Architectural design considerations

1. Fundamentals of Acoustics
S.Sunaksha
b.arch

2. The Nature of a Sound Event
 Sound consists of vibrations of air molecules
 Air molecules are analogous to tiny superballs
 Sound occurs when air molecules are disturbed
and made to ricochet off of each other

3. The Nature of a Sound Event
 The ricochets cause the density of the air
molecules to oscillate
Normal Compressed
Rarefied

4. The Nature of a Sound Event
 The ricochets cause the density of the air
molecules to oscillate back and forth

5. Wave Types
Sound consists of longitudinal waves
The wave’s
oscillation is in
the same
direction as its
propagation
propagation
oscillation
Water waves are transverse waves
The wave’s
oscillation is
perpendicular to
the direction of its
propagation
propagation
oscillation

6. Sound
Propagation
Sound waves
propagate in a sphere
from the sound
source (try to imagine
a spherical slinky).
Note that the
molecules themselves
are not travelling.
What spreads is the
energy of the wave.

7. Sound Perception
 When sound waves reach the eardrum, they
are transduced into mechanical energy in
the middle ear
 The mechanical motion is transduced into
electrical current in the inner ear. The
auditory nerves interpret the current as
sound
 Speed of sound (in air):
1128 ft./sec (344 m/sec)

8. Sound Wave Plots
 Sound waves are typically represented with
molecular density as a function of time
molecular density
compressed
normal
rarefied
time

9. Music vs. Noise
Musical sounds are typically periodic – the wave repeats regularly
Noise is aperiodic – there is no repeating pattern
Sine wave
Though they don’t exist in
nature, sine waves are often
useful for demonstrating
properties of sounds
Noise
repeats

10. Properties of a Musical Event
A musical event can be described by four properties.
Each can be described subjectively, or objectively (in
terms of measured properties)
Subjective Objective
Pitch Frequency
Volume Amplitude/Power/Intensity
Timbre Overtone content
Duration in beats Duration in time

11. Frequency/Pitch
Frequency is measured in cycles per second, or Hertz (Hz)
one second
f = 2 Hz
Wavelength (l), the
distance between
corresponding points
on the wave, is the
inverse of frequency.
l
l = c
f
=
1000 ft./sec.
2 cyc./sec.
= 500 ft./cyc.

12. Frequency/Pitch
Middle A = 440 Hz
l = 2.3 ft.
frequencies
audible to
humans
<
20 Hz < 20,000 Hz (20 kHz)
l = 50 ft. l = 0.05 ft.
Sound wavelengths are significantly larger than light wavelengths

13. Waves reflect from a surface if its
height/width is larger than the wavelength

14. Waves refract around surface if the
surface dimensions are smaller than
the wavelength
This explains why we can hear sound from around corners,
but cannot see around corners:
Light wavelengths are far too small to refract around any
visible surface

15. Our Pitch Perception is
Logarithmic
Equivalent pitch intervals are perceived according to an
equivalent change in exponent, not in absolute frequency
For example, we hear an equivalent pitch class with every
doubling of frequency (the interval of an octave)
55 x 20
55 x 21
55 x 22
55 x 23
55 x 24
55 x 25
55 x 26
Frequencies of successive octaves of concert A
55 110 220 440 880 1760 3520

16. Our Pitch System is Based on
Equal Division of the Octave
12 Tone Equal Temperament –
the octave is divided into twelve equal increments
We can describe an octave by:
n/12
for n = 0 to 11
• multiply it by 2
• choosing a starting frequency
A
220
x 2
220
0
12
A#
233
x 2
220
1
12
B
247
x 2
220
2
12
C
261.6
x 2
220
3
12
C#
277
x 2
220
4
12
D
293.6
x 2
220
5
12
D#
311
x 2
220
6
12
E
329.6
x 2
220
7
12
F
349.2
x 2
220
8
12
F#
370
x 2
220
9
12
G
392
x 2
220
10
12
G#
415.3
x 2
220
11
12
Higher octaves may be created by doubling each frequency
Lower octaves may be created by halving each frequency

17. Phase
Phase = “the position of a wave at a certain time”
If two waveforms at the same frequency do
not have simultaneous zero-crossings, we say
they are “out of phase”
Two waves at the
same frequency but
different phase
Wave 1
Wave 2
Wave 1 + Wave 2
In terms of sound perception, phase can be critical or imperceptible,
as we’ll see...

