1

Acoustic Energy and IntensityHongbin Ju

Department of MathematicsFlorida State University, Tallahassee, FL.32306

www.aeroacoustics.infoPlease send comments to: hju@math.fsu.edu

In this chapter we will discuss acoustic energy, its different components and theirrespective balance equations. Emphasis will be on definition of acoustic intensity.Acoustic intensity is an important quantity describing a sound field. It is the acousticenergy flux past through unit area during unit time. It is straightforward to define theinstantaneous acoustic intensity from acoustic energy balance equation. However, inengineering applications most of the concerns are time averaged energy flux. Theinstantaneous acoustic intensity needs to be simplified for a time steady process. Thoseterms with zero time average value should be dropped. And those terms that have timeaverage values but don't contribute to mean acoustic energy should also be dropped. Inthis way only fluctuation terms with even orders in the instantaneous acoustic intensityare to be kept in the final acoustic intensity definition.

Ideal (

†

s ij = -pdij ), isentropic medium (

†

k ∂T∂xi

= 0) will be assumed. No body force. The

unsteady flow is only caused by sound waves. And linear disturbance is assumed.

Acoustic Energy Balance Equations

Decomposition of Variables

All undisturbed flow variables are subscribed by ‘0’, such as,

†

p0,

†

r0,

†

r u 0. With soundwaves, any variable in the disturbed flow can be decomposed into a time averaged partand a fluctuation part,

†

p = p + p' , (1)

†

p = limT Æ•

1T

pdtt

t +T

Ú ,

†

p'= p - p ,

†

limT Æ•

1T

p'dtt

t +T

Ú = 0.

Linear acoustic waves are fluctuations around the undisturbed flow, so

†

p0 = p . (2)

For very strong waves the mean flow may be changed (acoustic streaming), which is outof the scope of this chapter. Only linear analysis is discussed here.

For a product of two (or more) variables, such as

†

pu ,

2

†

pu = limT Æ•

1T

p0u0 + p0u'+ p'u0 + p'u'( )dtt

t +T

Ú = p0u0 + p'u'. (3)

Euler Equations with Decomposed Variables

One can rewrite Euler equations using the time averaged and fluctuation variables inEq.(1).

The continuity equation is:

†

∂r∂t

+∂ ru j( )

∂x j

= 0, or, (4)

†

∂r'∂t

+∂

∂x j

r0u0 j + r'u0 j + r0u' j +r'u' j( ) = 0. (5)

The momentum equation is:

†

∂ rui( )∂t

+∂

∂x j

ruiu j + pdij( ) = 0 , or, (6)

†

∂∂t

r'u0i + r0u'i +r'u'i( ) +∂

∂x j

r0u0iu0 j + r'u0iu0 j + r0u'i u0 j + r0u0iu' j(

+r'u'i u0 j + u0ir'u' j +r0u'i u' j +r'u'i u' j + p0dij + p'dij ) = 0. (7)

The conservative form of energy equation is:

†

∂e∂t

+∂

∂x j

(eu j + pu j ) = 0 , (8)

where

†

e = rE +12

ruiui (9)

is the total energy in unit fluid volume. The total energy is composed of two types of

energies. One is the internal energy

†

rE ; the other is the kinetic energy

†

12

ruiui . Since the

perturbations are only acoustic waves, it has only the translational kinetic energy (norotational kinetic energy.)

†

I j = eu j + pu j (10)

3

is the instantaneous acoustic intensity. It also includes two parts. One is the energyconvection intensity

†

eu j ; the other is the energy production across the fluid elementsurface, i.e. work done by pressure

†

pu j .

The conservation form of energy equation (8) shows that, the total energy flux throughthe surface enclosing a fluid volume equals to the total energy increase rate in thisvolume. For a time steady process, the time average of the total energy in the volume isconstant. Therefore the time averaged equation of (8) is:

†

∂I j∂x j

= 0, (11)

In a steady process, the time average of total energy flux across the enclosing surface iszero.

Eq.(10) is a general definition of acoustic intensity. But it may have never been used. Wewill show that the acoustic intensity has many components. Each component has differentcontributions to the acoustic energy. Some of them can be dropped so that a simplifiedacoustic intensity formation can be obtained.

