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41.2 Radiation of Sound

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41.2.1 Point Source

41.2.1.1 Simple Source: Spherical Wave by a Monopole

Propagation of sound pressure wave pðx; y; z; tÞ is described by the following partial differential

equation for a medium where a field point is expressed by orthogonal coordinate system O-xyz; as shown

41-1

in Figure 41.1(a):

7 2p 2

›2p

›t2 ¼ 0;

7 2 ; ›2

›x2 þ

›2

›y2 þ

›2

›z2

􀁻 ! ð41:1Þ

where c denotes the velocity of sound propagation.

If the source region is compact and the generating

motion has no preferred direction, it will produce

a wave, which spreads spherically outwards. As the

medium is assumed infinite in extent, the waveform

will depend on the distance, r; from the

center of the source. The wave equation in this

case is

›

›r

r2 ›p

›r

􀀏 􀀐

›2p

›t2 ¼ 0 ð41:2Þ

When a monopole source of angular frequency v

is assumed, the simplest solution for the outward

propagating waveform is expressed as

pðr; tÞ ¼ pv ðrÞe2ivt ;

pv ðrÞ ¼

2vr

4pr

Sv eikr ; k ¼

c ¼

ð41:3Þ

where r denotes the density of the medium and k

denotes the wave number. Here, pv is used to

denote the sinusoidal component of the sound

pressure with angular frequency, v: The subscript

v on a variable typically indicates the sinusoidal

component of a variable, but the variables related

to sound energy and sound power, such as w; I; W ,

do not have the subscript v even if they mean the sinusoidal component. In this case, the source of sound

is taken as a “pulsating globe” of radius a and radial velocity Uv on the surface. Therefore, the flow

outward from the origin, Sv ; is related to Uv as follows, and as shown in Figure 41.1(a):

Sv ¼ 4pa2Uv ð41:4Þ

We should note that, while a pulsating globe with a finite radius a is assumed as the physical sound

source, the sound field by a monopole with infinitesimal small size and finite magnitude, as expressed

mathematically in Equation 41.3, is used.

The other quantities related to the spherical wave are described next [1]:

urv ¼

4pr2 ðikr 2 1ÞSv eikðr2ctÞ; radial velocity

w ¼ r

4pr2

􀀏 􀀐2

lSv l2 ðkrÞ2 þ

􀀒 􀀓

; energy density

I ¼ rc

4pr

􀀏 􀀐2

lSv l2 ¼

lpl2

; energy flux intensity

W ¼ ð4pr2ÞI ¼ rc

l2 lSv l2 ¼

rv2

4pc

lSv l2; total power

8>>>>>>>>>>>><

>>>>>>>>>>>>:

ð41:5Þ

FIGURE 41.1 Directivity of monopole and dipole,

(a) Spherical sound field by a monopole; (b) Axisymmetric

sound field by a dipole.

41-2 Vibration and Shock Handbook

We should note that the first term in the parentheses, kr; is negligible in the region where lkrl ,, 1 is

valid. Hence,

w ¼

􀀄

lSv l=4pr2􀀅2 ¼

u2r

v ¼

􀀏 􀀐

4pr2

is reduced. Conversely, when lkrl .. 1 is valid, then w ¼ rðk=4prÞ2lSv l2 ¼ I=c is valid.

41.2.1.2 Simple Source: Plane Wave by an Alternating Piston

Another example of simple sound wave is generated in the one-dimensional field of fluid medium, as

shown in Figure 41.1. Let us set the coordinate x along the axis of wave propagation, for example, the axis

of duct with a constant cross-sectional area. Then the wave equation is

›2p

›x2 2

›2p

›t2 ¼ 0 ð41:6Þ

The solution for a periodic source is given by

pðx; tÞ ¼ pv ðxÞe2ivt ; pv ðxÞ ¼ rcuv eikx ð41:7Þ

This is known as a plane wave, which is generated by the piston motion at the origin, the velocity of which

is expressed by

uðtÞ ¼ uv ðxÞe2ivt ð41:8Þ

The other quantities related with the plane wave are given below:

uv ðxÞ ¼ uv eikx ; particle velocity

w ¼ I ¼ rcluv l2 ¼ lpv l2=rc2; energy density; sound intensity

W ¼ SI ðS; cross-sectional areaÞ; total power

8>><

>>:

ð41:9Þ

A plane wave is generated in very limited situations, but its utility is rather wide since the sound wave

propagating through a duct or duct-like space with a gradually varying cross section is approximated as

the plane wave. Network theory is applied to the sound wave propagating through a branch and junction

by using the description of a plane wave.

