EE1.el3 (EEE1023): Electronics III
Dr Philip Jackson www.ee.surrey.ac.uk/Teaching/Courses/ee1.el3

Overview of Acoustics
• Lecture 12: Principles of sound
• Lecture 13: Human hearing, loudness & pitch
• Lecture 14: Measurement of sound & noise
• Lecture 15: Sound wave behaviour
• Lecture 16: Standing waves & room modes
• Lecture 17: Resonators and waveguides
• Lecture 18: Room acoustics
• Lecture 19: Musical acoustics
• Lecture 20: Sound localisation
• Lecture 21: Revision notes
• Lecture 22: Glossary
T.1

Lecture 12: Principles of sound
The 1-D wave equation for transverse vibration on a string:
ρ
L
∂
2
∂t y
2
∂
2 y
= T
∂x
2
⇒
∂
2 y
∂t
2
− v
2
∂
2 y
∂x
2
= 0 where waves travel along the string with speed v = q
T /ρ
L
The 1-D plane wave equation for sound in a medium:
∂
2 p
∂t
2
= c
2
∂
2 p
∂x
2
⇒
∂
2 p
∂t
2
− c
2
∂
2 p
∂x
2
= 0 where waves propagate longitudinally with speed c =
√
γrT
General solutions exist of the form: p ( x, t ) = g t − x c
+ h t + x c where g ( · ) and h ( · ) can be arbitrary waveforms
T.2

Lecture 13: Human hearing, loudness & pitch
Frequency range ≈ { 20 Hz to 20 kHz }
Frequency resolution determined by critical bands
Pitch perception
Loudness perception
4000
3000
2000
1000
0
0 5 10
Frequency (kHz)
15 20
T.3

Lecture 14: Measurement of sound & noise
Like Ohm’s law ( V = IR ), acoustic impedance z is defined p = uz where z = ρ
0 c for plane waves.
Sound from a point source with spherical wavefronts: p ( r, t ) =
1 g t − r r c
Sound power is the integral of intensity , over area
I ( t ) = p ( t ) u ( t ) ,
W =
I
S
I .dS
At radius r from point source, we integrate over a sphere:
W
I =
4 πr
2
T.4

Sound pressure, intensity and power levels
Sound intensity level : SIL = 10 log
10
I
I ref
!
where I ref
= 10
− 12
W m humans in the free field
− 2 is the threshold of hearing for
!
W
Sound power level : SWL = 10 log
10
W ref where W ref
= 10 square metre
− 12
W is the equivalent intensity over one
Sound pressure level : SPL = 20 log
10 p rms
!
p ref where p ref
= 2 × 10
− 5
Pa corresponds to I ref
≈ p
2 ref
/ρ
0 c for plane wave propagation in air
T.5

Lecture 15: Sound wave behaviour
Soundfield of a sinusoidal point source in complex form: p ( r, t ) =
Q e j ( ωt − kr ) r with source strength Q , and wave number is k = ω/c = 2 π/λ
Diffraction of the wavefronts occurs at wavelengths above
(frequencies below):
D ≈
λ crit
2 where D is characteristic dimension of the obstacle or gap.
Refraction occurs at the interface between two media, with sound speeds c
1 and c
2 sin θ
1
= c
1 sin θ
2 c
2 where angles from the normal are θ
1 and θ
2 respectively.
T.6

Lecture 16: Standing waves & room modes
The room modes occur at frequencies: f ( l, m, n ) = c
2 s l
X
2
+ m
Y
2
+ n
Z
2 with the mode number ( l , m , n ) in 3D, for l, m, n ∈ { 0 , 1 , 2 , . . .
} , which count the number of nodal planes along each axis.
Axial modes have one non-zero dimension and two zeros, e.g., (1-0-0).
Tangential modes have two non-zero dimensions and one zero, e.g., (1-2-0).
Oblique modes have all three non-zero dimensions, e.g., (1-1-1).
T.7

Lecture 17: Resonators and waveguides
Resonance frequencies of standing-wave modes in an open pipe: n c f ( n ) =
2 L
O in a semi-closed pipe: f ( m ) =
(2 m − 1) c
4 L
S
For transverse modes , the longitudinal wavenumber k z depends on the transverse wavenumbers, but is only real above the cut-on frequency : f cut-on
= c
2 π q k
2 x
+ k
2 y
T.8

Lecture 18: Room acoustics
As t → ∞ , the energy density reaches equilibrium:
E
∞
=
4 W
Sαc
The reverberation time , T
60
, is the time for the SPL in a room to drop by 60 dB:
T
60
=
KV
Sα where K = 24 / ( c log
10 e ) ≈ 0 .
16 sm
− 1 gives Sabine’s Eq.
The critical distance is defined as: d c
=
1 s
R
A
4 π where R
A
= Sα/ (1 − α ) is the room absorption constant.
T.9

Lecture 19: Musical acoustics
Acoustic wave-guide model of musical instruments
x(n)
h(m) s(n)
The Helmholtz resonance frequency : f
H c
=
2 π s
S
LV
T.10

Revision summary
• Read and digest lecture notes
– summarise topics
– identify areas for further study
• Practise on examples
– review worked examples
– complete exercises
– do additional exercises in books
• Rehearse exam technique
– attempt past exam paper
T.11