Acoustic Energy Components and Their Equations

For isentropic flow, pressure changes adiabatically as density:

†

p = rg ⋅ Constant , or

†

∂p /∂r = gp /r = a2, (12)

and the internal energy density is:

†

rE =p

g -1. (13)

For an isentropic process, the continuity and momentum equations (5)&(7) with equationof state (12) is a complete system for solving the sound field. Energy equation (8) is onlya linear combination of continuity and momentum equations (5)&(7). It is not needed forsolving the sound field. It is only used for energy balance analysis.

With the isentropic relation, continuity equation (5) can be rewritten as:

†

∂∂t

pg -1

Ê

Ë Á

ˆ

¯ ˜ +

∂∂x j

pg -1

u j

Ê

Ë Á

ˆ

¯ ˜ + p

∂u j

∂x j

= 0 . (14)

4

This is the equation for internal energy balance. The increase rate of internal energy is

equal to the internal energy convection into the fluid element

†

pg -1

u j , plus work done by

the pressure on the fluid element volume change

†

p∂u j

∂x j

.

Multiplying both sides of momentum equation (7) by

†

ui and making use of thecontinuum equation (4), we have:

†

∂∂t

12

ruiuiÊ

Ë Á

ˆ

¯ ˜ +

∂∂x j

12

ruiuiu jÊ

Ë Á

ˆ

¯ ˜ + u j

∂p∂x j

= 0 . (15)

This is the kinetic energy balance equation. The increase rate of kinetic energy is the sum

of convection of kinetic energy into the fluid

†

12

ruiuiu j and work done by the net force on

the movement of fluid element

†

u j∂p∂x j

.

Adding Eqs.(14) and (15) together one can obtain the total energy balance equation (8).

Acoustic Intensity in Stationary Ideal Homogeneous Medium

First we discuss the simplest case: quiescent mean flow

†

r u 0 = 0 .

Acoustic Energy Balance Equation

Since pressure changes adiabatically as density, we have from Eq.(12):

†

p = p0 +∂p∂r 0

r'+ 12

∂ 2 p∂r2

0

r'2 +L = p0 + a02r'+ g -1

2r0

a02r'2 +L,

i.e.,

†

p'= a02r'+ g -1

2r0

a02r'2 +L, (16)

where

†

a0 = ∂p0 /∂r0 = gp0 /r0 . Therefore, the internal energy

†

rE =p

g -1=

1g -1

p0 + a02r'+ g -1

2r0

a02r'2 +L

Ê

Ë Á

ˆ

¯ ˜ , (17)

The zero, first, and second order fluctuations of the internal energy are

†

p0 /(g -1) ,

†

a02r' /(g -1) , and

†

a02r'2 /(2r0) .

5

The kinetic energy to the second order is:

†

12

ruiui =12

r0u'i u'i. (18)

The total acoustic energy fluctuation to the second order is:

†

e'= 1g -1

a02r'+ 1

2r0

a02r'2 +

12

r0u'i u'i , (19)

and the acoustic energy balance equation is:

†

∂∂t

1g -1

a02r'+ 1

2r0

a02r'2 +

12

r0u'i u'iÊ

Ë Á

ˆ

¯ ˜ +

∂∂x j

p0

g -1u' j +

a02

g -1r'u' j + p0u' j + p'u' j

Ê

Ë Á

ˆ

¯ ˜ = 0. (20)

Decomposition of Acoustic Energy Balance Equation and Acoustic Intensity

From Eq.(20), by dropping first order terms one may define the time averaged acousticintensity as:

†

I j =a0

2

g -1r'u' j + p'u' j . (21)

We will show that the convection intensity,

†

a02

g -1r'u' j , contributes nothing to the total

acoustic energy and thus can be dropped.

