CAE 334/502 Lecture 2a, Spring 2014

8/12/2019 CAE 334/502 Lecture 2a, Spring 2014

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CAE 334/502Lecture 2b

Simple Sources, Sound Levels of SimpleSources,

Acoustic Spectra, Frequency Bands


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Objectives

Understand fundamental sound sources Understand the sound level relations for

simple sources

Understand the concept of an acoustic

spectrum

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Sources of SoundThere are three basic types of sound sources that we

use in acousticsnearly every real source isapproximated by one of these:

Point Source A source that is small in all dimensions compared to the

measurement distance From far away all sources act like point sources

Line Source A source that is long in one dimension compared to the

measurement distance Long trains, lines of traffic, long HVAC ducts

Plane Source A source that is long in two dimensions compared to the

measurement distance Close to large surfaces such as walls, windows, doors

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The Point Source A source that is small in all dimensions

compared to the measurement distance is calleda point source. The source can be considered to be a single point at

the center of the radiating object.

Sound energy from a point source spreads spherically

as it moves away from the source

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Omnidirectional Point Source

If the source radiates power equally in alldirections, we call it an omnidirectional pointsource The intensity will be constant on any sphere

surrounding the source and the area of a sphere is

S=4

r

2

so we can write

This is known as the inverse square law or law ofspherical spreading

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omni 2 2

1( ) 10log

4 4I W

W WI r L L

S r r


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The Far and Near Field

When we are far enoughaway from a source thatspherical spreadingoccurs and the sourcelooks like a point source,

we are said to be in thefar field.

When we are very close

to the source, sphericalspreading does not occurand we are said to be inthe near field. log r


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Free and Reverberant Field The region of the far field where reflections can be

ignored is called the free field. The region of the far field where reflections cannot be

ignored is called the reverberant field

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Most rooms

have a

reverberantfield far from

the source


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Waves from a Point Source

The waves that emanate from a point sourceare spherical and close to the source there is

much curvature to the waves.

But, far from the source the curvature is small

and so the waves turn into plane waves

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Initially

Spherical

Turn into

Plane Waves


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Point Source Sound Pressure Level

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2

0

0

2

2

10log400 4

110log

4

20log

0.1 dB ( in m)

10.5 ( in ft)

11 ( in m)

0.5 ( in ft)

For a point source, (like for a plane wave) so

and with

or

Notice that dro

rms

p I

p W

p W

p

r

r

r

r

pI

c

c WL L I

r

L Lr

L L r

L

ps 6 dB for every doubling of r


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Directional Point Sources

While energy from a non omnidirectional point source

spreads spherically, it is not necessarily the same in all

directions

We characterize this directionality by adding Q(,) to the

power-intensity relation.

When Q >1, more energy goes that way, if Q


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Directional Point Source Level

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2

2

,,

4

,, 10log4

, , 20log

0.1 dB ( in m)

10.5 ( in ft)

11 ( in m)0.5 ( in ft)

For a directional point source:

so

or

p W

p W

r

r

rr

WQI

r

QL Lr

L L D r


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Line or Cylindrical Source Sometimes a source is too big

in one dimension to look like apoint and instead looks like aline A long train or line of traffic

cars,

Sound from the sides of a longHVAC duct

We call this a line or cylindricalsource

At a distance rfrom a linesource of lengthL and power W,Iwill be given by:

This is called the Inverse Law orCylindrical Spreading

For a cylindrical source thesound pressure is then relatedto the sound power level by

2cyl

WI

r L

110log

2

10log

p W

p W

p

r

r

r

r

L L

rL

L L rL

L r

0.1 ( in m)

10.5 ( in ft)

11 ( in m)

0.5 ( in ft)

drops 3 dB for every doubling of


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Planar Source A general equation

relatingIto W at a

distancezfrom planar

source of area Sand

directionality Q is

GetLpfromLpLI

Far from the source

zterm dominates and

plane acts like a point

source

Close to the source

zterm is negligible and

the intensity is constant

with distance

2

4

4

WQI

SQz

24

WQI

z

WI

S

I(z)

z

Plane Area S


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Change in level from r1to r2

Point Source

For r2= 2r1, DLp= - 6 dB, a 6db drop per doubling of

distance

Line Source

For r2= 2r1, DLp,line= - 3 dB, a 3db drop per doubling ofdistance

2

1 1

2

2 2

1,

2

10 log 20 log

10log

p

p line

r rL

r r

rL

r

D

D


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Example 2.6

The sound pressure level of an omni-directional source is measured to be 80 dB

at a distance of 1m from its center.

A) What is the sound level at a distance of 5 m

B) What is the object sound power?

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Example 2.6 The sound pressure level of an omni-directional

source is measured to be 80 dB at a distance of1m from its center.

