LoudnessLoudness

October 18, 2006October 18, 2006

What is it??What is it??

The Process

At the Eardrum

Pressure wave arrives at the eardrum It exerts a force The drum moves so that WORK IS DONE The Sound Wave delivers ENERGY to the

EARDRUM at a measurable RATE. We call the RATE of Energy delivery a

new quantity: POWERPOWER

POWER

Wattsecond

Joule

second

energyPower Example: How much energy does a 60 watt light bulb consume in 1 minute?

J 3600 seconds 60second

joules 60

second

joules 60 watt 60

By the way …..

You BUY Joules from the power company.

We PAY for Kilowatt Hours

energytimetime

nergyKWH

e

We PAY for ENERGY!!Not Power/

More Stuff on Power

10 Watt

INTENSITY = power/unit area

10 Watts = 10 Joules per second.

Intensity

24

:

r

PI

sphereA

PI

Same energy (and power) goes through surface (1) as through surface (2)

Sphere area increases with r2 (A=4r2) Power level DECREASES with distance from the

source of the sound. Goes as (1/r2)

ENERGY

So….

To the ear ….

50m

30 watt

Area of Sphere =4r2

=4x3.14 x 50 x 50 = 31400 m2

Ear Area = 0.000025 m2

Continuing

watts.000000238power

EarAt

000025.0m

watt.00095

ear Power to

22

m

Scientific Notation = 2.37 x 10-7 watts

22

/00095.31400

30/ mw

m

wattAreaUnitPower

Huh??

Scientific Notation = 9.5 x 10-8

Move the decimal pointover by 8 places.

Another example: 6,326,865=6.3 x 106

Move decimal pointto the RIGHT by 6 places.

REFERENCE: See the Appendix in the Johnston Test

Scientific NotationAppendix 2 in Johnston

0.000000095 watts = 9.5 x 10-8 watts

The perception of loudness

The brain is not a “linear animal”. If the ear hears a sound, it sends a certain

“signal” (electrical) to the brain. The brain determines how loud the music is by

the size of this signal. The range of signals in the brain is limited and it

has to go over a huge range of loudness so it has to process the signal to be in a useful range.

It uses something called a logarithm.

Decibels - dB

The decibel (dB) is used to measure sound level, but it is also widely used in electronics, signals and communication.

It is a very important topic for audiophiles.

It is a LOGARITHMICLOGARITHMIC translation so it does what the brain does.

Decibel (dB)Suppose we have two loudspeakers, the first playing a sound with power P1, and another playing a louder

version of the same sound with power P2, but

everything else (how far away, frequency) kept the same.

The difference in decibels between the two is defined to be

10 log (P2/P1) dB       

where the log is to base 10.

?

What the **#& is a logarithm?

Bindell’s definition:

Take a big number … like 23094800394 Round it to one digit: 20000000000 Count the number of zeros … 10 The log of this number is about equal to the number

of zeros … 10. Actual answer is 10.3 Good enough for us!

Back to the definition of dB:

The dB is proportional to the LOG10 of a ratio of intensities.

Let’s take P1=Threshold Level of Hearing which is 10-12 watts/m2

Take P2=P=The power level we are interested in.

10 log (P2/P1)

An example:

The threshold of pain is 1 w/m2

1201210)10log(1010

1log 10

:PAIN of thresholdfor the rating dB

1212-

Another Example

01.1010

1

100

1

:

22

Example

Look at the dB Column

DAMAGE TO EARContinuous dB   Permissible Exposure Time      85 dB                           8 hours      88 dB                           4 hours      91 dB                             2 hours      94 dB                             1 hour      97 dB                             30 minutes    100 dB                             15 minutes    103 dB                             7.5 minutes    106 dB                             3.75 min (< 4min)    109 dB                             1.875 min (< 2min)    112 dB                              .9375 min (~1 min)    115 dB                              .46875 min (~30 sec)

Can you Hear Me???

Frequency Dependence

Why all of this stuff???

We do NOT hear loudness in a linear fashion …. we hear logarithmically

Think about one person singing.Add a second person and it gets a louder.Add a third and the addition is not so much.Again ….

Let’s look at an example. This is Joe the

Jackhammerer. He makes a lot

of noise. Assume that he

makes a noise of 100 dB.

At night he goes to a party with his Jackhammering friends.

All Ten of them!

Start at the beginning

Remember those logarithms? Take the number 1000000=106

The log of this number is the number of zeros or is equal to “6”.

Let’s multiply the number by 1000=103

New number = 106 x 103=109

The exponent of these numbers is the log. The log of {A (106)xB(103)}=log A + log B

9 6 3

Remember the definition

WattP

P

P

P

P

PP

mwattP

P

PdB

2

12

1212

2120

0

10

2)log(

20)log(10

120)log(10100

)10log(10)log(10100

)10log()log(10)10/log(10100

/10

log10

Continuing On

The power level for a single jackhammer is 10-2 watt.

The POWER for 10 of them is 10 x 10-2 = 10-1 watts.

110)10log(1010

10log10 11

12

1

dB

A 10% increase in dB!