Energy balance equation (20) can be separated into two equations respectively aboutinternal energy such as Eq.(14) and kinetic energy such as Eq.(15). To facilitate thedefinition of mean acoustic intensity, here we will decompose the energy equation basedon the work done by the pressure on the fluid element:

†

pu j = p0u' j + p'u' j . (22)

About the first order internal energy,

†

a02r' /(g -1) , from continuity equation (5), we have:

†

∂∂t

a02

g -1r'

Ê

Ë Á

ˆ

¯ ˜ +

∂∂x j

p0

g -1u' j +

a02

g -1r'u' j + p0u' j

Ê

Ë Á

ˆ

¯ ˜ = 0 . (23)

This shows the rate of the first order internal energy change due to the work done byambient pressure on the acoustic velocity

†

p0u' j . In this equation the time average of

6

†

a02

g -1r'u' j is not zero. However the time average of energy

†

1g -1

a02r' is zero. Which

means

†

a02

g -1r'u' j contributes no mean energy to the sound field and should be excluded

from the acoustic intensity.

Subtracting Eq.(23) from (20), we have:

†

∂∂t

12r0

a02r'2 +

12

r0u'i u'iÊ

Ë Á

ˆ

¯ ˜ +

∂∂x j

p'u' j( ) = 0 . (24)

This equation shows the changing rate of acoustic energy due to the work done by

acoustic pressure on the acoustic velocity,

†

p'u' j . Of the acoustic energy,

†

12

r0u'i u'i is the

acoustic kinetic-energy density, and second order internal energy

†

12r0

a02r'2 ª

12r0a0

2 p'2 is

the acoustic potential-energy density.

From equation (24), one can define the acoustic intensity as:

†

I j = p'u' j . (25)

One may see that this energy flux only affects the second order energy fluctuation

†

12r0

a02r'2 +

12

r0u'i u'i instead of the total acoustic energy fluctuation

†

e' . Since the time

average of the first order fluctuation internal energy

†

a02r' /(g -1) is zero, this definition of

acoustic energy flux can only work in the sense of time average. It is widely used forstationary medium.

Acoustic Intensity in Uniform Mean Flow

Acoustic Energy Balance Equation

The total energy per unit volume of the mean flow is:

†

e0 =p0

g -1+

12

r0u0iu0i, (26)

and to the second order the acoustic energy fluctuation in unit volume of the fluid is:

†

e'= a02

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'+r0u0iu'i +u0ir'u'i +

12r0

a02r'2 +

12

r0u'i u'i. (27)

7

The time averages of first order terms are zero.

The acoustic energy balance equation is:

†

∂e'∂t

+∂I j

∂x j

= 0, (28)

where

†

I j =a0

2

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u0 j +

p0

g -1+

a02

g -1r'+ 1

2ru0iu0i

Ê

Ë Á

ˆ

¯ ˜ u' j + p0u' j

+ru0iu'i u0 j + ru0iu'i u' j + p'u0 j

+1

2r0

a02r'2 +

12

r0u'i u'iÊ

Ë Á

ˆ

¯ ˜ u0 j + p'u' j

. (29)

Decomposition of Acoustic Energy

From continuity equation (5), we can have:

†

∂∂t

a02

g -1r'+ 1

2r'u0iu0i

Ê

Ë Á

ˆ

¯ ˜

= -∂

∂x j

a02

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u0 j +

p0

g -1+

a02

g -1r'+ 1

2ru0iu0i

Ê

Ë Á

ˆ

¯ ˜ u' j + p0u' j

È

Î Í

˘

˚ ˙

. (30)

This part of the first order fluctuation energy balance is due to the work done by theambient pressure on the acoustic velocity

†

p0u' j . The time average of this part of energy iszero, therefore this part of the energy flux should not be included in the acousticintensity.

From momentum equation (7) we have the energy balance equation:

†

∂∂t

r0u0iu'i +u0ir'u'i( )

= -∂

∂x j

ru0iu'i u0 j + ru0iu'i u' j + p'u0 j( ). (31)

This part of energy balance is due to the work done by the acoustic pressure on the meanvelocity

†

p'u0 j . The energy has second order term with nonzero time average.