18. Loudness
Loudness is related to three measurements:
• Pressure
• Power
• Intensity
All three are related to changes in sound
pressure level (molecular density)

19. Molecular Motion is Stationary
 As sound travels, molecules are not traveling with
the sound wave
 What is traveling is an expanding sphere of energy
that displaces molecules as it passes over them
 How strong is the force behind this energy wave?
 The more force is contained in a sound wave, the
greater its perceived loudness.

20. Power
Power = the amount of time it takes to do work
(exert force, move something)
Power is measured in watts, W
The range of human hearing encompasses many millions of watts.
Sound power level is also relative, not absolute. Air
molecules are never completely motionless.
Given these two difficulties, sound power levels are
measured on a scale that is comparative and logarithmic,
the decibel scale.
There are two difficulties in measuring sound power levels.

21. Logarithmic Scale
Logarithm = exponent
(an exponent is typically an integer, a logarithm not necessarily)
102 = 100 log10100 = 2
103 = 1000 log101000 = 3
102.698 = 500 log10500 = 2.698
102.875 = 750 log10750 = 2.875
Logarithms allow us to use a small range of
numbers to describe a large range of numbers

22. The Decibel Scale
 The decibel scale is a comparison of a
sound’s power level with a threshold level
(the lowest audible power level of a sine
tone at 1 kHz).
W = 10 watts
0
-12
L (dB) = 10*log (W/W )
0
10
W
Threshold (W0):
Power level of a given sound in watts, LW(dB):

23. Decibels
Typical power levels:
Soft rustling leaves 10 dB
Normal conversation 60 dB
Construction site 110 dB
Threshold of pain 125 dB
Halving or doubling sound power level results in a change of
3 dB.
For example, a doubling of the threshold level may be
calculated:
3.01 dB
LW(dB) =
Thus, a power level of 13 dB is twice that of 10 dB. A
power level of 60 dB is half that of 63 dB, and so on.

24. Pressure changes
Maximum change in sound pressure level
The amplitude level fluctuates with the wave’s oscillation.
Thus, power is the cause, pressure change is the result
(more generally: in a vibrating system, the
maximum displacement from equilibrium position)
The degree of fluctuation present in a vibrating object
Peak pressure level:

25. Pressure changes
Pressure level is measured in Newtons per square meter (N/m )
2
Threshold: 2 x 10 N/m (p )
-5 2
0
Also may be described as changes in sound pressure level
(molecular density).
For any propagating wave (mechanical, electric, acoustic, etc.)
the energy contained in the wave is proportional to the square
of its pressure change.
Pressure changes are also expressed in decibels, but in a way
that describes an equivalent change in power level:
L (dB) = 10*log10(W/W0)
W = 10*log10(p/p0)2 = 20*log10(p/p0)
This is how pressure is
measured
logmn = nlogm
There is a direct relationship between pressure and power
levels:

26. Pressure changes
In audio parlance, “amplitude” (the degree of
pressure change) is often equated with “loudness.”
The reason is that modifications to volume are made by
adjusting the amplitude of electrical current sent to an
amplifier.
But perceived loudness is actually based on power level plus
the distance of the listener from the source.

