Fundamentals Of Duct Acoustics 91556

FUNDAMENTALS OF DUCT ACOUSTICS

P. JOSEPH

FLUID DYNAMICS AND ACOUSTIC GROUP

WHY STUDY THE ACOUSTICS OF DUCTS

Ducts, also known as waveguides, are able to efficiently transmit sound over large distances. Some common examples: • Ventilation ducts • Exhaust ducts • Automotive silencers • Shallow water channels and surface ducts in deep water • Turbofan engine ducts

WAVE EQUATION

The acoustic pressure p(x,y,z,t) in a source-free region of space in which there is a uniform mean flow Ux in the x – direction satisfies the convected wave equation:

⎡ D2 2 2⎤ ⎢ 2 −c ∇ ⎥p = 0 ⎣ Dt ⎦

Dp ∂p ∂p = + Ux Dt ∂t ∂x

where c is the speed of sound. For simplicity, and without loss of generality, we shall only consider solutions to the wave equation in the absence of flow, Ux = 0.

SEPARABLE SOLUTIONS TO THE WAVE EQUATION IN CARTESIAN AND CYLINDRICAL COORDINATES

Cylindrical duct ba

Rectangular duct θr

Lz

x

x Ly

z y

∂2 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ∇ = 2+ ⎜r ⎟ + 2 2 ∂x r ∂r ⎝ ∂r ⎠ r ∂θ 2

∂2 ∂2 ∂2 ∇ = 2+ 2+ 2 ∂x ∂y ∂z 2

Assume harmonic (single-frequency) separable solutions of the form p = Ae − i (αkx −ωt )G (r )H (θ )

p = Ae − i (αkx −ωt )Y ( y )Z ( z )

SEPARABLE SOLUTIONS TO THE WAVE EQUATION IN CARTESIAN AND CYLINDRICAL COORDINATES (Continued)

Substituting into the wave equation and separating variables: Cylindrical duct ∂ 2 G 1 ∂G ⎧ 2 m2 ⎫ 2 + + ⎨k 1 − α − 2 ⎬G = 0 2 r r ∂ ∂ r r ⎭ ⎩

(

)

Rectangular duct ∂ 2Y 2 + k yY = 0 2 ∂ y

Bessel’s equation of order m

∂ H 2 + m H=0 2 ∂ θ 2

∂ 2Z 2 + k zZ =0 2 ∂ z

GENERAL SOLUTIONS

General solutions to these second order equations are: J m (κr )⎫ G (r ) = ⎪ Ym (κr )⎪ ⎪ 2 2 2 ⎬κ = k 1 − α ⎪ −imθ H (θ ) = e ⎪ ⎪⎭

(

( ) ( )

)

Transverse wavenumber is κ.

sin k y y ⎫ Y ( y) = ⎪ cos k y y ⎪ ⎪ 2 2 2 2 ⎬ k y + kz = k 1 − α sin( k z z ) ⎪ Z ( z) = cos( k z z ) ⎪ ⎪⎭

(

Transverse wavenumber is k y2 + k z2

)

HARD-WALLED DUCT EIGENVALUE EQUATION

The component of particle velocity un normal to the hard-walled duct vanishes:

∂G =0 ∂r

∂Z =0 ∂z

∂Y =0 ∂y

The solutions Ym(κr), sin(kzz) and sin(kyz) cannot satisfy these boundary conditions. Furthermore, only particular discrete values of transverse wavenumbers (eigenvalues) satisfy the boundary conditions given by. J m′ (κ mnb ) = 0 (prime denotes derivative of Bessel function)

(

)

sin kny Ly = 0

sin( knz Lz ) = 0

MODAL EIGENVALUES

′ /b κ mn = jmn

kny = n yπ / Ly

knz = nzπ / Lz

Bessel functions of order m = 0, 1 and 2

J0(x) J1(x) J2(x) Stationary values of the Bessel function m/n

x

0 1 2 3 4 5 6

1 0 1.8412 3.0542 4.2012 5.3176 6.4156 7.5013

2 3.8317 5.3314 6.7061 8.0152 9.2824 10.5199 11.734

3 7.0156 8.5363 9.9695 11.345 12.681 13.987 15.268

4 10.1735 11.7060 13.1704 14.5858 15.9641 17.3128 18.6374

5 13.3237 14.8636 16.3475 17.7887 19.1960 20.5755 21.9317

6 16.4706 18.0155 19.5129 20.9725 22.4010 23.8036 25.1839

7 19.6159 21.1644 22.6716 24.1449 25.5898 27.0103 28.4098

CUT OFF FREQUENCY

Earlier we saw that the transverse and axial wavenumbers of a single mode are connected by the dispersion relationships Cylindrical duct

