chapter 1 basic principles

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1

Basic Principles ofEngineering Acoustics

Topics:

• Basic Concepts• Sound Waves in Fluids

• Sound Waves in Solids

• Wave Equation



2

Basics of Sound:

Sound is a sensation of acoustic waves (disturbance/pressure

fluctuations setup in a medium)

Unpleasant, unwanted, disturbing sound is generally treatedas Noise and is a highly subjective feeling



3

• Sound is a disturbance that propagates through a mediumhaving properties of inertia ( mass ) and elasticity. The

medium by which the audible waves are transmitted is air.

Basically sound propagation is simply the moleculartransfer of motional energy. Hence it cannot pass through

vacuum.

Frequency: Number of pressure

cycles / time

also called pitch of sound (in Hz)

Guess how much is particle

displacement??

8e-3nm to 0.1mm



4

The disturbance gradually diminishes in strength as it travelsoutwards, since the initial amount of energy is gradually

spreading over a wider area. If the disturbance is confined to

one dimension ( tube / thin rod), it does not diminish as it

travels ( except for the loss of acoustic energy at the walls ofthe tube).



5

Basic Concepts

•Sound is a pressure wave that propagates through an elastic media.

•It is molecular transfer of motion - energy cannot transfer through vacuum.

•Elasticity and inertia are the desired characteristics of the medium.

•Fundamental mechanisms responsible for sound generation are

i) Vibration of solid bodies-structure born sound.

ii) Turbulence, unsteady flow induced sound- aerodynamic sound.



6

• Structure borne sound - region of interest is surrounding fluid.

• The source which generates sound is external to the medium.

• Aerodynamic sound - sources of sound are not readily identifiable.

• Region of interest is within fluid or external to it.

Basic Concepts



7

Speed of SoundThe rate at which the disturbance (sound wave) travels

Property of the medium

0

0

P c

γ

ρ =

T c

γ =Alternatively,

c – Speed of sound P 0, ρ 0 - Pressure and density

γ - Ratio of specific heats R – Universal gas constant

T – Temperature in 0K M – Molecular weight

Speed of Light: 299,792,458 m/s Speed of sound in air: 344 m/s

2

1

0273

1 ⎟ ⎠

⎞⎜⎝

⎛ += cT cc

smc /5.34325 =

smc /35540 =



8

Quantifying Sound

Root Mean Square Value (RMS) of Sound Pressure

Mean energy associated with sound waves is its

fundamental featureenergy is proportional to square of amplitude

1

22

0

1[ ( )]

T

p p t dt T

⎡ ⎤= ⎢ ⎥⎣ ⎦

∫

0.707 p a=

Acoustic Variables: Pressure and Particle Velocity

(for a harmonic wave)



9

Range of RMS pressure fluctuations that a human ear can

detect extends from

0.00002 N/m2 (20 µPa) (threshold of hearing)

to

20 N/m2

(sensation of pain) 1000000 times larger

Atmospheric Pressure is 105 N/m2

so the peak pressure associated with loudest sound

is 5000 times smaller than atmospheric pressure

The large range of associated pressure is one of the reasons weneed alternate scale.

RANGE OF PRESSURE



10

Sound Pressure Level20Log10P1= 20Log10P2 + 20Log10n

(1/2)

20Log10(P1/P2) = 20Log10n(1/2)

20Log10n(1/2) is still in deciBel, defined as Sound Pressure Level

A dB value is always relative to a reference. For Sound Pressure

Level (SPL) in acoustics, the reference pressure P2=2e-5 N/m2 or

20μPa.

SPL=20Log10(P1/2e-5) P1 is RMS pressure

n: Ratio of sound powers



11

Corresponding to audio range of Sound Pressure

2e-5 N/m2 - 0 dB

20 N/m2 - 120 dB

Normal SPL encountered are between 35 dB to 90 dB

For underwater acoustics different reference pressure is used

Pref = 0.1 N/m2

It is customary to specify SPL as 52dB re 20μPa

Sound Pressure Level



12



13

Sound Intensity



14

Sound IntensityA plane progressive sound wave traveling in a medium (say

along a tube) contains energy and,

rate of transfer of energy per unit cross-sectional area is

defined as Sound Intensity

0

1

T

I p u dt T = ∫

2

0

I

c

=

1010ref

I IL Log

I

=

2

1 0110 10 2

0

/( )20 10

2 5 (2 5) /( )

p c pSPL Log dB Log dB

e e c

ρ

ρ = =

− −

12 12

10 10 1012 2 2

0 0

10 1010 10 1010 (2 5) /( ) (2 5) /( )ref

I I SPL Log dB Log Log e c I e c ρ ρ

− −

−= = +

− −

For air, ρ 0c ≈ 415Ns/m3 so that 0.16 dBSPL IL= +

Holds true also for spherical

waves far away from source

I ref = 10-12

W/m

2



15

FREQUENCY & FREQUENCY BANDS

Frequency of sound ---- as important as its level

Sensitivity of ear Sound insulation of a wall

Attenuation of silencer all vary with frequency

<20Hz 20Hz to 20000Hz > 20000Hz

Infrasonic Audio Range Ultrasonic



16

MusicalInstrument

For multiple frequency composition sound, frequency spectrum is

obtained through Fourier analysis

Pure toneFrequency Composition of Sound



17

A

m p l i t u d e ( d B ) A1

f 1 Frequency (Hz)

