Intensity, Intensity Level, and
Intensity Spectrum Level
p a2 max
I 
or,
2c
p
2
a
2
a rms
p
I 

c
c
L  10 log
I
I ref
The spectrum level is the intensity level of the sound wave
within a 1 Hz band.
I  in 1 Hz band 
I  in 1 Hz band  1Hz
ISL  10 log
 10 log
I ref  in 1 Hz band 
I ref
A Sonar System
hydrophone
band
pass
filter
#1
band
pass
filter
#2
acoustic
pressure
band
pass
filter
#3
V2
fband Rref
V2
fband Rref
V2
fband Rref
Output
System Output
band
level
band #1
band #2
band #3
frequency
• Fourier Transform of the Time Domain Signal
• Frequency Analyzer
Example
1

p  t     5cos  2 110Hz  t   3cos  2  2 110Hz  t   4 cos 3  2  110Hz  t   Pa
2

1
f  t   a 0  a1 cos  t   a 2 cos  2t   
2
 b1 sin  t   b 2 sin 2  t   
Pressure
Pa
12.5
10
7.5
5
2.5
Time
0.01
-2.5
-5
0.02
0.03
0.04
0.05
sec
Fourier Coefficients
n
fn (Hz)
Pmax (Pa)
P2rms (Pa)
I (W/m2)
L (dB)
0
0
1
0.5
3.3 x 10-7
117
1
110
5
12.5
8.3 x 10-6
131
2
220
3
3
330
4
1

p  t     5cos  2 110Hz  t   3cos  2  2 110Hz  t   4 cos 3  2  110Hz  t   Pa
2

2
max
P
5
Pressure
(Pa)
a b
4
2
Prms
3
2
n
2
n
2
Pmax

2
2
2
2
Prms
Pmax
I 

c 2c
1
110
220
330
440
f (Hz)
Band Level
BL  10log
I  in a 1 Hz band  f
I tot
 10log
Iref
Iref
ISL
ISLave
BL  10 log
I  in a 1 Hz band  1Hz
f
 10 log
I ref
1Hz
BL  ISLave  10 log  f 
f1
f2
frequency
where Δf  f 2  f1
The Band Level is the intensity level over a band other than 1 Hz.
Example
60
Intensity
Spectrum
Level
(dB)
50
30
100
200
282
355
frequency (Hz)
• Using the plot of ISL as a function of frequency
shown, calculate
– the band level of every band
– the total band level
– the total band level in a band from 150 Hz to 300 Hz
Types of Spectrums
tonals
Spectrum
Level
broadband
noise
frequency
BL tot   ISLave  10 log  f BB    L tonal #1  L tonal #2  ...
Example
ISL (dB)
Spectrum containg both noise and tones
70
60
50
40
30
20
10
0
450
500
550
600
Frequency (Hz)
What is the total BL?
650
Common Bandwidths
• Constant Bandwidth
f = constant
• Proportional Bandwidth
– Octave Bandwidth
60
Spectrum
Level
(dB)
50
30
• f2 = 2 1 f1
– Half Octave Bandwidth
• f2 = 21/2 f1
• Center Frequency
f c  f1f 2
200
100
fc=141
282
355
frequency (Hz)
Preferred Octave Bands
Why do we care?
Source Level (SL) and Noise Level (NL)
are both examples of Band Levels (BL)
where the frequency band(s) are defined
by the frequencies of our Sonar System