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Samenvatting 7LS8M0 Architectural Acoustics (7lS8M0)
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Overview of the lectures of the course 7LS8M0 Architectural Acoustics (7lS8M0)
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7LS8M0 – Architectural acoustics01 – Fundamentals of Acoustics
Week 1 Lecture 1
Wave function (Harmonic motion)
A wave function can be compared with a harmonic motion. Wooden panels absorb sound in a certain
frequency range. This frequency range is determined by the mass spring frequency.
(1) 𝑠𝑠(𝑡𝑡) = 𝑠𝑠̂ cos (𝜔𝜔𝜔𝜔 + Φ) 𝑠𝑠(𝑡𝑡) = Displacement [m]
1 𝑡𝑡 𝑠𝑠̂ = Displacement amplitude
(2) 〈𝑠𝑠 2 〉 = ∫0 0[𝑠𝑠(𝑡𝑡)]2 𝑑𝑑𝑑𝑑 = 𝑠𝑠̃ 2
𝑡𝑡0 𝜔𝜔 = Angular frequency = 2πf
Φ = Phase shift
This formulae can be seen in a graph, where the displacement Φ is 〈𝑠𝑠 2 〉 = Effective value
shown. The sound pressure level is how loud we experience sound 𝑠𝑠̃ = 𝑠𝑠̃ /√2 = For harmonic signals
[dB]. When we look at Figure 1, the average sound pressure is zero.
However, we do not experience it as zero. What we hear is called the
effective value. Therefore the s(t) function needs to be squared in
formula 2.
Complex numbers
Complex numbers are essential in how we calculate sound: complex
number are an easy way to compute amplitude (effective value) and
phase of a harmonic signal. Equation 1 can then be written as
equation 3, with ej as a complex number. Complex numbers are a
tool to help in calculations. They do not have a physical signal, but
merely represent one.
Figure 1: Harmonic motion
(3) 𝑠𝑠(𝑡𝑡) = 𝑠𝑠̂ 𝑒𝑒 𝑗𝑗(𝜔𝜔𝜔𝜔+Φ)
(4) 𝑒𝑒 𝑗𝑗𝑗𝑗 = cos(𝑧𝑧) + 𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠(𝑧𝑧)
The amplitude of a complex number is 𝑠𝑠̂ visualised as a line in Figure
2. The length of the line is the amplitude. The phase of the signal is
equal to the angle (𝜔𝜔𝜔𝜔 + Φ). The x-axis is the projection of the line
on the ‘Real part’. Similarly, the projection on the line on the y-axis is
the ‘Imaginary part’.
(5) |𝑠𝑠(𝑡𝑡)| = �𝑅𝑅𝑅𝑅{𝑠𝑠(𝑡𝑡)}2 + 𝐼𝐼𝐼𝐼{𝑠𝑠(𝑡𝑡)}2
= �𝑅𝑅𝑅𝑅{cos (𝜔𝜔𝜔𝜔 + Φ)}2 + 𝐼𝐼𝐼𝐼{sin (𝜔𝜔𝜔𝜔 + Φ)}2
𝐼𝐼𝐼𝐼{𝑠𝑠(𝑡𝑡)} Figure 2: Representation complex numbers
(6) / s = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 � 𝑅𝑅𝑅𝑅{𝑠𝑠(𝑡𝑡)} � for 𝑙𝑙𝑚𝑚{𝑠𝑠(𝑡𝑡)} > 0
If we take the absolute value of a complex number we get the
amplitude of the signal and if we have the phase we get the angle.
Equation 5 shows that the absolute value of a complex signal can be
calculated by the real part squared by the imaginary part squared
taken the square root of that. To get the argument of s (the phase of
the signal) we take the arctan of the signal. The reason why complex
numbers are used is that extracting the amplitude and the phase of
Figure 3: Beats
,this signal would become very difficult. When adding up we have a Z = Impedance
F = Force put on a material
signal with frequency f modulated by another frequency. Adding 2
ν = Velocity (vibration of the material)
harmonic signals is easy when using complex numbers.
Impedance
Impedance is the resistance of a material against setting it into motion. A diamond is a very hard
material. The hardness of a material is a parameter that determines the impedance of a material. Thus,
a diamond has a very high impedance. We use impedance in acoustics to describe the behaviour of
materials against an acoustic wave (absorption).
F 1 1
(7) Z= (8) 𝑍𝑍𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚−𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑗𝑗𝑗𝑗𝑗𝑗 + 𝑟𝑟 + (9) 𝑓𝑓0 =
ν 𝑗𝑗𝑗𝑗𝑗𝑗 2𝜋𝜋√𝑚𝑚𝑚𝑚
Resonance
Resonance can occur when a material has a very high velocity (vibration in the material). When a certain
force is exited on a material, the vibration of that material might become infinite. This happened at the
bridge failure, which due to resonance started to show a wave movement. In buildings acoustics,
resonance systems are used.
Week 1 Lecture 2
Acoustic Power
Acoustic power is the amount of acoustic energy. Examples are acoustic
power radiated by a sound source (loudspeaker) or vibration energy
injected into a floor (while walking on it). The total power is the time-
independent part of this product (time-average of the oscillating part of
Figure 4: Damping of an oscillation
P is zero). Acoustic power is most efficient when the force and the
vibration are in phase (example: swing). In reality, an oscillation will damp over time. This is visualised
in Figure 4. This damping can be explained in equation 12. For every room, the signature e.g. damping
characteristic is different. This is called the transient response.