The equation for the rest of the acoustic energy is:

8

†

∂∂t

12r0

a02r'2 +

12

r0u'i u'iÊ

Ë Á

ˆ

¯ ˜

= -∂

∂x j

12r0

a02r'2 +

12

r0u'i u'iÊ

Ë Á

ˆ

¯ ˜ u0 j + p'u' j

È

Î Í

˘

˚ ˙

. (32)

The energy terms are all second order with nonzero time average. It is due to the work byacoustic pressure and acoustic velocity

†

p'u' j .

The three acoustic energy balance equations, (29), (30), (31), are decomposed bases on

†

pu j = p0u' j + p'u0 j + p'u' j .

Acoustic Intensity

According to the above analyses, the acoustic intensity in j direction can be defined as:

†

I j = r0u0i u'i u' j + u0iu0 j r'u'i +1

2r0

a02r'2 +

12

r0 u'i u'iÊ

Ë Á

ˆ

¯ ˜ u0 j + p'u' j . (33)

All first order fluctuation terms are averaged out. Third and higher order terms areignored. According to Lighthill (p.11), in linear acoustics first order quantities areretained and quadratic terms neglected in linear continuity and momentum equations,while in linear energy equation, first order terms should be absent (not neglected),quadratic terms retained, and cubic terms or above neglected.

A similar definition was proposed by Cantrell and Hart (Cantrell&Hart 1964, Eq.(13)) forisentropic uniform mean flows:

†

I j = r0u0i u'i u' j + u0iu0 j r'u'i +1

r0a02 u0 j p'2 + p'u' j . (34)

This definition has been widely accepted (Tester et.al.2004, Morfey1971, Möhring1971).One can prove from Eq.(15-0) in cht1.doc, that when there is no mean flow, or the wavenumber in mean flow direction is real, the time mean acoustic potential energy is equal to

the acoustic kinetic energy:

†

12r0

a02r'2 =

12r0a0

2 p'2 =12

r0 u'i u'i . With this property,

Eq.(34) is the same as Eq.(33).

Acoustic Intensity in Nonuniform Mean Flow

The analysis of acoustic energy in nonuniform flows is similar to that in uniform flows.The total mean flow energy per unit volume is described by Eq.(26), and to the second

9

order the acoustic energy in unit volume of the fluid is as in Eq.(27). The acoustic energybalance equation is Eq.(28). However the acoustic intensity in nonuniform flow is

†

I j = e0 + p0( )u0 j +a0

2

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u0 j + e0u' j +

a02

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u' j + p0u' j

+ru0iu'i u' j +ru0iu'i u0 j + p'u0 j

+1

2r0a02 p'2 +

12

r0u'i u'iÊ

Ë Á

ˆ

¯ ˜ u0 j + p'u' j

. (35)

instead of Eq.(29).

We may drop the odd order fluctuation terms and define the acoustic intensity as:

†

I j = e0 + p0( )u0 j +a0

2

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u' j +r0u0iu'i u' j +u0iu0 jr'u'i

+1

2r0a02 p'2 +

12

r0u'i u'iÊ

Ë Á

ˆ

¯ ˜ u0 j + p'u' j

. (36)

From continuity equation (5), we have:

†

∂∂t

a02

g -1r'+ 1

2u0iu0ir'

Ê

Ë Á

ˆ

¯ ˜

= -∂

∂x j

e0 + p0( )u0 j +a0

2

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u0 j + e0u' j +

a02

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u' j + p0u' j

Ê

Ë Á

ˆ

¯ ˜

+ru j∂

∂x j

a02

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜

. (37)

According to Bernoulli’s theorem, for a frictionless non-conducting flow,

†

DDt

a02

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ = 0. (38)

It is a constant along a streamline. If the flow energy at upstream is uniform, then Eq.(37)can be written as,

†

∂∂t

a02

g -1r'+ 1

2u0iu0ir'

Ê

Ë Á

ˆ

¯ ˜

= -∂

∂x j

e0 + p0( )u0 j +a0

2

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u0 j + e0u' j +

a02

g -1+

12

u0iu0i

Ê

Ë Á

ˆ

¯ ˜ r'u' j + p0u' j

È

Î Í

˘

˚ ˙

. (39)

10

Subtracting this equation from Eq.(28), then the acoustic intensity can be defined as:

†

I j = r0u0i u'i u' j + u0iu0 j r'u'i +1

2r0a02 p'2 +

12

r0 u'i u'iÊ

Ë Á

ˆ

¯ ˜ u0 j + p'u' j , (40)

which is exactly the same as Eq.(33).