41.2.1.3 Dipole and Multipoles and Their Sound Field

Let us return to the three-dimensional sound field. The second simple solution to Equation 41.1 is the

“dipole” sound field. Suppose that a pair of monopoles, close together, opposite in sign, and equal in

magnitude, Sv ; are located along the x-axis as shown in Figure 41.1(b). Since only a preferred direction is

assigned along the x-axis, the sound field is axisymmetric as represented by

pðr; u; tÞ ¼ 2k2Dv

rc cos u

4pr

1 þ

􀀏 􀀐

e2iðvt2krÞ ð41:10Þ

Dv is defined by

Dv ¼ Sv d ð41:11Þ

where d denotes the separation of the monopoles as shown in Figure 41.1(b). Mathematically, d tends to

zero, keeping Dv finite, and the preferred axis is the x-axis. Physically, a sound field is commonly realized

by a pair of monopoles with a finite separation that is short compared with the wavelength, l; as

illustrated next with realistic examples.

Source of Noise 41-3

The characteristic quantities relating with dipole sound field are described below [1]:

urv ¼ 2

k2Dv cos u

4pr

1 þ

k2r2

􀀏 􀀐

e2ivtþkr ; radial velocity

uuv ¼ i

k2Dv sin u

4pr2 1 þ

􀀏 􀀐

e2ivtþkr; peripheral velocity

w ¼ r

k2lDv l

4pr

􀁻 !2

cos2u þ

􀀏 􀀐2

􀀏 􀀐4

ð1 þ 3 cos2uÞ

􀀒 􀀓

; energy density

Ir ¼ rc

k2lDv l

4pr

􀁻 !2

cos2u; Iu ¼ 0; sound intensity

W ¼

rv4

12pc3 lDv l2; total power

8>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>:

ð41:12Þ

We should add the following notes on the dipole sound field.

When lkrl .. 1 is assumed, the second term in parentheses in Equation 41.10 is negligible. Then the

directivity for pðr; u; tÞ is expressed by cos u: A similar directivity is found on Ir and on urv ; uuv ; and w

with the assumption lkrl .. 1:

A pair of dipoles produces a quadrupole, a pair of quadrupoles produce an octopole, and so on.

These are called multipole in general. Out of multipoles, the quadrupole is common in

representing a sound field generated by mixing fluid flow, especially jet flow. More details on multipoles

are found in Ref. [1].

41.2.2 Sources of Finite Volume

41.2.2.1 Description of Sound Field by Green’s Function

In order to describe the sound field from distributed sources, source terms are introduced to the right

side of Equation 41.1. The partial differential equation with source term is derived from the equation

system representing the dynamics of fluid flow with periodic motion at angular frequency v:

7 2pv þ k2pv ¼ 2mv þ div Fv ð41:13Þ

where pv denotes the acoustic pressure amplitude according to pðx; y; z; tÞ ¼ pv ðx; y; zÞe2ivt ; mv denotes

the effective monopole source density expressed by 2ivrsv ; sv denotes the generalization of the point

source strength, Sv ; of Equation 41.4 for a distributed source, and Fv denotes the vector representation of

point force-density in the fluid. Introduction of Green’s function, gv ; of angular frequency, v; satisfying

the following equation is useful in general:

7 2gv þ k2gv ¼ 2dðx 2 x0Þdðy 2 y0Þdðz 2 z0Þeivt0 ð41:14Þ

Here, dðzÞ denotes the Dirac impulse (delta) function of the variable z; the coordinate ðx0; y0; z0Þ denotes

the position of unit source, r0; with periodic angular velocity, v; and t0 denotes the time pertaining to the

source. The solution of Equation 41.14 is

gv ðr; r0Þ ¼

4pR

eikr ; lr 2 r0l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðx 2 x0Þ2 þ ðy 2 y0Þ2 þ ðz 2 z0Þ2

ð41:15Þ

where r denotes the position vector of sound field with coordinates ðx; y; zÞ: The solution of Equation

41.13 is described by using gv ðr; r0Þ as follows:

pv ðrÞ ¼

ð ð ð

½mv ðr0Þ 2 div Fv ðr0Þ􀀉gv ðr; r0Þdx0dy0dz0 ð41:16Þ

41-4 Vibration and Shock Handbook

41.2.2.2 Radiation from Vibrating Small Body

A pulsating globe, as the simplest sound source, is

rarely realized in the real world. The approximated

monopole source required in most measurements

is an eight-sided polyhedron, as depicted in

Figure 41.2, where a loud speaker is installed on

each of the surfaces. The sound field radiated by a

thus approximated source is almost the same as

that by a pulsating globe at the far field, where

kr .. 1 is valid.

