Logarithms and Levels
Logarithms are used to compare two quantities to one another quickly with an easy frame of reference. It is particularly useful if there is a large difference in orders of magnitude between quantities as in acoustic pressure or acoustic energy calculations. We will see how useful logarithms can be in our next lesson. For now, let's concentrate on review of some of the basic principles leading up to our use of logarithms.
Unless otherwise stated, we will be working solely with logarithms that are in base 10
(Briggsian) . Some useful relationships to remember when working with logarithms are:
1. y = 10x then log10 ( y ) = x
2. log ( xy ) = log ( x ) + log ( y )

⎛x⎞
3. log ⎜ ⎟ = log ( x ) − log ( y )
⎝ y⎠

( )

4. 10 log x n = n10 log( x )

Intensity Level
In the last lesson, we defined the time average intensity in relation to the time average or rms pressure as well as the maximum acoustic pressure. p2 p2
I =
= max
2ρc
ρc
The intensity is a useful quantity because it quantifies the power in an acoustic wave, but because of the large variation in magnitudes of Intensity, it is more useful to use logarithms to compare intensities. The below table demonstrates the wide variation in Intensity for typical sounds in air.
We will start by defining a new quantity, L, the intensity level, which has units of dB.
I
L ≡ 10 log
I0
where: is the time average intensity of the sound wave.
I0 is the reference level used for comparison purposes.

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Source

Intensity (W/m2) Intensity Level (dB)

Jet Plane

100

140

Pain Threshold

1

120

Siren

1x10-2

100

Busy Traffic

1x10-5

70

Conversation

3x10-6

65

Whisper

1x10-10

20

Rustle of leaves

1x10-11

10

Hearing Threshold 1x10-12

1

The reference intensity in air is typically 1 x 10-12 W/m2. Using this simple definition you see that intensities spanning 14 orders of magnitude become intensity levels between 1 and 140.
This is an appealing scale because our ears seem to judge loudness on a logarithmic vice linear scale. Additionally, if you tried to graph various intensities, say as a function of frequency, your scale would likely only display the loudest noise with all others jammed along the abscissa.
When intensity levels are plotted, the graph becomes much more useful.

The units of decibels were constructed for intensity level definition. A “bel” was named after Alexander Graham Bell and defined:
I
"bel" ≡ log
I0
A “decibel” adopts the standard metric prefix and is 1/10th of a bel.

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Reference Intensity
We have already noted above that the reference intensity when calculating intensity level for sounds in air is conventionally 1 x 10-12 W/m2, the hearing threshold. This is not always the case. In fact for water, it is conventional to use a standard reference pressure, po. The most common reference pressure for water is 1µPa. This should not alarm you since the two can be converted using the specific acoustic impedance and assuming a plane wave.
2
p0
I0 = ρc Thus for water with a nominal density, ρ=1000 kg/m3 and the nominal speed of sound, c = 1500 m/s, the reference intensity would be:

(1 µPa )2
I0 =
= 6.67 x10 −19 W 2
3
m
(1000 kg m )(1500 m s )

Similarly, one can work backwards from the reference intensity in air and determine that the reference pressure is about 20 µPa (ρ = 1.21 kg/m3, c = 343 m/s).
Unfortunately, you must be very observant when using decibels to understand the reference level used in the calculation of an intensity level. While the numbers stated here for water and air are the most common today, up until the early 1970’s, the standard reference pressure level for sound in water was the microbar (µbar). To remove any ambiguity, intensity levels are generally stated with the reference included as follows:
L = 40 dBre 1 µPa
Of course, this puts additional burden on you when submitting answers on homework, tests and quizzes. Sound Pressure Level (SPL)
If the reference is provided as a pressure, and we know the about the pressure of the sound wave, we do not actually need to convert both to intensities because we can relate the pressure of a sound wave directly to the reference pressure using our basic rules for logarithms. p2 2
⎛ p2 ⎞ p2 I ρc ⎟
L = 10 log
= 10 log 2 = 10 log 2 = 10 log ⎜
⎜ p0 ⎟ p0 I0 p0 ⎜
⎟
⎝
⎠
ρc

A better equation for the intensity level is then:
⎛ p2
L = 20 log ⎜
⎜ p0
⎜
⎝

⎞
⎟ = 20 log p rms
⎟
po
⎟
⎠

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where, p rms =

p2 =

p2 max 2

In this form, the intensity level is often called the “sound pressure level.” The sound pressure p2 level and the intensity level must be equal provided the reference values correspond ( I0 = 0 ). ρc ⎛X ⎞
Note that the form 10 log⎜ 1 ⎟ is used for energy quantities (power, intensity). These are
⎜X ⎟
⎝ 2⎠
⎛x ⎞ sometimes called “mean squared” quantities. The form 20 log⎜ 1 ⎟ is used for acoustic pressure
⎜x ⎟
⎝ 2⎠ and other “root mean squared” quantities such as voltage.

