], m = mass [kg], g = 9.81 m/s2 and h = height [m].
Potential energy of a spring is calculated with:
𝟏𝟏
𝑷𝑷𝑷𝑷 = 𝑲𝑲𝒚𝒚𝟐𝟐
𝟐𝟐
Where PE = potential energy [J], K = spring constant [N/m], y = displacement of the spring [m].
The potential energy of a bottle of gas whose pressure exceeds atmospheric pressure by a small
amount is calculated with:
𝟏𝟏 𝑽𝑽 𝟐𝟐
𝑷𝑷𝑷𝑷 = 𝒑𝒑
𝟐𝟐 𝑷𝑷𝟎𝟎
Where PE = potential energy [J], V = volume [m3], P0 = atmospheric pressure [Pa], p = amount of
exceeded pressure [Pa].
The potential energy of a displaced guitar string at its midpoint is calculated with:
𝟐𝟐𝟐𝟐 𝟐𝟐
𝑷𝑷𝑷𝑷 = 𝒚𝒚
𝑳𝑳
Where PE = potential energy [J], T = tension of the string [N], L = length [m] and y = displaced
distance [m]. When the string is released this energy is changed into kinetic energy, and after
this it changes back and forth between KE and PE.
Analysis of motion is facilitated by considering the way in which one form of energy is
converted into another. For example, lifting an object (gives PE) and then dropping it (acquires
KE). The speed of this can be calculated with an simplification of KE = PE:
𝒗𝒗 = �𝟐𝟐𝟐𝟐𝟐𝟐
Where v = speed [m/s], g = 9.81 m/s2 and h = height [m].
1.10 Power
Power is the rate at which work is done and is calculated with:
𝑾𝑾
𝑷𝑷 =
𝒕𝒕
Where P = power [W] or [J/s], W = work [J] and t = time [s]. For electrical equipment we use
Watt as well, a 100 W lamp convert electrical anergy (to heat and light) at the rate of 100 J/s.
Another unit to measure big amount of this is the kWh, the same formula is used as above but W
= work [KJ] and t = time [hour].
8|The Science of Sound
,Chapter 2: Vibrating systems
All vibrating systems (e.g. motion of tides and trees swaying in the wind) have some things in
common. The motion repeats in each regular time interval; the period, and a type of force
constantly acts to restore the system toward its point of equilibrium (evenwicht).
2.1 Simple harmonic motion
To calculate the force to stretch most springs not too far you use:
𝑭𝑭 = −𝑲𝑲𝑲𝑲
Where F = force [N], K = spring constant [N/m] and y = displacement of the spring [m]. This
system is in equilibrium when the spring is vertical and the Fg stretches the amount by a
constant amount and the force of the spring upwards is the same. The – sign shows us that when
the mass is below its equilibrium position, the F will be upward and thus positive. In this case
the force could be called restoring force. This always acts in a direction to restore its
equilibrium.
This motion is called a simple harmonic motion. In this system the period is independent of the
amplitude (maximum value of y) of the vibration. The frequency is the number of oscillations
per second which is calculated with:
𝟏𝟏
𝒇𝒇 =
𝑻𝑻
Where f = frequency [Hz] and T = period of one vibration [s].
When a mass-spring system is vibrating you calculate the frequency with:
𝟏𝟏 𝑲𝑲
𝒇𝒇 = �
𝟐𝟐𝟐𝟐 𝒎𝒎
Where f = frequency [Hz], K = spring constant [N/m] and m = mass [kg].
2.2 Energy and damping
In figure 3 you see that v2 is maximal when y2 is minimal
and vice versa. This means that the total mechanical
energy is constantly changing from kinetic to potential
and back to kinetic.
All real vibrating systems lose mechanical energy as a
result of friction or e.g. heat. Unless this energy is
renewed, the amplitude will decrease. When a system
loses energy during the amplitude will lover. The dashed
line in figure 4 shows this change in amplitude. This is
Figure 3. Graphs of simple harmonic motion:
called the envelope or decay curve. A vibrating system upper graph is displacement vs time and lower
whose amplitude decreases in this way is said to be graph is speed vs time.
damped and the rate of decrease is het damping constant.
2.3 Simple vibrating systems
Three examples of simple vibrating systems are:
1. Pendulum (small angle). You can calculate this with:
Figure 4. Damped vibrator.
9|The Science of Sound
, 𝟏𝟏 𝒈𝒈
� 𝒇𝒇 =
𝟐𝟐𝟐𝟐 𝒍𝒍
Where f = frequency [Hz], g = 9.81 m/s2 and l = length of string [m]. As long as 𝑥𝑥 ≪ 𝑙𝑙, in
which x is the distance from the 0 point.
2. A spring of air. See figure 5, you can calculate this with:
𝟏𝟏 𝜸𝜸𝜸𝜸𝜸𝜸
𝒇𝒇 = �
𝟐𝟐𝟐𝟐 𝒎𝒎𝒎𝒎
Where f = frequency [Hz], 𝛾𝛾 = heat capacity ratio (1.4 for air) [-], p =
gas pressure [Pa], A = area cylinder [m2], m = mass of the piston
(zuiger) [kg] and l = length cylinder [m]. Figure 5. Piston.
3. A Helmholtz resonator. This is used to analyze musical sounds. The
mass of the air in the neck serves as the piston and the air in the
larger volume V as the spring, see figure 6. You calculate f with:
𝒗𝒗 𝒂𝒂
𝒇𝒇 = �
𝟐𝟐𝟐𝟐 𝑽𝑽𝑽𝑽
Where f = frequency [Hz], v = speed of sound (344 m/s), a = area of
the neck [m2], V = volume of the resonator [m3] and l = length of the
neck [m]. Figure 6. Helmholtz
resonator.
The Helmholtz resonator can also be thought of as having a mass and
𝝆𝝆𝒂𝒂𝟐𝟐 𝒗𝒗𝟐𝟐
spring constant that are 𝒎𝒎 = 𝝆𝝆𝝆𝝆𝝆𝝆 and 𝑲𝑲 = where 𝜌𝜌 = the density of air.
𝑽𝑽
The resonators can come in multiple shapes and sizes, e.g. blowing air across an empty
pop bottle.
2.4 Systems with two or three masses
The previous systems all had a single coordinate which is sufficient to describe their motion, or
in other words they have one degree of freedom. Systems with more degrees of freedom have
more than one natural mode, with each its own frequency.
Consider the systems a and b in figure 7. This system has 2 ‘normal’ or independent modes.
Assuming equal masses and springs with the same K the frequencies can be calculated with:
𝟏𝟏 𝑲𝑲 𝟏𝟏 𝟑𝟑𝟑𝟑
𝒇𝒇𝒂𝒂 = � 𝒇𝒇𝒃𝒃 = �
𝟐𝟐𝟐𝟐 𝒎𝒎 𝟐𝟐𝟐𝟐 𝒎𝒎
Figure 7. Two modes of vibration of a two mass vibrator.
There are many combinations possible of the 2 modes in figure 7. Depending on the system, the
springs can move transversal(move at right angle of the spring) or longitudinal(move in the
direction of the springs). In a transversal vibration the masses can move in the same but also in
the opposite direction.
The same things apply to 3 mass systems. The independent mode is often called the normal
mode. Getting the system to vibrate in a single normal mode requires special care. This can be
10 | T h e S c i e n c e o f S o u n d