Complex Noise Pattern

No discrete tones- infinite frequencies

Better to group them in frequency bands – total strength in

each band gives measure of sound

Octave Bands commonly used (Octave: Halving / doubling)

produced by exhaust of Jet Engine, water at base of

Niagara Falls, hiss of air/steam jets, etc



18

OCTAVE BANDS1= 1

1x2=2

2x2=4

4x2=8

8x2=16

16x2=32

32x2=64

64x2=128

128x2=256

256x2=512

512x2=1024

10 bands(Octaves)

For convenience Internationally accepted ratio is

1:1000 (IEC Recommendation 225)

Center frequency of one octave band is 1000Hz

Other center frequencies are obtained by continuously

dividing/multiplying by 103/10 starting at 1000Hz

Next lower center frequency = 1000/ 103/10 ≈ 500Hz

Next higher center frequency = 1000*103/10 ≈ 2000Hz

c U L f f f =

International Electrotechnical Commission



19

Octave Filters



20

Octave and 1/3rd Octave

band filters

mostly to analyse relatively

smooth varying spectra

If tones are present,

1/10th Octave or Narrow-bandfilter be used



21

For most noise, the instantaneous spectral densityℑ (t) is a time varying quantity, so that ℑ in this

expression is average value taken over a suitable

period τ so that ℑ =< ℑ (t)>τ

So, many acoustic filters & meters have both fast (1/8s) and slow (1s)integration times (For impulsive sounds some sound meters have I

characteristics with 35ms time constant)

I n t e n s i t y

I

f 1 Frequency (Hz) f 2

INTENSITY SPECTRAL DENSITY

Acoustic Intensity for most sound

is non-uniformly distributed over time and frequency

Convenient to describe the distribution through spectral density

2

1

f

f

I

f

I df

Δℑ =

Δ

= ℑ∫

ℑ is the intensity within the frequency band Δ f=1Hz



22

DeciBel measure of ℑ is the Intensity Spectrum Level (ISL)

.1

10logref

z

ISL I

⎛ ⎞ℑ

= ⎜ ⎟⎜ ⎟⎝ ⎠If the intensity is constant over the frequency

bandwidth w (= f 2- f 1),then total intensity is just I= ℑ w and

and Intensity Level for the band is

1 .1

w I Hz

Hz = ℑ×

10log IL ISL w= +

Intensity Spectrum Level (ISL)

If the ISL has variation within the frequency band (w),

each band is subdivided into smaller bands so that in each band ISL

changes by no more than 1-2dB



23

IL is calculated and converted to Intensities I i and then total

intensity level ILtotal is

10log

i

i

total

ref

I

IL I

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

∑10logi i i L ISL w= +

as SPL and IL are numerically same, 10logSPL PSL w= +

PSL (Pressure Spectrum Level) is defined over a 1Hz interval – so the SPL of a tone is same as its PSL

101010log 10

i IL

total

i

IL⎡ ⎤

= ⎢ ⎥⎣ ⎦∑10log

i

i

total

ref

IL

I

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

∑Can be

written as

Thus, when intensity level in each band is known, total intensity level can be estimated



24

Combining Band Levels and Tones

SPL = PSL + 10 log w

For pure tones, PSL = SPLSPL of the two tones is 63 & 60 dB

For the broadband noise,

SPL = PSL + 10 log w

= 40 + 10 log (600 -500)SPL = 60 dB

Thus the overall band level

= Band level of broadband noise + Level of tones

= 60 + 63 + 60 = 64.7 + 60≈ 66 dB



25

Sound Power Intensity : Average Rate of energy transfer per unit area

2

2 W/m

4

W I

r π

=2

2 2

0

4 4 Watt p

W r I r c

π π ρ

= =

Sound Power Level: 1010log

ref

W SWL

W

=

Reference Power W ref =10-12 Watt

dB

Peak Power output:

Female voice – 0.002W, Male voice – 0.004W,

Soft whisper – 10-9W, An average shout – 0.001W

Large orchestra – 10-70W, Large Jet at takeoff – 100,000W

15,000,000 speakers speaking simultaneously generate 1HP



26Figure: Gas pipe line model which gives both air and structure born sound

Basic Concepts



27

Engineering Acoustics

Topics:

• Basic Concepts

• Sound Waves in Fluids


• Wave Equation



28

• While sound propagating through gases and liquids, longitudinal elastic

waves can exist.

• Longitudinal waves are characterised by particle velocities parallel to the

direction of propagation.

Figure: Longitudinal waves

Sound Waves in Fluids



29

Longitudinal waves

i) Plane waves

ii) Spherical waves

iii) Cylindrical waves

Plane waves

• Plane waves are characterised by

– Points of same sound pressure (for example, in the cross-section ofthe duct) form parallel planes, called wave front

– Points of same particle velocity form parallel planes

• Example: Infinite duct with harmonically moving piston at one end




31

Figure: Plane waves in infinite duct

• Wave length is given by




32

• Wave propagation without reflection i.e. for free longitudinal waves, soundpressure p and particle velocity are in phase

Where is the density in the undisturbed medium.

c is the speed of sound.