In this case, the sound field is represented

by Equation 41.16, assuming that Fv ðr0Þ ¼ 0;

mv ðr0Þ ¼ 2ivrUvdðnÞ; and dx0dy0dz0 ¼ dndS;

where the source mv ðr0Þ is distributed on a thin

layer of thickness, dn; on the spherical surface

element, dS: Substitution of these relationships

into Equation 41.16 yields the formula

pv ðrÞ ¼ 2

ð ð

ivrUv

4pR

eikRdS

¼ 2

ivr

4pR0 ð4pa2ÞUv eikR0 ð41:17Þ

where

ÐÐ

S dS means integration on the approximately

spherical surface, and then R ; lr 2 r0l ø

R0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ y2 þ z2

is applied. The final reduction of Equation 41.17 gives the same expression as is

deduced from Equation 41.3 and Equation 41.4.

The sound field generated by a monopole source is illustrated in Figure 41.1. This field is unchanged

with the presence of a rigid plane AB in Figure 41.3. It is clear, on account of symmetry, that the presence

of the rigid plane does not alter the sound field in any way, because only the tangential component of

particle velocity is induced on the plane. This utilization of the symmetry and construction of the semiinfinite

field is applied to the expression of the sound field generated by the baffled structure, as shown in

Figure 41.4.

Let us imagine a sound field generated by an oscillating small body in an infinite medium, as shown in

Figure 41.4(a). Since the oscillation is caused by an external force, the sound source is modeled by the

force 2Fv ðr0Þ: Therefore, the sound field is described by Equation 41.16, assuming mv ðr0Þ ¼ 0: After

applying a law of vector analysis, such as gv ðr; r0Þdiv Fv ðr0Þ·grad0 gv þ div0ðgv Fv Þ; the sound field is

FIGURE 41.2 Practical monopole sources.

Wavefronts

FIGURE 41.3 Field of a pulsating spherical source.

Source of Noise 41-5

represented by

pv ðrÞ ¼

ððð

Fv ðr0Þ·7 0gv ðr; r0Þdx0dy0dz0 ð41:18Þ

where

ÐÐÐ

div0ðgv Fv Þdx0dy0dz0 ¼

ÐÐ

S gv Fv dSnðx0; y0; z0Þ ¼ 0 is applied. Assume that a small sphere of

radius a0 is oscillating along the x-axis with angular frequency v: Instead of an oscillating

sphere with velocity Ut ¼ Uv e2ivt ; the sound field is generated by the concentrated body

force, FðtÞ ¼ 2mðd=dtÞU ðtÞ ¼ ivmUv e2ivt ; (m is the mass of the sphere), at the origin, r0 ¼ 0: Then

Fv ðr0Þ is expressed by Fv ðr0Þ ¼ ðFxv iÞdðx0Þdðy0Þdðz0Þ; Fxv ¼ imvUv : This approximate reduction is

appropriate when ka0 ,, 1 is valid. Substituting the approximation, Equation 41.18 is rewritten as

pv ðrÞ ø Fxv

›

›x0

gv ðr; r0; uÞ ¼ 2k2Dv

4pr

cos u 1 þ

􀀏 􀀐

eikr ð41:19aÞ

Dv ¼

krc

Fvx ð41:19bÞ

As the sound field is axisymmetric about x-axis, the sound pressure, pv ðr; uÞ; depends only on r and u;

where r and u are defined in Figure 41.4(b). The expression is applicable to the sound field generated by

an oscillating small body in a free space.

41.2.3 Radiation from a Plane Surface

41.2.3.1 Radiation from a Small Body in Infinite Plane Surface

The introduction of an infinite rigid plane surface to the sound field, as shown in Figure 41.3, simplifies

the formulation of the sound field generated by an oscillating body adjacent to a large plane.