Rectangular duct

′ / kb ) α mn = 1 − ( jmn

2

α nynz

(

= 1 − ⎡ n yπ / kLy ⎣⎢

) + (nz π / kLz ) ⎤⎦⎥ 2

2

These expressions make explicit the existence of threshold frequencies ω mn = ck mn at frequencies below which α is purely imaginary and the mode decays exponentially along the duct. The mode is said to cut off, or evanescent.

CUT OFF FREQUENCY

This cut-off frequency follows from the above as Cylindrical duct

Rectangular duct

( ⎣

ω nynz = cπ ⎡⎢ n y / Ly

′ /b ωmn = cjmn

2 ⎤ − 1/ 2

) + (nz / Lz ) ⎥⎦ 2

In of cut-off frequency:

(

α mn = 1 − ω mn / ω

)

2

(

α nynz = 1 − ω nynz / ω

)

2

MODE COUNT FORMULAE

At a single frequency only a finite number of modes N(kb) and N(kLx) are cut on and able to propagate along the duct without attenuation. The rest decay exponentially along the duct. In the high frequency limit, ka → ∞: Cylindrical duct

Rectangular duct

N (kb ) → kb + ( kb ) 1 2

1 2

2

N (kLx ) →

(R + 1) kL π

x +

where R = Ly Lx ≥1

R

π

(kLx )2

MODE COUNT FORMULAE

A comparison of this mode-count formula for circular ducts with the exact count (histogram) is presented below. N (kb ) → 12 kb + ( 12 kb )

2

180

Number of propagating modes

160 140 120 100 80 60 40 20 0

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Non-dimensional frequency kb

MODES AND MODE SHAPE FUNCTIONS

In seeking a solution for the pressure field in a duct we obtained, not a single unique solution, but a family of solutions. The general solution is a linear superposition of these ‘eigenfunction’ solutions: Cylindrical duct

pˆ (r ,θ , x ) =

∞

∞

Rectangular duct mn Φ mn (r , θ )e

∑ ∑A

m = −∞ n =1

Φ mn (r , θ ) = J m (κ mn r )e − imθ

−iα mn kx

∞

pˆ ( y, z , x ) = ∑

∞

−iαkx ( ) A Φ y , z e ∑ nxny nxny

ny =1 nz =1

(

)

Φ nynz ( y , z ) = cos kny y cos( k nz z )

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.

MODE SHAPE FUNCTIONS Φ Cylindrical Duct Mode Shape Functions 10

12

+

10 (ka)11=1.8412

1

-

+

(ka)21=3.0542

3

2

+ -

nm

+

(ka)03=7.0156

(ka)02=3.8317

(ka)01=0

2

23

-

-

+

-

(ka)13=8.5363

(ka)12=5.3314

+

+

-

(ka)22=6.0706

+ - +

-

+

-

-

+

-

+

nm

+-

+

-

- +

(ka)23=9.9695

+ - + - + - + - + -+

nb. ka used here. Should be ka.

MODE SHAPE FUNCTIONS Φ Rectangular Duct Mode Shape Functions

ny = 0, nz = 0

ny = 0, nz = 1

ny = 1, nz = 1

ny = 1, nz = 2

MODAL CUT - OFF RATIO

Earlier we saw that each mode the axial wavenumber is αmnk and the wavenumber in the direction of propagation is k (=ω/c)

κmn

k

φx αmnk

By simple geometry, αmn equals the cosine of the angle φx between the local modal wavefront and the duct axis. It may therefore be interpreted as a measure of how much the mode is cut on. A more common index of ‘cut on’ is specified by the cut-off ratio ζ defined by ζ mn = k κ mn

CATEGORIES OF MODAL BEHAVIOUR

ζmn < 1.