Applications of Acoustic Intensity

In this section we will give an application example of the acoustic intensity (33) or (34)in hard wall circular duct.

From Eq.(11), the time averaged acoustic energy flux across an enclosed surface is:

†

r I ⋅ d

r S Ú = 0. (41)

For a circular duct in Fig.1, we choose the enclosing surface shown by the shaded area.At cross section A, the sound power (sound energy across this section in unit time) is:

†

WA = I xrdfdr0

2p

Ú0

R

Ú . (42)

Fig1, Control surface in the circular duct.

Sound powers across section B,

†

WB , and sound power across the duct wall,

†

WW , can bedefined similarly. From Eq.(41), we have:

†

WA -WB = WW . (43)

This equation means from sound power distribution in axial direction one can computethe sound power absorbed by the duct wall. This turns out to be very useful for liner ductanalysis.

11

Sound Power in Axial Direction for Hard Wall Circular Duct

Dimension scales used here are: length: duct diameter D, velocity: sound speed

†

a0,density:

†

r0, pressure:

†

r0a02 , energy flux:

†

r0a03 , and sound power:

†

r0a03D2 .

Acoustic intensity in axial direction from (34) (Cantrell&Hart) is:

†

I x = 1+ M 2( )p'u' + M p'2 + u'2( ) . (44)

Assume the next form of the waves in the duct for circular frequency

†

w :

†

p'(x,r,f,t) = Real ˆ p mn (x)Jm (bmnr)ei(mf-wt )

n=1

•

Âm=-•

•

ÂÈ

Î Í

˘

˚ ˙ ,

u'(x,r,f,t) = Real ˆ u mn (x)Jm (bmnr)ei(mf-wt )

n=1

•

Âm=-•

•

ÂÈ

Î Í

˘

˚ ˙ .

(45)

m, n are respectively the circumferential and radial mode numbers. mnth mode will beused to indicate the mode with circumferential mode number m and radial mode numbern.

†

Jm (bmnr) is the first kind Bessel function. For hard wall,

†

bmn is real.

By means of orthogonality of Bessel function, one can show that:

†

p'u'rdrdf0

R

Ú0

2p

Ú = Qmn1T

Real ˆ p mn (x)ei(mf-wt )[ ]Real ˆ u mn (x)ei(mf-wt )[ ]dfdt0

2p

Út

t +T

Ún

ÂmÂ

= p QmnReal ˆ p mn (x) ˆ u mn* (x)[ ]

nÂ

m .

(46)

where * represents complex conjugate, and,

†

Qmn = Jm2 (rbmn )rdr =

12

R2, for m = 0,n =1;

12

R2 - m2 /bmn2( )Jm

2 (Rbmn ), other m,n.

Ï

Ì Ô

Ó Ô 0

R

Ú (47)

Similar equations hold for

†

p'2rdrdfdt0

R

Ú0

2p

Ú and

†

u'2rdrdf0

R

Ú0

2p

Ú .

Therefore, the total sound power is the sum of all sound powers calculated from eachmnth mode:

†

W = Wmnn

Âm , (48)

12

†

Wmn = pQmn 1+ M 2( )Real ˆ p mn ˆ u mn*( ) + M ˆ p mn

2+ ˆ u mn

2( )[ ] .

Next we will discuss the sound power for a single mnth mode.

mnth Mode in One Axial Direction

†

ˆ p mn = Amneiymneiamnx,

ˆ u mn =amn

w - Mamn

ˆ p mn . (49)

†

Amn and

†

ymn are real numbers.