A) What is the sound level at a distance of 5 m

B) What is the object sound power?

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1

2

,2 ,1

2

0.1( 120)

120 log 20 log 145

80 14 66

1 110 log 80 10 log 914 4

10 1.3

dB

dB

dB

mWW

p

p p p

W p

L

rLr

L L L

L Lr

W

D

D


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Tonal and Broadband Sources

Most sound sources are not pure tones Tonal sources combine distinct strong tones

Voice and most musical instruments

Tonal characteristics change quickly in time

Rotating machinery Tones related to rotation rates and gear ratios

Broadband sources have no distinct strong tones

Flow induced hisses, Wind and Surf noise

The plot of energy or amplitude vs frequency

is called the spectrumof a source


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Broadband and Tonal Spectra

Broadband

Tonal


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Tonal Sources

The base frequency is called fundamental or 1stharmonic

Higher multiples are called overtones or the higher harmonics

(2nd, 3rd)

For rare sources the overtones might not be not direct multiples

(i.e. are not harmonic)

The spectrum has distinct peaks that indicate the fundamentaland the overtones of the source

Brain picks these out and a tonal noise is usually more annoying

than a broadband noise of the same level

Musical Instruments are examples of tonal sounds

The relative amplitude and phase (time synchronization) of theovertones determines the sound or timbre (pronounced tamber) of

the instrument

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Fourier Analysis

The mathematician/physicist Joseph Fourier

showed that all periodic signals can be describedby a superposition of a fundamental andharmonics of varying amplitudes and phases Fourier synthesis was the basis of all early electronic

synthesizers

This concept was extended to include non-periodicsignals as well, which are described a continuoussmear of frequencies and are analyzed using integrals(the Fourier Transform)

Fourier analyzers which compute the spectrumusing Fourier Transforms are some of the mostimportant tools used in acoustics and vibration

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Fourier Transform

The Fourier Transform is used to obtain thespectrum from a general time signal by

correlating (multiplying and averaging) the

time signal with a complex exponential of a

given frequencythe result is the spectralamplitude (i.e. amount of energy) at that

frequency:

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( ) ( ) j tP p t e dt


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Middle C (262 Hz) for 4 Instruments

Spectra (amplitude vs frequency)

1000 2000 3000 4000

-60

-40

-20

0

Trumpet

1000 2000 3000 4000

-60

-40

-20

0

Clarinet

1000 2000 3000 4000

-60

-40

-20

0

Flute

1000 2000 3000 4000

-60

-40

-20

0

Organ


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Middle C (262 Hz) for 4 Instruments

Spectra (amplitude vs frequency)

1000 2000 3000 4000

-60

-40

-20

0

Trumpet

1000 2000 3000 4000

-60

-40

-20

0

Clarinet

1000 2000 3000 4000

-60

-40

-20

0

Flute

1000 2000 3000 4000

-60

-40

-20

0

Organ


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0 500 1000-0.5

0

0.5Trumpet

0 200 400-0.5

0

0.5Clarinet

0 200 400-0.5

0

0.5Flute

0 200 400-0.5

0

0.5Organ

Middle C for Different Instruments

Amplitude Vs Time


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Band Analysis

The audible frequency range is too big Webreak it into regions called bands defined bytheir lower frequency,fL, upper frequency,fU, and center frequencyfC

fCis the geometric mean offLandfU

Power quantities are computed as totalenergy in the bands Total W, I, and p2nottotal LW, LI, or Lp Totalnot average. As bands get wider, band

levels increase

C L Uf f f


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Octave Bands

Octave bands have center frequencies that

differ by an octave which is a factor of 2

The standardized octave band centerfrequencies (defined in ANSI S1.6) are at 16,31.5, 63, 125, 250, 500, 1 k, 2 k, 4 k, 8 k, 16kHz The six bands from 125 Hz to 4 kHz are the most

used Arch. ACS

Just remember there is a band at 1kHz and work

from that.

See Table 2.1 of LAA for the upper, lower, and centerfrequencies for octave bands

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Octave Band Analysis Filters

Below is a plot of the spectrum of some of the

standard octave band filters used in acoustics

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125 250 500 1000 2000 4000

-80

-60

-40

-20

0

f [Hz]

A

ttenuation

[dB

]


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1/nth octave bands

For a 1/nth octave band the center

frequencies differ by a factor of 21/n

fL=fC*2-1/2n fU=fC*2

1/2n

Octave bands have n=1

1/3 octave bands which have n=3

1/3 octave bands are used when higherresolution is needed

See Table 2.1 of LAA for the upper, lower, and centerfrequencies for 1/3 octave bands

