CAE 334/502 Lecture 2b from Spring 2014

8/12/2019 CAE 334/502 Lecture 2b from Spring 2014

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2/16/2014 CAE 334/502 - Week 2 1

CAE 334/502 – Lecture 2a

Phasors, Acoustic Velocity, Impedance,Power, Intensity and Sound Levels



Objectives

• Understand the basic concept of a phasor• Understand the difference between acoustic

velocity and speed of sound

• Understand the concept of acoustic

impedance

• Understand the concepts of acoustic powerand intensity

• Understand basics of sound levels

2/16/2014 CAE 334/502 - Week 2 2



The course so far

• Last time in CAE 334/502 …

• Sound is a wave characterized by

– Frequency f , wavelength , and speed of sound c

c = f

– Also characterized by =2 f and k = 2/

• A pure tone, plane wave can be written as

2/16/2014 CAE 334/502 - Week 2 3

( , ) sin 2 / sin p x t A f x c t A kx t



2/16/2014 CAE 334/502 - Week 2 4



2/16/2014 CAE 334/502 - Week 2 5

Phasors

• The text introduces and briefly discusses theconcept of phasors to describe waves andsimple harmonic motion – Phasors are indeed quite powerful and a standard

method of math notation used throughout physics

and engineering. They are used extensively fordescribing waves or really any quantity thatundergoes time harmonic variation

– You may have seen this in Phys 221 or 224 andwill have seen it in CAE 383 if you have taken it.

– Luckily, if your math skills are a bit lacking incomplex numbers, we don’t use them much andfull understanding is not essential



2/16/2014 CAE 334/502 - Week 2 6

Phasor Basics

• Idea: Represent sinusoidal motion as acomponent of circular motion and circular motionas a rotating vector described by a complexnumber

• Key math relation: Euler’s Equation

• Then we could write a complex pressure

( ) ( )

where -1cos sin

ˆ = so

ˆcos Re{ }

ˆsin Im{ }

j

j t kx j j t kx

je j

p Ae Ae e

A t kx p

A t kx p



Phasors as Projections

• A cosine wave is a horizontal projection of arotating phasor (i.e. the real part)

• A sine wave is the vertical projection (i.e. theimaginary part, as seen below)

2/16/2014 CAE 334/502 - Week 2 7



Leading and Lagging

• We may use the terms leading and lagging todescribe a phase shift.• In the animation, the blue vector is leading the red

vector (+) (or, alternately, the red vector is laggingthe blue vector)

• Notice the fixed angle between the red and bluevectors – that angle is the phase difference

2/16/2014 CAE 334/502 - Week 2 8



2/16/2014 CAE 334/502 - Week 2 9

Why Phasors?

So why would I want to take nice simple realnumbers and make them complex? Because itcan actually simplifiy the math

• Consider Integration and Differentiation

multiplying by jk and dividing by j are much

easier than converting sines to cosines andvice-versa and trying to remember your +/-’swhich is what you need to do with trig functions

( ) ( )ˆd 1ˆ j t kx j t kx p

jkAe pdt Aedx j



Acoustic Particle Velocity: u

• The speed of sound c tells us how fast thewave propagates

– c is a medium property that depends on densityand pressure

• There is another velocity, u, which is theacoustic particle velocity

– u is the speed of the air molecules moving back

and forth – u varies sinusoidally at the same frequency as

pressure (amplitude and phase differ)

– The amplitude of u is related to amplitude of p

2/16/2014 CAE 334/502 - Week 2 10



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Why u is important

•

u is important because u is continuous at theinterface at the interface between materials

– If a fluid is in contact with a moving surface, thefluid molecules must be at the same velocity as

the moving surface or a vacuum is created• Recall that frequency of wave is the frequency of thegenerating vibration, i.e. frequency of surface velocity

– What this means is that a velocity is imposed on

the gas by a moving boundary.

• Luckily, there is a simple relation between pressure andparticle velocity for pure tones – the impedance



2/16/2014 CAE 334/502 - Week 2 12

Specific Acoustic Impedance

• Another important property of a medium is the idea ofimpedance. Impedance is a measure of resistance tomotion of the medium.

• The specific acoustic impedance, z , is the ratio ofsound pressure, p, to particle velocity, u,

• For a plane wave in a fluid, z= 0c where 0 is density.

This is a medium dependent constant and called thecharacteristic impedance

3 [Pa s/m=N s/m =mks rayls] p

z u

0 412= rayls for air at room temp p

z cu



Example 2.1

• The RMS sound pressure at the surface of apanel of large tractor in the middle of a field ismeasured as 10 Pa.

What is the RMS surface velocity of thetractor?

Assume plane waves are leaving tractorpanel and T = 20 C.

