2.065/2.

066 Acoustics and Sensing

Lecture 4

Massachusetts Institute of Technology

Professor Nicholas Makris

*
 Acoustic Intensity
◮ Acoustic Intensity
Power
~ = ℜ{p~u∗ } |Intensity| =
hIi Area
~ ~
Power = Force · Velocity
◮ Power and Intensity
ZZ Note:
W = ~ · dS
hIi ~ Power = rate of change of work over time

◮ Light Bulb - omnidirectional

W = I1 S1 = I2 S2
S1 = 4πr12
S2 = 4πr22
W W
W W = IS ⇒ I= S
= 4πr 2
Source |hI~1 i| =
S1
W
|hI~2 i| =
S2
 Omni-directional Point Source
   ∗   ∗ 
~ p j pp j
hIi = ℜ p 1+ îr = ℜ 1− îr
ρ0 c kr ρ0 c kr
 2   
|p| j |p|2 j |p|2
=ℜ 1− îr = ℜ 1− îr = îr
ρ0 c kr ρ0 c kr ρ0 c
ZZ
W = ~ · dS
hIi ~
ZZ
|p|2 ~ A
= îr · dS, p = ej(kr−ωt) (for point source)
ρ0 c r
ZZ 2
A
= r 2 dΩ, dΩ = sin θdθdφ
ρ0 cr 2
4πA2
=
ρ0 c

4πA2
Power from a point source W = ρ0 c
 Decibels (dB)

Sound Pressure Level

  log10 10a = a log10 10 = a
|p|2
Lp = 10 log 10 dB re pref log10 y a = a log10 y
|pref |2
|p|2 |p|2
|pref |2
Lp Lp
|pref |2
1 0 20 13
2 3 30 15
3 5 100 20
4 6 800 29
5 7 1000 30
6 8 e ∼ 4.343
8 9 ∼ 1.26 1
9 10 ∼ 1.12 0.5
10 10
 Intensity

For point source and plane

wave:
|p|2
I∼
ρ0 c

Weber-Fechner Law
S - human sensitivity to stimulus
I - received intensity

dI
dS = = const.
Z I Z
1
dS = dI
I
S = ln I + constant
(Optics Letters 95)
 Reference Pressure
What is pref ?
In Water: pref = 1 µPa = 10−6 Pa
In Air: pref = 20 µPa = 2 × 10−5 Pa

bars = B micro bars = µB

1 µB = 0.1 Pa = 105 µPa
   2
 p 2  p 1 µB 
 1 µPa  =  1 µB · 1 µPa 

 
 p 2
y dB re 1 µPa = 10 log 
1 µPa 
   
 p 2 1 µB 2
= 10 log  + 10 log
1 µB  1 µPa
= z dB re 1 µB + 10 log(105 )2
= z dB re 1 µB + 100 dB
 Simple Sonar Equation
(free space, omni-directional point source)
  
 2
Sound Pressure Level Lp = 10 log  ppref  dB re pref

Omni-directional point source in free space

A j(kr−ωt) 2
p= re ⇒ |p|2 = Ar2 Typically, rref = 1 m
 2   2   2 
r
Lp = 10 log r2Ap2 = 10 log r2Ap2 + 10 log rref2 dB re pref
ref ref ref

Source Level
 2 
LS = 10 log r2Ap2 dB re rref and pref
ref ref
 
r2
Transmission Loss H = 10 log 2
rref
dB re rref

Simple Sonar Equation Lp = LS − H

 Power Level
(free space, omni-directional point source)

Omni-directional point source:

|p|2

A2
I= ρ0 c = r 2 ρ0 c
 Power for omni-directional
 4πA2
⇒ W = point source

 ρ0 c
W W
I= Area = 4πr 2

 
W
Power Level LW = 10 log Wref dB re Wref

Typically, Wref = 1 Watt

Relationship between Power Level and Source Level

 
W
LW = 10 log dB re Wref
Wref
 
4πA2
= 10 log dB re Wref
ρ0 cWref
   2 2 
A2 pref rref 4π
= 10 log 2 p2 + 10 log dB re Wref
rref ref Wref ρ0 c
| {z } | {z }
LS depends on pref , ρ0 and c,
which differ across different
fluid media

LW = LS − 171 dB re 1 Watt
Water:
LS = LW + 171 dB re 1 m and 1 µPa
LW = LS − 108 dB re 1 Watt
Air:
LS = LW + 108 dB re 1 m and 20 µPa