FIGURE 41.4 Dipole field generated by an oscillating small body: (a) oscillating small body; (b) oscillating sphere

along x-axis.

41-6 Vibration and Shock Handbook

The configuration discussed in this section relates to a source in the presence of an infinite plane barrier,

so that the medium is confined to one side of the plane.

By taking the effect of the image caused by the rigid plane surface, the Green’s function, gv ðr; r0Þ; in this

case simplifies as given below, to what is called Rayleigh’s formula [2]:

gv ðr; r0Þ ¼

2pR

eikr ; R ¼ lr 2 r0l ð41:20Þ

where r0 denotes the projection position of the source on the surface. Therefore, Equation 41.16 is

rewritten as follows:

pv ðrÞ ¼

ð ð ð eikR

2pR ½mv ðr0Þ 2 div Fv ðr0Þ􀀉dx0dy0dz0 ð41:21Þ

41.2.3.2 Radiation from a Circular Piston

Let us consider the sound field generated by a rigid circular piston of radius a mounted flush with

the surface of an infinite baffle and vibrating with simple harmonic motion of angular frequency, v:

The solution of this example is applicable to a number of related problems, including the radiation

from the open end of a flanged organ pipe.

The coordinate system is shown in Figure 41.5, where the infinite baffle and the piston are placed in the

Oxy-plane. Note that the observation point, P, is denoted by r ¼ OP

􀀊!; while the source point, Q, is

denoted by r0 ¼ OQ

􀀊!:

Since sound is observed only in the semi-infinite plane where z . 0 is valid, only mv ðr0Þ remains

nonzero in Equation 41.21. As the velocity of the piston is denoted by Uv e2ivt along the z-axis,

mv ðr0Þ ¼ 2ivrUv dðz0Þ; is distributed only on the circular piston. Finally, Equation 41.21 is rewritten as

pv ðrÞ ¼

2ivrUv

ð ð

eikR

dx0dy0 ð41:22Þ

a 0

Q (x0,y0,0)

P (x,y,z)

P′(x,y,0)

FIGURE 41.5 The rigid circular piston and the coordinate system.

Source of Noise 41-7

Assuming a ,, r; the following approximation

can be made:

R ø r 2 ðx0 cos w þ y0 sin wÞ sin u;

ø 1

ð41:23Þ

where w and u denote the angles defining the

observation point, P, as shown in Figure 41.5. By

changing from Cartesian coordinates, x0; y0; to

polar coordinates, r; w0; such that x0 ¼ r cos w0;

y0 ¼ r sin w0 in the integral above, we rewrite

Equation 41.22 as

pv ðrÞ ¼ 2

ivrUv

2pR

eikr

ð2p

dw0

􀀐

ða

exp½2kr cosðw0 2 wÞ sin u􀀉rdr

ð41:24Þ

The integration is performed by introducing the

Bessel function of the first order, J1ðzÞ as follows:

pv ðrÞ ¼ 2

ivr eikr

2pr

fv ðuÞ;

fv ðuÞ ¼ pa2Uv

2J1ðka sin uÞ

ka sin u

􀀒 􀀓 ð41:25Þ

Note that dependency on w has disappeared in the integration process, which follows from the

axisymmetry of the sound field. The corresponding intensity at r is given by

Ir ðuÞ ø

lpv l

rc ¼

rcU2v

4r2 ðkaÞ2 2J1ðka sin uÞ

ka sin u

􀀒 􀀓2

ð41:26Þ

Figure 41.6 illustrates the dependency of Ir on u for two cases of ka: Note that, for the smaller

ka ¼ 2pa=l; Ir is almost independent on u, which is similar to the dependence of the monopole.

41.2.3.3 Radiation from a Rectangular Plate

An normal velocity distribution, uv ðx0; y0Þ; is prescribed over a baffled planar radiator located in the

plane z0 ¼ 0 in the region 2Lx # x0 # Lx ; 2Ly # y0 # Ly ; as shown in Figure 41.7.