For a cut-on mode,

†

w ≥ 1- M 2bmn ;

†

amn± =

-wM ± w 2 - (1- M 2)bmn2

1- M 2 is real;

†

Wmn = pQmn Amn2 1+ M 2( ) amn

w -amn M+ M 1+

amn

w -amn M

2Ê

Ë Á Á

ˆ

¯ ˜ ˜

È

Î

Í Í

˘

˚

˙ ˙ . (50)

For a cut-off mode,

†

w < 1- M 2bmn ;

†

amn± =

-wM ± i (1- M 2)bmn2 -w 2

1- M 2 is complex;

†

Wmn = pQmn Amn2 e-2xImg(amn ) 1+ M 2( )Real amn

w -amn MÊ

Ë Á

ˆ

¯ ˜ + M 1+

amn

w -amn M

2Ê

Ë Á Á

ˆ

¯ ˜ ˜

È

Î

Í Í

˘

˚

˙ ˙ .

But since

†

1+ M 2( )Real amn

w -amn MÊ

Ë Á

ˆ

¯ ˜ + M 1+

amn

w -amn M

2Ê

Ë Á Á

ˆ

¯ ˜ ˜ = 0 ,

we have

†

Wmn = 0. (51)

Wave with cutoff mode is called evanescent wave. An evanescent wave transports no netacoustic energy (in stationary media), for pressure and axial velocity are 900 out of phase.

13

There is an abrupt change of acoustic intensity from cut-on modes to cut-off modes(Pierce, p316). Physics meaning of evanescent waves (cut-off modes) in duct is: thewaves decay very fast away from its source. But the medium is ideal and thus the decayis not due to viscous dissipation. Each fluid element oscillates with sound energy. Thisenergy is established at the unsteady stage when there is transmitted energy across theduct section. After it gets to the steady stage, there is no net transmitted sound across ductsection.

mnth Mode in Both Axial Directions

†

ˆ p mn = Amn+ eiymn

+

eiamn+ x + Amn

- eiymn-

eiamn- x,

ˆ u mn = Bmn+ Amn

+ eiymn+

eiamn+ x + Bmn

- Amn- eiymn

-

eiamn- x,

(52)

†

Bmn± =

amn±

w - Mamn± .

For a cut-on mode,

†

w ≥ 1- M 2bmn ,

†

amn± =

-wM ± w 2 - (1- M 2)bmn2

1- M 2 ,

†

Wmn = Wmn+ + Wmn

- + Wmnc ,

†

Wmn± = pQmn Amn

±( )2

1+ M 2( ) amn±

w -amn± M

+ M 1+amn

±

w -amn± M

2Ê

Ë Á Á

ˆ

¯ ˜ ˜

È

Î

Í Í

˘

˚

˙ ˙ .

The cross term is zero:

†

Wmnc = pQmn Amn

+ Amn- cos(ymn

+ -ymn- + amn

+ x -amn- x) 1+ M 2( ) Bmn

+ + Bmn-( ) + 2M 1+ Bmn

+ Bmn-( )[ ]

= 0

Therefore,

†

Wmn = Wmn+ + Wmn

- . (53)

For a cut-off mode,

†

w < 1- M 2bmn ,

†

amn± =

-wM ± i (1- M 2)bmn2 -w 2

1- M 2 ,

†

Bmn± =

-Mbmn2 ± iw (1- M 2)bmn

2 -w 2

w 2 + M 2bmn2 ,

†

Bmn+( )

*= Bmn

- ,

†

Bmn-( )

*= Bmn

+ .

14

†

Wmn = Wmn+ + Wmn

- + Wmnc ,

Since

†

Wmn+ = Wmn

- = 0 ,

we have

†

Wmn = Wmnc . (54)

†

Wmnc = 2pQmn Amn

+ Amn-

⋅ 1+ M 2( )Real Bmn+ ei(ymn

+ -ymn- )[ ] + M cos(ymn

+ -ymn- ) + Real Bmn

+ 2ei(ymn+ -ymn

- )( )[ ]{ },

In Summary

(1)Total sound power is the sum of sound powers calculated from each individual mnthmode;(2)For each cut-on mnth mode,

†

Wmn ≠ 0 is constant in axial direction;(3)For each cut-off mnth mode, if the wave only ‘propagates’ in one direction,

†

Wmn = 0;if the wave ‘propagates’ in both directions,

†

Wmn ≠ 0 is constant in axial direction.