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Narrowband Analysis

With the advent of microprocessors it became

practical to determine frequency spectra of

signals using Fourier Analysis

Fourier Analysis gives rise naturally to narrow

bands of equal width (something like 1 Hz, 2 Hz,5 Hz, etc). Because these are narrow, this is

usually called narrowband analysis

The important thing to note is that narrow-bands

are of equal frequency width at all frequencies

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Combining Band Levels

Bands combine as incoherent sources

Converting 1/3 to Octave bands (OB)

Convert three 1/3 OB levels toI orW, add the three and

then convert the sum back to a level

OB level will always be higher than 1/3 OB levels

Combine octave bands to single number

Convert octave bands toI orW,add, convert back

This level is what a simple sound level meter would read

Combine narrowband values within each nth octave

to get the octave band levels the same way


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Example 2.7

The sound pressure level of a source is

measured in the 16 Hz to 8 kHz octave bands

as below.

What is the overall, broadband, sound

pressure level?

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16 31.5 63 125 250 500 1000 2000 4000 8000

Lp 72 68 50 46 39 35 31 27 25 25


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Example 2.7

Lp,total= 73.5 dB

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16 31.5 63 125 250 500 1000 2000 4000 8000

Lp 72 68 50 46 39 35 31 27 25 25

Lp/10 7.2 6.8 5.0 4.6 3.9 3.5 3.1 2.7 2.5 2.5

0.1,

1

7.2 6.8 5.0 2.5 2.5

,

10log 10

10log 10 10 10 10 10 73.5 dB

i

N

Lp total

i

p total

L

L


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Band Analysis Example

Let us look at an octave, 1/3 octave and

narrow band spectrum for the same sound.

What can we tell about the source as we look

at each?

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Analysis Example

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125 250 500 1000 2000 4000 800020

30

40

50

60

70

80

f [Hz]

Lp

[dBr

e20

Pa]

Spectrum

Octave


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Analysis Example

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125 250 500 1000 2000 4000 800020

30

40

50

60

70

80

f [Hz]

Lp

[dBr

e20

Pa]

Spectrum

1/3 Octave

Octave


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Analysis Example

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125 250 500 1000 2000 4000 800020

30

40

50

60

70

80

f [Hz]

Lp

[dBr

e20

Pa]

Spectrum

Narroband

1/3 Octave

Octave


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What is it?

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Its a Vacuum Cleaner

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125 250 500 1000 2000 4000 800020

30

40

50

60

70

80

f [Hz]

Lp

[dBr

e20

Pa]

Spectrum

Narroband

1/3 Octave

Octave


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Use of wide frequency bands limits accuracy Tonal characteristics of sound are lost in summation or

averaging and very different sounds can have identical

octave band levels

What is the difference between the sounds?

They both appear to be broadband as they have equalpower in each octave band

Limitations of Octave Band Analysis

Sound 1 Sound 2

125 250 500 1000 2000 40000

20

40

f [Hz]

Power[dB]

Octave spectrum

125 250 500 1000 2000 40000

50

100

f [Hz]

Power[dB]

c ave spec rum


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With 1/3 octave band analysis we can see that sound1 has tones and sound 2 still has equal power in

each 1/3 octave band so its broadband


Sound 1 Sound 2

0

20

40

Power[dB]

One-third-octave spectrum

125 250 500 1000 2000 40000

20

40

f [Hz]

Po

wer[dB]

Octave spectrum

0

50

100

Power[dB]


125 250 500 1000 2000 40000

50

100

f [Hz]

Power[dB]

Octave spectrum


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With narrowband analysis we can see that Sound 1 istonal and Sound 2 is pink noise with a 1/f spectrum,

which gives equal power in all 1/nthoctave bands

(hence the flat 1/3 octave and octave band spectra)


Tonal Pink Noise

0

20

40Tone Mix Narrowband Spectrum

Power[dB]

0

20

40

Power[dB]


125 250 500 1000 2000 40000

20

40

f [Hz]

Power[dB]

Octave spectrum

0

20

40Narrowband Spectrum

Power[dB]

0

50

100

Power[dB]


125 250 500 1000 2000 40000

50

100

f [Hz]

Power[dB]

Octave spectrum


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Three Broadband Noises

White Noise

Equal energy at each frequency which means

narrowband spectra is flat, octave band rises

Sounds quite hissy

Pink Noise Equal energy in each band which means flat

octave band spectra, falling narrowband

Sounds like rain or surf noise

Brown Noise Octave band energy drops 3db per octave

Sound like thunder or waterfall

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For Next Time

Read Chapter 3 of LAA

Reach Chapter 2 as well if you havent already

HW #1 Due

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CAE 334/502 Lecture 2a, Spring 2014

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Transcript of CAE 334/502 Lecture 2a, Spring 2014