In this case, the sound pressure field is represented by the following equation, similar to the previous

section:

pv ðrÞ ¼

2ivr

ðLy

2Ly

dy0

ðLx

2Lx

Uv ðx0; y0ÞeikR

dx0 ð41:27Þ

Assuming 2Lx ; 2Ly << r; the same approximation in as Equation 41.23 is acceptable. Therefore,

pv ðrÞ ¼ pv ðr; u; fÞ takes the form [2]:

pv ðr; u; fÞ ¼

2ivr eikr

2pr

ðLy

2Ly

dy0

ðLx

2Lx

Uv ðx0; y0Þ exp½2ik sin uðx0 cos f þ y0 sin fÞ􀀉dx0 ð41:28Þ

The result of the above integration, assuming Uv ðx0; y0Þ ¼ Uv ¼ const; which means the rigid

rectangular piston oscillates with amplitude Uv along the z-axis, is

pv ðr; u; fÞ ¼

ivrUv

2pr ð4Lx Ly Þeikr SðkLx sin u cos fÞSðkLy sin u sin fÞ ð41:29aÞ

−2 −1 1 2

λ/Δ = 10

λ/Δ = 2

FIGURE 41.6 Directivity of sound intensity generated

by an oscillating piston.

41-8 Vibration and Shock Handbook

where the function SðzÞ is defined by

SðzÞ ¼

sin z

z ð41:29bÞ

Now, let us discuss the case of flexural plate, which has the same lengths, 2Lx and 2Ly ; but Uv ðx0; y0Þ

represents the flexural vibration mode. Flexural mode patterns of rectangular panels take the general

form of contiguous regions of roughly equal area and shape, which vary alternately in vibration phase,

and are separated by nodal lines of zero vibration.

For simply supported edges, the normal vibration velocity distribution is (see Chapter 4)

Uv ðx0; y0Þ ¼ Umn sin

mpx

2Lx

􀀏 􀀐

sin

npy

2Ly

􀁻 !

; ð0 # x # 2Lx ; 0 # y # 2LyÞ ð41:30Þ

Note that the coordinate system O-x0y0z0 shifts its origin from the center point of the panel to the edge

at the leftmost and frontmost point in Figure 41.7.

The approximation represented by Equation 41.23 is valid as well in this case. The result of integration

is rather simple as given below [3]:

pv ðr; u; fÞ ¼

2ivrUmn ekr

2pr

4Lx Ly

mnp2 ð21Þm eia 2 1

ða=mpÞ2 2 1

" #

ð21Þn eib 2 1

ðb=npÞ2 2 1

" #

ð41:31Þ

where a and b are defined by

a ¼ 2kLx sin u cos f; b ¼ 2kLy sin u sin f ð41:32Þ

The corresponding intensity, Ir ðu; wÞ; at r is expressed by

Ir ðr; u; fÞ ¼

lpv l2

rc ¼ 4rclUmnl2 4kLx Ly

p3rmn

􀀏 􀀐2

cos

sin

􀀏 􀀐cos

sin

􀀏 􀀐

½ða=mpÞ2 2 1􀀉½ðb=npÞ2 2 1􀀉

8>>>><

>>>>:

9>>>>=

>>>>;

ð41:33Þ

P(x,y,z)

Q(x0,y0)

2Lx

2Ly 0

fο f

FIGURE 41.7 A flexural rectangular plate and the coordinate system.

Source of Noise 41-9

where cosða=2Þ is used when m is an odd integer, and sinðb=2Þ is used when m is an even integer; cosðb=2Þ

is used for even n and sinðb=2Þ for odd n:

41.2.4 Estimation of Noise-Source Sound Power

41.2.4.1 Power Conversion Factor of Machinery

It is often necessary to estimate the expected sound power that a particular machine might introduce into

an environment [4]. One way where such an estimate may be approached for a particular class of

machine is by means of the sound power conversion factor, hn: This factor is defined as

hn ¼

Pm ð41:34Þ

where P ¼ sound power of the machine (W), and Pm ¼ power of the machine (W). This relationship is

valid for both mechanical and electrical machinery. The conversion factors for some common noise

sources are given in Table 41.1.

Example 41.1

Estimate the sound power level of a typical 1-kW electric motor that operates at 1200 rpm.

Solution

From Table 41.1, we find that for typical electric motors, hn ¼ 1 £ 1027: Thus, using Equation 41.34, we

obtain

P ¼ hnPm ¼ ð1 £ 1027Þ £ 1000 ¼ 1024ðWÞ

as the total sound power of motor. Then using Equation 37.7, LW is given by

LW ¼ 10 log

1024

10212

􀁻 !