Mode is cut-off and decays exponentially along the duct. Pressure and particle velocity are in quadrature and zero power is transmitted.

ζmn = 1.

Mode is just cut on (or cut-off) and propagates with infinite phase speed (and zero group velocity). No modal power is transmitted.

ζmn > 1.

Mode is cut on and propagates at an angle cos−1 (1 / ζ mn ) to the duct axis. Transmitted modal sound power . Axial phase speed greater than c and group velocity less than c.

CATEGORIES OF MODAL BEHAVIOUR

ζmn >> 1.

ζmn > 1.

ζmn = 1.

AXIAL PHASE SPEED

Each mode propagates axially along the duct as pmn ∝ e − iα mn kx . The axial modal phase speed cmn is given by ω α mn k

= c / α mn

cmn 1 = 2 c 1 − (ω mn / ω ) The axial phase is infinite at the cut-off frequency, tending to c as the frequency approaches infinity.

cmn/c

cmn =

ωmn/ω

CIRCUMFERENTIAL PHASE SPEED IN A CIRCULAR DUCT

At a fixed position along the duct, points at the wall of constant phase are given by η (θ , t ) = ωt − mθ . The circumferential phase speed at the wall is therefore

cθmn = a

∂θ ωa = ∂t m

A property of Bessel functions Jm for large m is that km1a ≥ m . Combining this result with the cut on condition ω / c > k m1 gives ωa and hence cθmn > c >1

mc

For propagation, the circumferential (as well as axial) modal phase speed modes must be supersonic.

MODAL RADIATION FROM HARD WALLED CIRCULAR DUCTS

The acoustic pressure p(R,θ , φ , t ) in the far field of a semi-infinite circular hard walled duct may be expressed as

e − imθ −ikR +iωt pmn (R,θ , φ , t ) = Amn Dmn (k , a, φ ) R

;kR >> 1

MODAL RADIATION FROM HARD WALLED CIRCULAR DUCTS

Some representative directivity plots (in decibels) for the (m,n)=(40,3) mode at three frequencies is presented below

(m,n)=(40,3) ka = 60, ζ=20

ka = 95, ζ=2

ka = 137, ζ=1.1

MODAL RADIATION FROM HARD WALLED CIRCULAR DUCTS: SIMPLE RULES • The angle φx of the principal radiation lobe equals φ x = cos −1 (1 / ζ mn ) which is identical to the axial propagation angle within the duct. • Modal radiation becomes progressively weaker as the frequency approach cut-off from above tending to zero exactly at cut-off. • No major or minor lobes occur in the rear arc. • Zeros (or nulls) in the radiation pattern occur at angles = cos−1 jmj ′ , j≠n . Angles of the minor lobes occur roughly mid-way between the angles of the zeros. The number of zeros and minor lobes increase roughly as the frequency squared. • Symmetrical angles exist θs beyond which modal radiation is extremely weak. These are referred to as shadow zones (or cones of silence) and occur at φs = sin −1 m ka

LINERS FOR THE ATENUATION OF DUCT BORNE NOISE Reactive and dissipative attenuators

Sound transmission in ducts may be attenuated, either by reflection by means of the introduction of impedance discontinuities (REACTIVE ATTENUATORS), or by the conversion of sound energy into heat (RESISTIVE OR DISSIPATIVE LINERS). Many duct attenuators combine both attenuation mechanisms. Reactive attenuators are most effective at low frequencies (less than 100Hz).

REACTIVE AND DISSIAPTIVE ATTENUATORS weak transmission

Incident wave Strong reflection

REACTIVE SILENCER

Incident wave weak reflection

weak transmission

DISSIPATIVE SILENCER

DISSIPATIVE ATTENUATORS MAY BE RESONANT OT NON-RESONANT

LOCAL AND EXTENDED REACTION

Liners are either locally reactive, where the wave motion is confined to the normal to the surface, or bulk reacting, where the wave are free to propagate in the liner.

LOCALLY REACTING LINER

BULK REACTING LINER