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Example 2.1

• The RMS sound pressure at the surface of a panel oftractor in the middle of a field is measured as 10 Pa.What is the RMS surface velocity of the tractor?

• In the middle of a field there are no walls so

there are only waves leaving the tractor.

• The waves are traveling in air at 20 C so

2/16/2014 CAE 334/502 - Week 2 14

0 01.2 , 343 412

100.24 24

412

rmsrms

c z c

pu

z

3

kg Pa sms mm

m mms sPa s

m

Pa



2/16/2014 CAE 334/502 - Week 2 15

So, why do we care about z?

1. It is a relation between u and p• If we find one we can find the other

2. Acousic energy is always reflected at achange in z

– We can cause sound reflection with changes in z which means …

– We can significantly reduce sound transmissioneither with very large changes in z (high mass

walls) or by constructing a partition with manylayers since each layer reflects some energy(multi-layered walls)



2/16/2014 CAE 334/502 - Week 2 16

Acoustic Power

• Acoustic Power ( W ) [watts=W] – It is the time rate of acoustic energy flow i.e. the

power required to get medium moving

– It is a property of vibrating object – has nothing to

do with room or observers distance from object• Think of it is analagous to the power rating of a 60Wlightbulb. It draws 60W regardless of the room but lightlevel depends upon the location and room reflectance

– Typical sound powers of objects ranges from nW

(pin drop) to MW (rocket launch)• A typical voice in conversation has W 10 μW

• A loud stereo has W 100 mW

• A loud car horn has W 1 W



2/16/2014 CAE 334/502 - Week 2 17

Acoustic Intensity

• Acoustic Intensity ( I ) [W/m2] – Acoustic intensity is the acoustic power flow through a unit

area and is found from the average product of p and u

– Acoustic intensity is dependent upon the sourcedirectivity, source location, and room properties

– Typical sound intensities range from nW/m2 (pin drop @

1m) to MW/m2 (rocket launch near platform) – For a plane wave we would find that

*

0

1 1ˆ ˆ

Re2

means complex conjugateT

I pu dt puT

22

0

0

rmsrms rms rms

p I p u cu

c



2/16/2014 CAE 334/502 - Week 2 18

Power – Intensity relation

• Since intensity is a power per unit area, we can findthe sound power of an object by integrating theintensity on any surface surrounding the object

• If we can measure average I i on each patch of asurface of N patches surrounding a source we canestimate W as

( )S

W I S dS

1

N

i i

i

W I S



Powers and Intensities Add

• For incoherent sources, powers andintensities add (RMS pressures do NOT)

Wtot = W1 + W2 Itot = I1 + I2

2/16/2014 CAE 334/502 - Week 2 19

W = 0.1 W W = 30.1=0.3 W



Example 2.2

• The tractor panel of Example 2.1 is 2 m x 1 min size. How much acoustic power is radiatedby the panel?

2/16/2014 CAE 334/502 - Week 2 20



Example 2.2

• The tractor of Example 2.1 has a panel that is2 m x 1 m in size. How much acoustic poweris radiated by the panel?

2/16/2014 CAE 334/502 - Week 2 21

10 0.024 0.24

(2 1) 0.24 0.48

2

W

m

W

rms rms I u p

W S I



2/16/2014 CAE 334/502 - Week 2 22



Range of Sound Pressures

Sound

Pressure

(Pa)

Threshold of Hearing 0.00002

Quiet Studio 0.0002

Quiet Classroom 0.002

Conversation 0.02

Raised Voice 0.2

Jackhammer 2.0

Threshold of Pain 20Instant Damage 200

• Notice the hugerange of magnitudesof sound pressures

• Notice how thegradation levelfollows a power of

10 increase

2/16/2014 CAE 334/502 - Week 2 23



Linear vs Log Plots

• Notice how thelinear scale is tooscrunched for useful

analysis of levels

• The logarithmicscale naturally

separates orders ofmagnitude

2/16/2014 CAE 334/502 - Week 2 24

Instant Damage

Threshold of Pain

Jackhammer

Raised Voice

Conversation

Quiet Classroom

Quiet Studio

Threshold of Hearing0

20

40

60

80

100

120

140

160

180

200

p a s c a l s

( P a )

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000



2/16/2014 CAE 334/502 - Week 2 25

Decibels or dB

• Because of the huge range of p, I , and W and the factthat our ear is non-linear we need a different way torepresent p, I , and W

• The decibel scale can compare powers of vastlydifferent orders of magnitude using logarithms

difference in dB of two powers= 10*log10(W 1/ W 2)

Or

Power Level in dB = 10*log10(W /W reference)

• Note: the dB has no units (units cancel in the ratio) – you must pay attention to what you are dividing



Note for dB the log is base 10

• When working with sound levels and decibels(or dB) you often have to work with logs andanti-logs.