• Time averaged sound power is given by

• Sound intensity is

• Thus for plane wave, sound intensity is proportional to the mean square

value of the pressure.




33

Spherical waves

• Spherical waves will be produced

– If source is spherical

– Surface vibrates with same amplitude and phase at all points

– If source characteristic dimension is small compared to sound wavelength

Figure: spherical wave propagation Figure: sound passing through a hole smallin diameter compared to

generates spherical wave




34

• Examples are loud speakers and outlets of exhaust of outlet pipes etc.

• With increasing radius, the curvature of wave front decreases – can beapproximated as a plane wave front

– Spherical sound pressure wave can be locally approximated as a plane waves

• Time averaged sound intensity is

where is the mechanical power emitted into the medium by the source.




35

• If the source is an infinite long cylinder, and entire surface vibrates with

same amplitude and phase – Cylindrical waves will be generated

Figure: Infinite long cylinder generating

cylindrical waves

•Sound intensity can be expressed as


where W’ is the mechanical power per unit length emitted into the mediumby the line source.



36


Topics:

• Basic Concepts



• Wave Equation



37

• Solid media can sustain both normal and shear stresses

– Not only longitudinal but transverse waves also exists.

– Both waves in combination can built bending waves

Figure: longitudinal waves- particles move along

direction of propagation

Figure: Transverse waves- particles move

perpendicular to direction of propagation

Sound Waves in Solids



39

Quasi-Longitudinal waves in barsThe speed of sound in a bar is given by:

, where E is Young´s modulus.

d

c L

c L

λ LU n d e fo rm e d b e am

E x p a n s i o nC o m p r e s s i o n

L

E c

ρ =

, 5100 m/s L steel c =



40

Transversal waves insolids

Wave Propagation

Particle displacement



41

A pure transversal (S or shear) wave

The wave speed is given by: , where G

is the shear modulus. S

Gc

ρ =

, 3100 m/sS steel c =

An infinite solid



42

Mixed waves in solidsA pure transversal or longitudinal wave only exists in aninfinite solid medium. In plate or beam structures thesewave types are mixed and form different waves types.One important is bending waves involving a pure bendingdeformation of the cross-section.



43

Bending waves in thin plates

The wave speed for bending waves cB is frequency

dependent:

where υ is Poisson´s ratio and h is the thicknessof the plate.

This means that different harmonics will travelwill different speed, i.e., a given wave form willchange its shape over time. This phenomenon iscalled dispersion.

2

42

( )12(1 )

B

Ehc ω ω

υ ρ =

−



44

plate

air

Bending waves couple well to asurrounding medium and can radiatesound

plat

e



45


Topics:

• Basic Concepts



• Wave Equation



46

• Assumptions for deriving wave equation

– The medium is homogenous and isotropic, i.e., it has the same properties at all points and in all directions.

– The medium is linearly elastic, i.e., Hooke’s law applies.

– Viscous losses are negligible.

– Heat transfer in the medium can be ignored, i.e., changes of state can be assumed

to be adiabatic.

– Gravitational effects can be ignored, i.e., pressure and density are assumed to be

constant in the undisturbed medium.

– The acoustic disturbances are small, which permits linearization of the relations

used.

Wave Equation



47

Equation of continuity

• The following quantities are considered:

– Pressure:

Wave Equation



48

•the equation of continuity gives a relation between density and particle

velocity.

•We consider in and outflow of mass in the x- direction at a given pointin time, for a volume element ΔV =Δ xΔ yΔ z fixed in space,

Wave Equation



49

Figure: Mass flow in the x-direction through a volume element fixed in space.

Wave Equation



50

– the mass in the volume element t is

– the mass flow into the element is

– the mass flow out is

– The net flow in the element is therefore

– Must equal the mass change

Wave Equation



51

• for small variations about the undisturbed equilibrium state

• This can be simplified to

• In the generalized three dimensional case

Wave Equation



52

– the del operator

– simplified expression of the continuity equation

– Considering undisturbed density which is independent of time and position

– The linearzed equation is given by

Wave Equation



53

Equation of motion

– Consider a specific fluid particle, with a fixed mass

– And a fixed volume

Figure: Force in the x-direction on a particular fluid particle moving with the medium.

Wave Equation



54

• The force in the x-direction is

Here is constant

Wave Equation



55

• In three dimensions, the force vector becomes

• The del operator is

• Using the relation

Wave Equation



56

– At time t

position

velocity is

– At a later instant

– Position is

– Velocity is

– Acceleration can be written as

Wave Equation



57

• By using Taylor series

• The acceleration of the fluid particle becomes,

Wave Equation



58

• With simplifying notation this is

• For acoustic fields with small disturbances

• Making use of above equation, and

Wave Equation



59

• the equation of motion can be formulated as

• then the linear, inviscid equation of motion is

Wave Equation



60

• The homogenous linearized wave equation

Wave Equation

chapter 1 basic principles

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