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8.2 LUMPED-PARAMETER ANALYSIS

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The lumped-parameter model is utilized in many areas of physical analysis.

The advantage of the lumped-parameter model is that the governing equations

are either ordinary differential equations or algebraic equations. The

properties of the lumped-parameter elements (coefficients) usually have a

physical interpretation, and the acoustic lumped-parameter model has analogues

in terms of corresponding mechanical or electric systems. Lumpedparameter

models in acoustic analysis are generally valid for situations in

which ka < 1 or fa=c < 1=2_ _ 0:16, where a is a characteristic dimension of

the physical system and f is the frequency.

8.2.1 Acoustic Mass

For a mechanical system, Newton’s second law of motion may be written in

terms of the change in velocity with respect to time:

Fnet ј m

(8-1)

The quantity m is the mass being accelerated and v is the velocity of the

mass. This expression is identical in mathematical form to the relationship

for the voltage change across an inductive element in an electrical circuit:

_e ј LE

(8-2)

The quantity LE is the mutual inductance (units: henry), and i is the electrical

current (amperes). The electric voltage is analogous to the net mechanical

force, and the electric current is analogous to the velocity of the mass.

The relationships given by Eqs (8-1) and (8-2) suggest that an analogous

relationship may be developed for acoustic systems in which a portion

of the system is accelerated. In the acoustic systems, however, it is more

convenient to use acoustic pressure instead of the force and the acoustic

volume velocity рU ј SuЮ instead of the particle velocity. Newton’s second

law of motion for an accelerated mass may be written in the following form:

Fnet ј S_p ј m

dрU=SЮ

(8-3)

_p ј

dt ј MA

(8-4)

332 Chapter 8

The quantity MA is the acoustic mass (units: kg/m4). The physical systems

analogous to mechanical mass are shown in Fig. 8-1. The acoustic pressure

is analogous to the electric voltage (or mechanical force), and the volumetric

flow rate is analogous to the electric current (or velocity of a mass).

The expression for the acoustic mass for a tube, as shown in Fig. 8-2,

may be developed. The mass of gas within the tube is given by:

m ј _oр_a2LЮ (8-5)

The quantity a is the radius of the tube and L is the tube length. Making the

substitution for the mass into Eq. (8-4), we obtain the following expression:

MA ј

_a2L_o

р_a2Ю2 ј

_oL

_a2 (long tube) (8-6)

Actually, there is an additional mass of gas at each end of the tube that

is also accelerated. This additional mass must be added to the mass within

Silencer Design 333

FIGURE 8-1 Inertance elements: (a) mechanical mass, (b) electrical inductance, and

the tube to determine the correct acoustic mass. There are two cases that

may be considered for the end of the tube, as shown in Fig. 8-3:

(a) Flanged end (Pierce, 1981):

_m ј р8=3Юa3_o ј _o_a2 _L1 (8-7)

The additional equivalent length to account for the mass of gas at the end

that is accelerated is given by the following expression:

_L1 ј

(flanged end) (8-8)

(b) Free end:

_m ј 0:613_a3_o ј _o_a2 _L2 (8-9)

334 Chapter 8

FIGURE 8-2 Mass fluid with a density _o within a circular tube of radius a and

length L.

FIGURE 8-3 End conditions for a tube: (a) flanged end and (b) free end.

The additional equivalent length to account for the mass of gas at the end,

for a tube with a free end, that is accelerated is given by the following

expression:

_L2 ј 0:613a (free end) (8-10)

The total acoustic mass for a tube may be written in the form:

MA ј

_oLe

_a2 (8-11)

The equivalent mass for the tube is given by the following:

Le ј L ю _La ю _Lb (8-12)

The quantities _La and _Lb are the additional equivalent end corrections

for each end of the tube, depending on the type of end termination.

If the instantaneous volumetric flow rate is sinusoidal, we may write

the following:

UрtЮ ј Um e j!t (8-13)

If we make this substitution into Eq. (8-4), we find the following relationship

between the instantaneous acoustic pressure difference and the instantaneous

volumetric flow rate:

_pрtЮ ј j!MAUm e j!t ј _pm e j!t (8-14)

The acoustic reactance for a mass element may be defined, as follows:

XA ј

U ј j!MA (8-15)

The quantity ! ј 2_f is the circular frequency for the sound wave.

8.2.2 Acoustic Compliance

The change in force acting on a mechanical spring element (an energy

storage element) is proportional to the displacement of the ends of the

spring, or the velocity integrated over the time during which the displacement

occurs:

_Fs ј KS

u dt ј

u dt (8-16)

The quantity KS is the spring constant, and CM ј 1=KS is the mechanical

compliance (units: m/N). The corresponding relationship for an electrical

capacitor is as follows:

Silencer Design 335

_e ј

i dt (8-17)

The quantity CE is the electrical capacitance (units: farad).

The corresponding capacitance element for the acoustic system is a

volume of gas that is compressed and expanded by the gas entering the

volume at a volumetric flow rate U, as shown in Fig. 8-4. For a fixed volume

V, we may write the conservation of mass principle for the gas as follows:

dt ј _oU (8-18)

The acoustic compression/expansion process is thermodynamically

reversible and adiabatic (isentropic) for small amplitudes. The pressure–

density relationship for such a process is as follows:

p=__ ј constant ј Po=__

o (8-19)

dp ј рPo=__

oЮ____1 d_ (8-20)

336 Chapter 8

FIGURE 8-4 Compliance elements: (a) mechanical spring, (b) electrical capacitor,

and (c) acoustic compliance.

