Duct acoustics

WHY STUDY THE ACOUSTICS OF DUCTS?

Ducts, also known as waveguides, are able to efficiently channel sound over

large distances. Some common examples are:

· Ventilation Ducts

· Exhaust Stacks

· Automotive Silencers (Mufflers)

· Aircraft Turbofan Engines

· Shallow Water Channels and Surface Ducts in Deep Water

TYPES OF DUCT SILENCERS

Sound radiated from duct systems can be reduced by the use of reactive or

dissipative silencers.

Reactive silencers – the transmitted sound power is reduced by reflecting some

of the incident sound power (usually by changes in the geometry of the duct).

Dissipative silencers – the transmitted sound power is reduced by dissipating

some of the the incident sound power as heat.

Acoustic

liner Rigid wall

Reactive silencer Dissipative silencer

REACTIVE SILENCERS

PLANE WAVE PROPAGATION

Sound Waves in a Duct

j tFe

j tVe SArea =

0Acoustic Impedance aP

Z cV

FP

S

Sound Waves in a Duct

j tFe j t kxAe

kc

j t kxBe

jkxBe jkxP x Ae

The Wavenumber (propagating waves)

time

T

2

T

Temporal frequency

distance

2k

Spatial frequency

(wavenumber)

k is the phase change per unit distance jkxAe

Sound Waves in a Duct

j tFe j t kxAe

j t kxBe

jkx jkxP x A e Re

where R is the reflection coefficient

Sound Waves in a Duct

0 0

jkL jkL

a

jkL jkL

ZP Ae Be

cV c Ae Be

Or in terms of an impedance at the end

0

0

0

tan

1 tan

L

a

L

Zj kL

Z c

Zcj kL

c

Closed Duct

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-40

-30

-20

-10

0

10

20

30

40

50

L

(dB)

aZstiffness

mass

damping

0 1 2 3 4 5 6 7

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

1

tan

aZ

c j kL

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

-40

-30

-20

-10

0

10

20

30

40

Open Duct

L

(dB)

aZstiffness

mass

damping

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

tanaZ

j kLc

Intake and Exhaust Noise

Intake and Exhaust Noise

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ÃÃmÄÄ

ÃÃmÄÄ

ÃÃVÄÄ

local strain model

transmitted force ÃÃFÄÄ

ÃÃkÄÄ

o

Muffler Performance

Acoustic Filters

pS

ˆ p

Ss

S

2

0ˆ

10log 12

s cT

ZL

Z S

10-3

10-2

10-1

100

-5

0

5

10

15

20

25

30

35

Acoustic Filters – High-Pass

L

L

2S a

pS

ˆ p

Ss

S

2ˆ

10log 12

sTL

kL

ˆ

4

L s

TL

dB

1.5 L L a

ˆ 1s

Acoustic Filters – Quarter Wave Resonator

L

L

2S a

pS

ˆ p

Ss

S

2ˆ tan

10log 12

s kLTL

TL

dB

ˆ 1s0 0.5 1 1.5 2

0

5

10

15

20

25

30

ˆ 1s

0.5 1 1.50

5

10

15

20

25

30

Acoustic Filters – Helmholtz Resonator

L

̂

2S a

pS

12 4

pL S

VS

2

22

ˆ10log 1

ˆ4 1

TL

TL

dB

1

1.7 L L a

V

Acoustic Filters – Band-Pass

0 0.5 1 1.5 20

5

10

15

L

L

cS pS

ˆ c

p

Ss

S

2

2 21 1ˆ10log cos sinˆ4

TL kL s kLs

TL

dB

ˆ 10s 2

max

1 1ˆ10log

ˆ4

TL ss

Mufflers

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Muffler Performance - effect of absorption

Acoustic Filters

ÃÃmÄÄ

ÃÃmÄÄ

ÃÃmÄÄ

ÃÃVÄÄ

local strain model

force

ÃÃkÄÄ

o

HIGHER-ORDER MODE PROPAGATION

MODES AND MODE SHAPE FUNCTIONS

At frequencies greater than f > 1.84c / D, higher order modes can propagate. The general solution is a linear superposition of these ‘mode

shape function’ solutions:

1

,,,ˆn

kxi

mnmn

m

mnerAxrp

immnmmn erJr ,

11

,,,ˆnz

kxi

nxnynxny

ny

ezyAxzyp

nynz ny nzy z k y k z, cos cos

The resultant acoustic pressure in the duct is the weighted sum of fixed

pressure patterns across the duct cross section. Each of which

propagate axially along the duct at their characteristic axial phase

speeds.