As a quick example, a sound wave in water with an rms pressure of 100 µPa would have an intensity level or sound pressure level (in dB):
100 µPa
L = 20 log
1 µPa
L = 40 dB re 1 µPa
As stated above, the reference pressure is given in this answer so that we know the intensity level is a comparison of the intensity to the reference pressure. In the future, all intensity levels for sound in water can be assumed to be referenced to 1 µPa unless otherwise stated. For sound in air, the standard reference pressure is 20 µPa.

About the Decibel (dB)
A couple of things to note about this new unit, dB:
1) Remember that decibels are often used to deal with values that differ over many orders of magnitude thus allowing for much smaller differences in dB. For instance, a hydrophone with a source level of 120 dB emits a sound wave with a rms pressure of 1,000,000 Pa. A hydrophone with a source level of 100 dB emits a sound wave with an rms pressure of
100,000 Pa. Thus in this instance, a difference of 20 dB equals a difference of 900,000
Pa.
2) Every time you see the units of dB, you should think of a ratio. By definition, a level in dB is related to the ratio of rms pressure to a reference pressure (in water pref = 1µPa).
When expressing a sound pressure level referenced to 1 µPa, the units are noted as dB/1 µPa or dB re 1 µPa. Sound levels in air use 20 µPa as the reference level, the average human hearing threshold at for a 1 kHz signal. Acoustic signals in water were originally referenced to 1 µbar. You can show that sound levels referenced to the new 1 µPa reference level are therefore 100 dB higher than those referenced to 1 µbar.
⎛ 105 µ Pa ⎞
20 log ⎜
⎟ = 100 dB
⎝ 1 µ Pa ⎠
Later we will see how a difference of levels of two sources, in dB, is related to the ratio of the pressure (or intensity) of the two sources.

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I
, a 10 dB intensity level means that I is 10 times greater than I0
I0
and a 3 dB intensity level increase corresponds to a doubling of the energy level.

3) Since L = 10 log

4) Intensity levels and sound pressure levels both use the symbol, L. As we move through the course, we will discuss source levels, noise levels, and reverberation levels. A common procedure in the Navy is to assign a subscript such as LS for a source level.
Several standard textbook have adopted the convention of putting the subscript before the
“L.” In this case, SL would mean source level.

Working with intensity levels
For this course, we will need to work with intensity levels in many ways. Some examples of using intensity levels are given below:
Subtracting Intensity Levels

Finding the difference between two intensity levels is a little bit different. The difference in the two intensity levels represents the ratio of the intensities or pressure:

L 2 − L1 = 10 log

I2
I0

− 10 log

I1
I0

L 2 − L1 = 10 log I 2 − 10 log I 0 − [10 log I1 − 10 log I 0 ]
L 2 − L1 = 10 log I 2 − 10 log I1
L 2 − L1 = 10 log

I2
I1

or substituting in the definition for the intensity:

p2
2
L 2 − L1 = 10 log

ρc
2
p1 ρc L 2 − L1 = 10 log
L 2 − L1 = 20 log

3-5

p2
2
2 p1 p2 p1 So if a noisy sub was emitting a source level of 140 dB and a quiet sub was emitting a source level of 80 dB, the difference between the two intensity levels would be:

L 2 − L1 = 140 dB − 80 dB = 60 dB
For perspective, this represents a ratio of the intensity of both submarines To find the actual ratio (not in dB):

L 2 − L1 = 10 log

I2 or I1

L 2 − L1
I2
= 10 10 or
I1

I 2 = 10 6 ∗ I1
Or ratio of the acoustic pressures emitted:
L 2 − L1 = 20 log

p2 p1 or

L 2 − L1 p2 = 10 20 or p1 p 2 = 103 ∗ p1

In other words, the acoustic pressure of the sound wave from the louder sub is 3 orders of magnitude or a thousand times greater than that of the quiet sub.
This example illustrates why it is so much more efficient to reference all intensities or pressures to intensity levels to provide an easier comparison between numbers that can be so many orders of magnitude different.
Adding Incoherent Intensity Levels

Noise in the ocean is the combination of noise from many different sources. How can we add two intensity levels together? We want to add two intensity levels, L1 and L2, where:

L1 = 10 log

I1
I0

and L 2 = 10log

"L1 + L 2 " = 10 log but I tot = I1 + I 2

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I tot
I0

I2
I0

First, let's rewrite equation, L1 = 10 log

I1 to solve for I1
I0
I
L1
= log 1
10
I0
L1

I1

1010 =

similarly, 10

L2
10

I0

=

therefore I1 = I010

I2
I0

L1
10

and I2 = I010

L2
10

L2
⎛ L1
⎞
so I tot = I0 ⎜10 10 + 10 10 ⎟
⎝
⎠

⎛I ⎞
L tot = L1 ⊕ L 2 = 10 log ⎜ tot ⎟
⎝ I0 ⎠
L1
L2
⎛ 10
⎞
L tot = 10 log ⎜10 + 10 10 ⎟
⎝
⎠
We used the notation with a circle around the plus sign to represent the power sum of two decibel quantities.