The quantity Po is atmospheric pressure and _ is the specific heat ratio for

the gas. The time rate of change of the density of the gas within the volume

may be found from Eq. (8-20):

dt ј

_oPo___1

(8-21)

Combining Eqs (8-18) and (8-21), we obtain the following expression

for the volumetric flow rate of gas into the volume:

U ј

Vр_o=_Ю__1

_Po

dt _

_Po

(8-22)

The expression for the speed of sound for an ideal gas рPo ј _oRTЮ is given

by Eq. (2-1):

c2 ј _RT ј _Po=_o (8-23)

Making this substitution into Eq. (8-22), we obtain the following relationship

for the volumetric flow rate:

U ј

_oc2

(8-24)

If we separate variables and integrate Eq. (8-24), we obtain the analogous

relationship for an acoustic compliance element:

_p ј

_oc2

U dt ј

U dt (8-25)

The quantity CA is the acoustic compliance (units: m3/Pa or m5/N):

CA ј

_oc2 (8-26)

If we make the substitution for the volumetric flow rate from Eq.

(8-14) into Eq. (8-25), we may obtain the following expression for the

instantaneous acoustic pressure difference for a compliance element:

_pрtЮ ј

j!CA

e j!t ј _

jUm

!CA

e j!t ј _pm e j!t (8-27)

The acoustic reactance for a compliance element may be defined as follows:

XA ј

U ј _

!CA

(8-28)

Silencer Design 337

8.2.3 Acoustic Resistance

The mechanical resistance in a system is provided by a damper, often considered

as a viscous or linear damper, in which the force on the damper is

directly proportional to the velocity, as shown in Fig. 8-5:

Fd ј RMv (8-29)

The quantity RM is the mechanical resistance or damping coefficient (units:

N-s/m).

The analogous electrical quantity is the electrical resistance RE,

defined by Ohm’s law:

_e ј REi (8-30)

The quantity RE is the electrical resistance (units: ohm).

The analogous acoustic resistance is defined in a similar manner:

_p ј RAU ј RASu (8-31)

338 Chapter 8

FIGURE 8-5 Resistance elements: (a) mechanical damper, (b) electrical resistor, and

The quantity RA is the acoustic resistance (units: Pa-s/m3 ј N-s/m5). The

units for the acoustic resistance (Pa-s/m3Ю are sometimes called acoustic

ohms. In contrast to the electrical resistance, the acoustic resistance is

often a function of the frequency of the sound wave. In acoustic systems,

the resistance may be provided by restrictions, such as screens.

The acoustic resistance of tubes depends on the size of the boundary

layer near the tube wall relative to the tube radius or diameter. This ratio is

given by the following expression:

rv ј aр2_f _=Ю1=2 (8-32)

The quantity is the viscosity of the gas in the tube and a is the tube radius.

The frictional resistance for the gas within a tube is given by the following

expressions:

(a) small tube, rv < 4

ffiffiffi

2p ј 5:66:

RA ј

_a4 (8-33)

(b) intermediate tube, rv > 5:66:

RA ј р4_f_Ю1=2

_a2

a ю2

_ _

(8-34)

The acoustic resistance of an orifice of negligible thickness is given by

the following expression:

RA ј

2_f 2_

(for ka <

ffiffiffi

2 p

Ю (8-35a)

_a2 (for ka >

ffiffiffi

2p Ю (8-35b)

8>><

>>:

The quantity k ј 2_f =c is the wave number and a is the orifice radius.

8.2.4 Transfer Matrix

The transfer matrix approach to silencer design has been used extensively

since large-capacity digital computers have become available. Let us consider

the acoustic mass element shown schematically in Fig. 8-6. The input

and output variables are the acoustic pressure and volumetric flow rate. The

following set of equations may be written for steady-state operation:

p2 ј p1 ю j!MAU1 (8-36a)

U2 ј U1 (8-36b)

Silencer Design 339

These two relationships may be written as a single matrix equation:

_ _

1 j!MA

0 1

_ _

T11 T12

T21 T22

_ _

(8-37)

The T matrix is called the transfer matrix. For a mass element, the transfer

matrix is obtained from Eq. (8-37):

ЅT_mass ј

1 j!MA

0 1

_ _

(8-38)

The transfer matrices for the compliance and resistance element may be

written in a similar manner:

ЅT_comp ј

1 _j=!CA

0 1

_ _

(8-39)

ЅT_rest ј

1 RA

0 1

_ _

(8-40)

For two or more elements in series, as shown in Fig. 8-7, the output of

element A is the input to element B:

_ _

ј ЅT_A

_ _

ј ЅT_AЅT_B

_ _

(8-41)

We observe from Eq. (8-41) that the overall transfer matrix for two or more

elements in series is the matrix product of the individual transfer matrices:

ЅT_o ј ЅT_AЅT_B (8-42)

340 Chapter 8

FIGURE 8-6 Transfer element schematic for an acoustic mass. The ‘‘outputs’’ are

the acoustic pressure p2 and the acoustic volumetric flow rate U2; the ‘‘inputs’’ are

the corresponding values of p1 and U1.

FIGURE 8-7 Acoustic elements in series.

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