• Make sure you are working in base 10 and notbase e (i.e. natural log or ln). This is particularlyimportant when working with things like MATLABor Mathematica

• We’re going to drop the “10” in the log notation.

If we need natural log I’ll use ln instead of log

2/16/2014 CAE 334/502 - Week 2 26



Some Simple Log Math

2/16/2014 CAE 334/502 - Week 2 27

20

10

10

log 10

20 log 10

log log log

log / log log

y

y y x x

y x x

y ax y a x

y a x y a x

If then

If then

If then

If then



Sound Power Level, LW

For sound power we choose a reference level thatis a nice power of 10 and is about the powerrequired for an object placed near outer ear to justbe audible, 1 pW

We we will use 1 pW=10-12 W for W ref so then

• Sound Power Level, LW (called PWL some books)

LW = 10 log10 (W /10-12) [dB] LW = 10 log10(W ) + 120 [dB]

2/16/2014 CAE 334/502 - Week 2 28



Sound Intensity Level, L I

Since 1 pW is the reference for sound power andintensity is power for per unit area, we’ll set the

intensity reference to I ref = 1 pW/m2

• Sound Intensity Level, L I (called IL in some books)

L I = 10 log10 ( I /10-12) [dB]

L I = 10 log10( I ) + 120 [dB]

2/16/2014 CAE 334/502 - Week 2 29



Sound Pressure Level, L p

• Sound power is proportional to p2

rms not p

rms

so we compute the sound pressure level as10log10( p

2rms/ p

2ref )= 20log10( prms/ pref )

• We choose the reference level as 2x10-5

Pasince that is the about threshold of hearing.

2/16/2014 CAE 334/502 - Week 2 30

2

10 25

10 105

10log

2 10

20log 20log 942 10

rms p

rms p rms

p L

p L p

[dB]



Decibel vs Log vs Linear Scales

• Notice how thedecibel scalespreads levelsnicely like logs

but also givesus values thatare easier to

remember

2/16/2014 CAE 334/502 - Week 2 31

Instant Damage

Threshold of Pain

Jackhammer

Raised Voice

Conversation

Quiet Classroom

Quiet Studio

Threshold of Hearing0

20

40

60

80

100

120

140

160

180

200

p a s c a l s

( P a )

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

0

20

40

60

80

100

120

140

Decibels (dB)

Linear Log Decibel



Sound Pressure Level Notation

• Notice that I have been writing L p, L

I , L

W for

sound pressure, intensity and power level.

– Your textbook uses this notation as well.

– This is the proper notation as defined by ASTM

E634-09 and ANSI S1.1 and ANSI S1.13

• Note: Many, many, many older textbooks usethe terms SPL, PWL, IL instead.

– These are not the proper notation according to thestandards so I discourage you from using them.

2/16/2014 CAE 334/502 - Week 2 32



2/16/2014 CAE 334/502 - Week 2 33

Typical Sound LevelsSound Pressure Level dB Sound Power Level

Rocket Launch at PadHead trauma 180 Saturn V Rocket

Jet Engine on RunwayNearly instant deafness

140 Jet Engine

Rock Concert PeakThreshold of Pain

130 Orchestra at Peak

Jack Hammer @ 2m 100 Typical Home Stereo

Vacuum Cleaner @ 1 m 80 Raised Voice

Conversation @ 1 m 60 Typical Voice

Background a Library 40 Ticking Watch

Background Level ofRecording Studio 20 Pin Drop

Threshold of Hearing 0Power at outer ear forLp=0 dB at eardrum



Converting Back

• We often need to convert back from levels torms pressures, intensities, or powers.

2/16/2014 CAE 334/502 - Week 2 34

0.1 12012 10

0.1 12012 10

0.05 945 20

10 10 10

10 10 10

2 10 10 10

W

W

I

I

p

p

L L

L L

L L

rms

W

I

p



2/16/2014 CAE 334/502 - Week 2 35

Relations between Sound Levels

• Since I= p2 / 0c we can relate L

p and L

I in a

plane wave as

L p = L I + 10log( 0c / 400)

and, since 0

c 400, L p L

I

• If L I is constant on a surface of area S thatsurrounds an object then I is constant and

since I=W/S and L I and LW are related as L I = LW -10log(S )



Sound Level Units

• All sound levels have the units of decibels(dB) so you must be careful about checkingwhat kind of level is being presented.

• Sound power levels are occasionally reportedin bels (B). If you see a sound level listed as5.2 B then you have 52 dB and it’s probably a

sound power level.

