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Basic Principles ofEngineering Acoustics
Topics:
• Basic Concepts• Sound Waves in Fluids
• Sound Waves in Solids
• Wave Equation
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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
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• 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
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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).
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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.
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• 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
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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 =
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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)
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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
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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
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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
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Sound Intensity
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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
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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
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MusicalInstrument
For multiple frequency composition sound, frequency spectrum is
obtained through Fourier analysis
Pure toneFrequency Composition of Sound
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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
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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
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Octave Filters
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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
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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
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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
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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
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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
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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
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26Figure: Gas pipe line model which gives both air and structure born sound
Basic Concepts
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Engineering Acoustics
Topics:
• Basic Concepts
• Sound Waves in Fluids
• Sound Waves in Solids
• Wave Equation
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• 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
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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
Sound Waves in Fluids
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Figure: Plane waves in infinite duct
• Wave length is given by
Sound Waves in Fluids
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• 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.
Sound Waves in Fluids
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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
Sound Waves in Fluids
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• 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.
Sound Waves in Fluids
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• 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
Sound Waves in Fluids
where W’ is the mechanical power per unit length emitted into the mediumby the line source.
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Engineering Acoustics
Topics:
• Basic Concepts
• Sound Waves in Fluids
• Sound Waves in Solids
• Wave Equation
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• 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
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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 =
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Transversal waves insolids
Wave Propagation
Particle displacement
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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
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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.
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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 =
−
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plate
air
Bending waves couple well to asurrounding medium and can radiatesound
plat
e
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Engineering Acoustics
Topics:
• Basic Concepts
• Sound Waves in Fluids
• Sound Waves in Solids
• Wave Equation
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• 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
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Equation of continuity
• The following quantities are considered:
– Pressure:
Wave Equation
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•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
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Figure: Mass flow in the x-direction through a volume element fixed in space.
Wave Equation
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– 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
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• for small variations about the undisturbed equilibrium state
• This can be simplified to
• In the generalized three dimensional case
Wave Equation
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– 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
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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
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• The force in the x-direction is
Here is constant
Wave Equation
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• In three dimensions, the force vector becomes
• The del operator is
• Using the relation
Wave Equation
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– At time t
position
velocity is
– At a later instant
– Position is
– Velocity is
– Acceleration can be written as
Wave Equation
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• By using Taylor series
• The acceleration of the fluid particle becomes,
Wave Equation
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• With simplifying notation this is
• For acoustic fields with small disturbances
• Making use of above equation, and
Wave Equation
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• the equation of motion can be formulated as
• then the linear, inviscid equation of motion is
Wave Equation
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• The homogenous linearized wave equation
Wave Equation
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