27. Power combined with distance is intensity, I,
measured in watts per square meter (W/m ).
2
Intensity
Power corresponds to the sphere of energy expanding
outward from the sound source
The power remains constant, spread evenly over the
surface of the sphere
Perceived loudness depends primarily on the sound
power level and the distance from the sound event
Intensity is also measured in decibels:
L (dB) = 10*log (I/I )
0
10
I I = 10 W/m
0
-12 2

28. Timbre
The perceived difference in sound quality when two
different instruments play at the same pitch and loudness
Sine waves are useful as demonstrations because they are a
wave with one frequency only, thus they are often termed
pure tones
Natural sounds are composed of multiple frequencies
To understand how a wave can be composed of multiple
frequencies, we can consider the behavior of a wave in a
bounded medium, such as a string secured at both ends (or
air vibrating within a pipe)

29. Timbre
When we pluck a string, we initiate wave motion
The wavelength is twice the length of the string
The perceived pitch is the fundamental, the speed
of sound divided by the wavelength

30. Timbre
This curved shape represents the
string’s maximum deviation
It’s more accurate to think of it as a
series of suspended masses (kind of like
popcorn strung together to hang on a
Christmas tree).

31. Timbre
Each suspended mass can vibrate independently.
Thus, many simultaneous vibrations/frequencies
occur along a string.
When a string is first plucked, it produces a
potentially infinite number of frequencies.

32. Timbre
Eventually, the bounded nature of the string confines wave
propagation and the frequencies it can support
Only frequencies that remain in phase after one propagation
back and forth can be maintained; all other frequencies are
cancelled out
Only frequencies based on integer subdivisions of the string’s
length, corresponding to integer multiples of the fundamental,
can continue to propagate

33. Timbre
…etc.
NOTE:
These frequencies are
equally spaced
Therefore, they do
not all produce the
same pitch as the
fundamental
Therefore, other
frequencies are
introduced
These frequencies are called harmonics

34. Timbre
 Harmonics are well known to many
instrumentalists
– Strings
– Brass

35. Timbre
 The first six harmonics are often the
strongest:
220
Fundamental
440
Octave
660
Perfect fifth
880
Octave
1100
Major third
1320
Perfect fifth
 People can learn to “hear out” harmonics

36. Timbre
 Instruments and natural sounds usually
contain many frequencies above the
fundamental
 These additional frequencies, as part of the
total sound, are termed partials
 The first partial is the fundamental

37. Timbre
 The first partial is the fundamental
 Other terms are also used
 Overtones are partials above the
fundamental (the first overtone is the
second partial)
 Harmonics are partials that are integer
multiples of the fundamental

38. The Spectrum
 Jean Baptiste Fourier (1768-1830) discovered a
fundamental tenet of wave theory
 All periodic waves are composed of a series of
sinusoidal waves
 These waves are harmonics of the fundamental
 Each harmonic has its own amplitude and phase
 The decomposition of a complex wave into its
harmonic components, its spectrum, is known as a
Fourier analysis

39. The Spectrum
It is often more useful to represent complex
waveforms with a spectral plot as opposed to a time
domain plot
=
time domain
amplitude as a function of time
spectral domain
amplitude as a function of frequency

40. Sound in Time
 Our perception of sound and music events is
determined by the behavior of frequency
and loudness over time

41. Sound in Time
 All instruments can be characterized by
changes in amplitude over time (the
envelope)
time
loudness
trumpet bowed violin harp
Changes in amplitude often correspond with changes in
frequency content...

42. Sound in Time
 Most instrument’s sound begins with an initial
transient, or attack, portion
 The transient is characterized by many high
frequencies and noise
 Example: the scraping of a bow or the chiff of
breath
 An instrument’s distinctiveness is determined
primarily by the transient portion of its sound

43. Sound in Time
 Following the transient, instruments usually
produce a steady-state, or sustained, sound
 The steady state is characterized by
– Periodicity
– Harmonic spectrum

44. The Spectrogram
Most natural sounds (and musical instruments) do not
have a stable spectrum.
Rather, their frequency content changes with time.
The spectrogram is a three-dimensional plot:
Vibraphone note at 293 Hz (middle D)
1) time
2) frequency
3) power of a given frequency (darkness level)
The instrument’s sound is characterized by the fundamental at 293 Hz and the fourth harmonic at 1173 Hz.
The attack also contains noise below 2 kHz, the tenth harmonic at 2933 Hz and the seventeenth harmonic at 4986 Hz.
Once the steady state portion sets in, the highest harmonic fades first, followed by a fading of the fundamental.