¼ 80ðdBÞ

41.2.4.2 Fan Noise

We are familiar with noise nuisance caused by a domestic ventilating fan. The mechanical power of the

fan is expressed by

Pm ¼ pTQ ð41:35Þ

where pT denotes the total pressure rise through the fan and Q denotes the volumetric flow rate. According

to the law of sixth power of flow velocity deduced by the aeroacoustics theory, sound power of the fan is

proportional to pT2:5 Q: Therefore, the specific ratio kT defined below is more useful than hn for the fan:

kT ¼

pT2:5 Q ð41:36Þ

TABLE 41.1 Estimated Sound Power Conversion Factors for Common Noise Sourcesa

Noise Source Conversion Factor

Low Midrange High

Compressor, air (1 – 100 hp) 3 £ 1027 5.3 £ 1027 1 £ 1026

Gear trains 1.5 £ 1028 5 £ 1027 1.5 £ 1026

Loud speakers 3 £ 1022 5 £ 1022 1 £ 1021

Motors, diesel 2 £ 1027 5 £ 1027 2.5 £ 1026

Motors, electric (1200 rpm) 1 £ 1028 1 £ 1027 3 £ 1027

Pumps, over 1600 rpm 3.5 £ 1026 1.4 £ 1025 5 £ 1025

Pumps, under 1600 rpm 1.1 £ 1026 4.4 £ 1026 1.6 £ 1025

Turbines, gas 2 £ 1026 5 £ 1026 5 £ 1025

a Total sound power for the four octave bands from 500 to 4000 Hz.

Source: Irwin, J. D. and Graf, E. R. 1979. Industrial Noise and Vibration Control, Prentice Hall,

Englewood Cliffs, NJ.

41-10 Vibration and Shock Handbook

Furthermore, usually the specific fan noise level, KT; as given below is more useful than kT:

KT ¼ LW 2 10 logðpT2:5 QÞdB ð41:37Þ

where pT denotes the total pressure rise in mmAg, Q denotes the flow rate in m3/min, LW denotes the total

sound power level in dB, and KT represents the radiation efficiency of the fan noise. This efficiency varies

with the type of the fan and with the flow rate when the model is assigned. In particular, KT will be the

lowest at the flow rate at which the aerodynamic power efficiency is the highest. Therefore, we will have the

best advantage on the sound environment when the fan is operated at the highest efficiency. For

convenience of design, the LW is often evaluated by A-weighted total sound level (dB-A) not by linear total

level (dB). Table 41.2 gives the specific fan noise level evaluated with dB-A for five types of the low-pressure

fans.

Example 41.2

Consider a ventilating axial-flow fan, the specifications of which are: pT ¼ 10 mmAq; Q ¼ 30 m3=min;

D ¼ 30 cm (diameter of the duct containing fan rotor). Estimate the directional distribution of

intensity Ir ðuÞ; assuming r ¼ 3 m and f ¼ 150 Hz for the main component of the fan noise, and

KT ¼ 279:0 dB-A from Table 41.2. The fan noise is assumed to be radiated as a plane sound wave at the

mouth of the duct.

Solution

First, we modify the KT in dB-A to that of linear scale. From the frequency response for A-weighting

network shown in Figure 2.5, for f ¼ 150 Hz; the modification is found as DKT ¼ 15 dB: Then, the

modified specific fan noise level is KT ¼ 279:0 þ 15:0 ¼ 264:0 dB: By using Equation 41.37, LW ¼

KT þ 25 log PT þ 10 log Q ¼ 264 þ 25 þ 15 ¼ 224 dB is obtained. This means the emitted total

sound power is W ¼ 10224=10 ¼ 3:98 £ 1023 W: Since we assume a plane sound wave at the mouth of

the duct for the fan noise, we can use the sound radiation model of a circular piston with a ¼ 15 cm

for the radiated sound wave. In our case ka ¼ 2pa=ðc=f Þ ¼ 0:415; or l=a ¼ 15:1: A monopole model will

be valid for the directional distribution of intensity from Figure 41.6. Therefore, Ir ¼ W =2pr2 ¼

0:704 £ 1024 W=m2 or lpl2 ¼ Irrc ¼ 2:92 £ 1022ðPa2Þ: This is the same as the sound pressure level

Lp ¼ 10 loglpl2=pref ¼ 78:6 dB or LpðAÞ ¼ Lp 2 15 ¼ 63:6 dB-A:

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