Cylindrical duct Rectangular duct

EXAMPLES – RIGID CIRCULAR-DUCT MODES

Wave characteristics

Flexural structural

waves

Acoustic plane

waves

1

2 α c

α c

RESISITIVE SILENCERS

TYPES OF ACOUSTIC LINER

Single degree of freedom

(SDOF) cavity liner

Two degree of freedom

(2DOF) cavity liner

Bulk absorber liner

Partitions

(honeycomb)

Porous septum

Rigid backplate

Porous face-sheet

Fibrous

material Wave propagation

Bulk absorber

liner

Rigid backplate

Rigid backplate

Porous face-sheet

Porous face-sheet

Partitions

(honeycomb)

Two degree of freedom

(2DOF) liner

Single degree of freedom

(SDOF) liner

TYPES OF ACOUSTIC LINER

EXAMPLE OF A PRACTICAL LINER

(Photograph of a Rolls-Royce fan rig, from AIAA paper no. 2001-2268 )

Acoustic

lining

“Hard” liner

splices

Cavity liners (SDOF, 2DOF etc) consist of a sandwich construction. Each layer consists of a porous sheet and a cellular separator such as honeycomb. The backplate is rigid.

Cavity liners can be tuned to the frequency band of interest (by varying the partition depth). The bandwidth can be extended by the addition of more layers.

The acoustic properties of cavity liners depend on the resistance of the porous sheet(s), and the depth of the partition(s).

Cavity liners are “locally-reacting” liners, because the cellular structure prevents lateral sound propagation within the lining.

TYPES OF ACOUSTIC LINER

COMMENTS

Bulk absorber liners are a single layer construction. A fibrous material (e.g. glass

fibre, mineral wool etc) fills the gap between a porous face-sheet and a rigid

back plate.

Bulk absorbers tend to have the widest bandwidth, and are best suited to absorb

broadband and low-frequency noise.

Bulk absorbers are “bulk-reacting” liners, because sound can be transmitted

within the lining.

BULK ABSORBERS

SPECIFIC ACOUSTIC IMPEDANCE

Specific acoustic impedance z, at a point in a single-frequency sound field, is the

complex ratio

At a boundary, the acoustic particle velocity is the component normal to the wall.

It is convenient to nondimensional the value of z as follows:

is the characteristic impedance of the fluid (e.g. air).

u

pz

ˆ

ˆ

Acoustic pressure – Acoustic particle velocity – tiep ˆ

tieu ˆ

R – Resistance

X – Reactance

Units or SI rayl 1msPa

iXRc

zZ

00

00c

CREMER OPTIMUM IMPEDANCE - EXAMPLE

mn contour map

10kb 0xM

1,0 nmmn ,

Optimum

impedance

iZ 25.195.2

EIGENVALUES – rigid duct

bkmnx

only0 m

10kb 4.0xM

Propagating

(cut-on) modes

21 xx MkM

EIGENVALUES – lined duct

iZ 5.00.1

10kb 4.0xM

bkmnx

only0 m

EXAMPLES: LINED-DUCT MODES

ABSORPTION OF SOUND IN LINED DUCTS

The transmission loss of mode (m,n) across a lined section of length l, in

decibels, is

The attenuation of a mode (m,n) is proportional to .

elk

lxp

xpmnxmn 1010

logIm20ˆ

0ˆlog20

lmn6859.8

(The axial decay rate is denoted by .) mnx

kIm

mn

mn

Modes near cut-off – well absorbed

Modes not near cut-off – poorly absorbed

LINER ATTENUATION VERSUS MODAL PROPAGATION ANGLE

VISUALISATION OF PRESSURE FIELD