2/16/2014 CAE 334/502 - Week 2 36



2/16/2014 CAE 334/502 - Week 2 37

Accuracy of Sound Levels

Since you need to use your calculators or computer in

evaluating and working with sound levels, you may belazy with significant figures and accuracy

• Most inexpensive sound level meters are accurate to no more than

1.5 dB, Expensive sound level meters are only accurate to 0.5 dB• Humans cannot notice differences of much less than about 1 dB at

most frequencies

• As a result, you should NEVER report a level to anaccuracy of more than 0.1 dB (i.e. you can write 95 dB or95.2 dB but never 95.21 dB) – I will take points off on your homework or exam if you give results with

too many significant digits. Really.



Sound Level Apps

• There are a number of sound level apps outfor the Iphone, Ipod touch, Ipad and androidphones.

• These work surprisingly well but the accuracy

is usually much worse than 1.5 dB – Most are probably more like 4dB

• The most complete are probably these

– SPL Meter from Studio Six Digital $8.99

– SoundMeter from Faber Acoustical $19.99

2/16/2014 CAE 334/502 - Week 2 38



Sound Level Summary

• Sound Pressure Level

– Reference value is pref =20 Pa

L p = 20 log( prms) + 94 dB

• Sound Intensity Level

– Reference value is I ref =1 pW/m2

L I = 10 log( I rms) + 120 dB

• Sound Power Level

– Reference value is W ref =1 pW

LW = 10 log(W rms) + 120 dB

2/16/2014 CAE 334/502 - Week 2 39



Example 2.3

• What is the sound pressure level if themeasured RMS pressure is prms = 10 Pa?

2/16/2014 CAE 334/502 - Week 2 40



Example 2.3

• What is the sound pressure level if themeasured RMS pressure is prms = 10 Pa?

2/16/2014 CAE 334/502 - Week 2 41

2 12

10 10 102 2 25 5

12

10

10 10

100 1010log 10log 10log

22 10 2 10

10log 0.25 10 114

20log 94 20log 10 94 20 1 94 114

Pa

Pa Pa

dB

Alternatively

dB

rms p

p rms

p L

L p



Example 2.3

• What is the RMS pressure if the measuredsound pressure level is L p=100 dB?

2/16/2014 CAE 334/502 - Week 2 42



Example 2.2

• What is the RMS pressure if the measuredsound pressure level is L p=100 dB?

2/16/2014 CAE 334/502 - Week 2 43

10

10

10

0.3

100 20log 94

20log 6

log 0.3

10 1.995 2

dB

Pa Pa

p rms

rms

rms

rms

L p

p

p

p



2/16/2014 CAE 334/502 - Week 2 44

Combining Sound LevelsRecall that for incoherent sources acousticpowers and intensities (and hence p2) add

– pressures and sound levels do not add

• In general, for N different sources

• For N identical sources

1

0.1

1

0.1

1

10log 10

10log 10 10log( )

i

N L

Total

i

L

Total

L

L N L N



Decibel Addition

• The chart at rightcan be used foradding two sourcesquickly

• Here is how you can remember that dB ofsources do not add directly: Two people

talking at 70 dB is not as loud as being 50 ftaway from jet takeoff (140 dB), so we cannot

just add the two 70 dBs

2/16/2014 CAE 334/502 - Week 2 45



Amplification and Attenuation

• While we never add the L p

of two sourcesdirectly, we may need to add a change insound level, L p to and from sources.

– A positive L p indicates an increase or

amplification of sound pressure – A negative L p indicates a decrease or attenuation

of sound pressure.

• Later this class we will spend much time

learning about how to estimate L p to accountfor various attenuations to sound waves

L p2 = L p1 + L p

2/16/2014 CAE 334/502 - Week 2 46



Example 2.4

• The sound pressure level at the wall of amechanical room is 75 dB. If the wallprovides 20 dB of attenuation, what is thesound level in the office on the other side of

the wall?

2/16/2014 CAE 334/502 - Week 2 47



Example 2.4

• The sound pressure level at the wall of amechanical room is 75 dB. If the wallprovides 20 dB of attenuation, what is thesound level in the office on the other side of

the wall?

L p2 = 75 – 20 = 55 dB

2/16/2014 CAE 334/502 - Week 2 48

E l 2



Example 2.5

• A factory has five machines all equidistantfrom a worker. By themselves they producesound levels of 78, 82, 85, 82, and 83 dB.

• What sound level will they produce if all are

turned on at once?

2/16/2014 CAE 334/502 - Week 2 49

E l 2 5



Example 2.5

• A factory has five machines all equidistantfrom a worker. By themselves they produceLp of 78, 82, 85, 82, and 83 dB at the worker.

• What Lp will they produce at the worker if all

are turned on at once?

2/16/2014 CAE 334/502 - Week 2 50

0.1

,

1

7.8 8.2 8.5 8.2 8.3

6

10log 10

10log 10 10 10 10 10

10log(8.958 10 ) 89.5

i

N L

I I Total

i

Lp L L

Lp

Lp

and

dB

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CAE 334/502 Lecture 2b from Spring 2014

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