45. Localization
 The auditory system localizes events through
interaural time delay – the sound wave reaches the
nearer ear a few milliseconds before it reaches the
farther ear
 For stereo systems, using delay for localization is
impractical because it requires people to listen
from a “sweet spot”
 Localization effects are simulated through
differences in loudness

46. Localization
 In a multi-speaker system, a sound emanating
from one speaker will be localized at that speaker
 A sound produced at equal volume from two
speakers will be perceived as a “phantom image”
placed in space between them
 Changing the volume balance between two
speakers will cause the phantom image to “drift”
towards the louder speaker

47. Measurement and Perception
 Our perception of auditory events is based
on all these measurements in combination
 And more
 An auditory event may be more than the
sum of its parts

48. Measurement and Perception
 Changing the phase of components in a steady-
state tone produces no perceptible change in
sound, although the shape of the wave may change
noticeably
Phase

49. Measurement and Perception
 The behavior of components in the attack segment is likely
to be far more complex than in the steady state segment
 Changing the phase of attack components can change the
character of the attack
 Solo performance sounds different from group
performance because no two players can ever sound at
exactly the same time; thus the attack is blurred
 Since an instrument’s characteristics are defined primarily
by the attack, the phase of attack components is critical
Phase

50. Measurement and Perception
 We have discussed timbre as the result of overtone content
 It is also judged by the sound’s envelope
 Research in sound synthesis has shown the envelope shape
to be more definitive than an exact match of overtone
content
 The attack portion is critical—a faster attack can be
confused with “brightness” (more high frequency overtones)
 Considerable research has gone into the creation of “timbre
space,” a multi-dimensional plot in which timbres are
classified according to overtone content, envelope and
attack time
Timbre

51. Measurement and Perception
Loudness
While intensity is the measurement most closely correlated to
loudness, the perception of volume is based on a number of
factors, not all of them entirely measurable.

52. Measurement and perception
Perceived loudness is frequency-dependent
Equal loudness curves (Fletcher, Munson, 1930s).
Perceived equal
loudness of sine
tones
This is why
many receivers
have a
Loudness knob
Loudness

53. Measurement and perception
Perceived loudness is frequency-dependent
Loudness
Within close frequency ranges, perceived loudness is
proportional to the cube root of intensity
Two violins playing the same pitch will generate twice the
intensity of one violin, but will not sound twice as loud
To achieve twice the volume, eight violins are required

54. Measurement and perception
Perceived loudness is bandwidth-dependent
Loudness
Increasing the bandwidth (component frequency
content) of a sound makes it sound louder, even if the
intensity remains constant
Despite many efforts, no one has suceeded in
creating a definitive perceptual scaling system for
loudness

55. Measurement and Perception
Loudness
Some have argued that estimation of loudness is not
automatic (measurable), but depends on a number of
higher-level estimations of distance, import, context, etc.
…we are exceedingly well trained in finding out by our
sensations the objective nature of the objects around us,
but we are completely unskilled in observing these
sensations per se; and the practice of associating them
with things outside of us actually prevents us from being
distinctly conscious of the pure sensations.
Hermann Helmholtz, On the Sensations of Tone (1885):

56. Measurement and Perception
Conclusion
Objective measurements can tell us more about sound
events
By the same token, they give us insight into what we
don’t know
This course will examine music in technical terms
This examination will give us some new insights
It will also give us an idea of where music crosses the
barrier from the objective (acoustics) to the subjective
(magic?)