L tot = L1 ⊕ L 2

To add intensity levels, there are two shortcuts that can be used for some problems to make it easier than using the above equation:
L1
L2
⎛ 10
⎞
1. if L1 = L 2 then L tot = 10 log ⎜10 + 10 10 ⎟ = L1 + 3 dB this is because:
⎝
⎠
L1
L1
L1
⎡ ⎛
⎛
⎞
⎞⎤
⎡ L1 ⎤
L tot = 10 log ⎜10 10 + 10 10 ⎟ = 10 log ⎢ 2 ⎜ 10 10 ⎟ ⎥ = 10 log ⎢10 10 ⎥ + 10 log [ 2] = L1 + 3 dB
⎢ ⎝
⎝
⎠
⎠⎥
⎣
⎦
⎣
⎦
The rules of logarithms, specifically the second rule above, tells us the only time it is appropriate to actually add dB would be when intensity was multiplied by some quantity as in the case of the gain provided by an amplifier. In this example, if an amplifier had doubled the intensity as if there were two source intensities, we say the amplifier provided a 3 dB increase and we simply add the 3 dB to the initial intensity level.
2. if L1>>L2 (or vice versa), then Ltot≈L1 (or vice versa). Here, “much more than” is defined as
10 dB or I1>10*I2,

3-7

Problems
1. Using the rules for logs:
a) Simplify the following relationship for this empirical sound level of a signal in water
⎛ xa yb ⎞
(dBre 1µPa): L = 20 log10 ⎜ c ⎟ dB re 1µPa .
⎜ z ⎟
⎠
⎝
b) If x=5, y=8, z=10, a=10, b=5, and c=11, what is the resulting level (in dBre 1µPa).
2. Cavitation may take place at the face of a sonar transducer when the sound peak pressure amplitude being produced exceeds the hydrostatic pressure in water.
a) For a hydrostatic pressure of 600,000 Pa, what is the highest intensity that may be radiated without producing cavitation?
b) What is the intensity level in dB re 1 µPa?
c) How much acoustic power is radiated if the transducer face has an area of 1/3 m2?
3. If P2 rms = 100 µPa and P1 rms = 25 µPa, what is:
I2
a)
=?
I1
b) L2 – L1 =?

4. If L1 = L2 = 60 dBre 1 µPa, L3 = 57 dBre 1 µPa, L4 = 50 dBre 1 µPa, and L5 = 65 dBre 1 µPa, what is Ltot, the some of all the levels.
5. What is the intensity of a 0 dB(re 1 µPa) acoustic wave in water?
6. If L1 = L2, then prove that Ltot = L1+3 dB (the 3 dB rule).
7.
a)
b)
c)
d)
e)

If P1,rms = 200 µPa and P2,rms = 10µPa, determine (assume P0 = 1µPa):
L1,
L2,
L1⊕L2,
L1-L2,
What does the previous result tell us?

8. If L1=30 dB re 1µPa and L2=65 dB re 1µPa, what is P2/P1?
9. Show that a plane wave having an effective acoustic pressure of 1 µbar in air has an intensity level of 74 dB re 0.0002 µbar.
10. Find the intensity (W/m2) produced by an acoustic plane wave in water of 120 dB sound pressure level relative to 1 µbar.

3-8

11. What is the ratio of the sound pressure in water for a plane wave to that of a similar wave in air of equal intensity? cair = 343 m/s, ρair = 1.21 kg/m3, cwater = 1500 m/s, ρwater = 1000 kg/m3. 12. If the intensity level in seawater is 160 dB re 1 µPa, what is the rms acoustic pressure in µPa? a) What is the rms acoustic pressure in µPa if the intensity level is 160 dB re 1 µbar?
b) What is the rms acoustic pressure in µPa if the intensity level is 160 dB re 0.0002 µbar?
13. Over a certain band of frequencies in the deep ocean basis, the noise level due to surface water turbulence (due to wind) is 62 dB and the noise level due to distant shipping is 65 dB. What is the total noise level?
14. The rms pressure from a low frequency sound source is 200 µPa. What is the combined rms pressure for both sources? What is the combined source level in dB re 1 µPa?

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