1
(10) 𝑃𝑃 = 𝐹𝐹 ∗ 𝑣𝑣 (11) 𝑃𝑃𝑎𝑎 = 𝜈𝜈̂ 2 𝑅𝑅𝑅𝑅{𝑍𝑍} (12) 𝑠𝑠(𝑡𝑡) = 𝑠𝑠̂ 𝑒𝑒 𝛿𝛿𝛿𝛿 𝑒𝑒 𝑗𝑗(𝜔𝜔𝜔𝜔+Φ)
2
Fourier Transform
Joseph Fourier is a mathematician who designed the Fourier transform. When working with a periodic
(non-harmonic) signal, the Fourier transform can represent the signal by a sum of harmonics. Each
harmonic has a different weight (amplitude), which can be called the Fourier coefficients (Cn). In
equation 13 it can be seen that the signal is built up of harmonics. n is a real number; for different n
values we have different frequencies. For non-periodic signals, the Fourier transform can also be
applied. Then, an integral is applied instead of a sum: see equation 15 and 16. Instead of the Fourier
coefficient (Cn) we now have the transform signal to the frequency domain (C(ω)). This gives a
continuous function, which is called the spectrum of sound. In analysis of sound the frequency content
(spectrum of sound) is often used. In short, the Fourier transform is the transformation from time to
frequency domain and back. The possibility to analyse the frequency domain of a signal makes the
Fourier transform very relevant in acoustics.
2𝜋𝜋
(13) 𝑠𝑠(𝑡𝑡) = ∑∞
𝑛𝑛=−∞ 𝐶𝐶𝑛𝑛 𝑒𝑒
𝑗𝑗𝜔𝜔0 𝑛𝑛𝑛𝑛
(14) 𝜔𝜔0 =
𝑇𝑇
∞ 1 ∞
(15) 𝑠𝑠(𝑡𝑡) = ∫−∞ 𝐶𝐶(𝜔𝜔) 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 𝑑𝑑𝑑𝑑 (16) 𝐶𝐶(𝑤𝑤) = ∫ 𝑠𝑠(𝑡𝑡) 𝑒𝑒 −𝑗𝑗𝑗𝑗𝑗𝑗 𝑑𝑑𝑑𝑑
2𝜋𝜋 −∞
In the Fourier transform the sampling frequency is important. Even up to 2 points per wavelength, the
Fourier transform can successfully determine the frequency domain. With less points, however, it goes
, wrong. Therefore, the rules is used that in a sampling frequency, the highest frequency that can be
captured is the half of sampling frequency. The frequencies that we are able to hear are between 20
and 20.000 Hz. So if we want to record a signal that captures all these frequencies we need at least a
signal of 20.000 * 2 = 40.000 Hz. That is why 44.100 or 480.000 Hz are sampling frequencies that can
be used to record audio.
A property of Fourier transform is that if the time signal gets more narrow, the frequency content gets
wider. Clapping your hands in a room is a short time signal, and therefore allows to record a wide
frequency domain of the room. This narrow signal is called a Delta function. A Delta function ideally
only has a value at 1 instance in time (nor realistic, therefore it only exists in theory).
Impulse response and Transfer function
The impulse response is the frequency domain of an acoustic system that is obtained after the
generation of a Delta response. The impulse response shows the properties of an acoustic system (e.g.
a room). By analysing and modifying the impulse response of an acoustical system, the acoustics of the
space can be improved. The impulse response can also be used to play sound in a space, which is called
The convolution operation. With this convolution operation we can make any signal in a room audible
with the impulse response of that room. (=) Convolve a signal with the impulse response of a room to
hear the signal in the room. This is also called ‘oralise’ sound in a space. The impulse response of a room
is independent of the source that excites the room, it is a property of the room.
∞ 𝜔𝜔 1
(17) 𝑠𝑠 ′ (𝑡𝑡) = ∫−∞ 𝑠𝑠(𝜏𝜏)𝑔𝑔(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝑑𝑑 = 𝑔𝑔(𝑡𝑡) ∗ 𝑠𝑠(𝑡𝑡) (18) 𝑓𝑓 = =
2𝜋𝜋 𝑇𝑇
Week 2 Lecture 1
Wave equation in fluids
When looking at an acoustic longitudinal wave, areas of high and low density can be observed (high and
low amplitude). Individual air molecules do not follow the wave, but simply oscillate around their local
position. Sound energy is being transferred, not the molecules itself. This is fundamentally different for
wind flow, where air molecules are transported. The period in time of the piston results in a period in
space, called the wavelength of that frequency. It is important to know that the frequency of the piston
in time is a property of the piston (or of the source of sound). The wavelength of the wave in the
medium is a property both the medium and the frequency of the piston.
An impedance tube is used to measure the reflection from an absorption patch inside the tube. The
tube is designed in such a way that the sound generated by the speaker can be considered as a 1D
wave. The wave equation can be derived from 2 laws:
1. The force balance says that the force exited on the element (difference between the pressure on
the left and the right) is equal to the mass times the acceleration: Newton’s second law (F=m*a).
In this formulae the mass is expressed by the density of air (equation 19).
2. The mass balance says that the total mass inflow should be equal to the rate of change of density
(net mass inflow = rate of change of density, equation 20).
𝜕𝜕𝜕𝜕 𝜕𝜕𝑣𝑣𝑥𝑥 𝜕𝜕𝑣𝑣𝑥𝑥 𝜕𝜕𝜕𝜕
(19) = −𝜌𝜌0 (20) 𝑝𝑝0 =−
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
Wave propagation in air works differently than CFD around buildings. Therefore, equation 19 and 20
do not resemble the Navier-Stokes equations. Assumptions in acoustics are: no viscous effects, no mean
flow effects, temperature fluctuations but no heat flow (adiabatic conditions), amplitude of acoustic
variables are small (pressure amplitude < 1 Pa) such that linearization applies, we consider the
𝑣𝑣𝑥𝑥 = x-component of acoustic particle velocity
𝜌𝜌0 = Ambient density
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