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Acoustic energy in ducts: Further observations
.Journal
qf Sound
and Vibration
ACOUSTIC
(1979) 62(4), 517-532
ENERGY IN DUCTS:
FURTHER
OBSERVATIONS
W. EVERSMAN~ Department
of’Mechanical Engineering, (Received
University qf Canterhurv,
22 April 1978, and in revised,form
Christchurch.
New Zcuhd
8 August 1978)
The transmission of acoustic energy in uniform ducts carrying uniform flow is investigated with the purpose of clarifying two points of interest. The two commonly used definitions of acoustic “energy” flux are shown to be related by a Legendre transformation of the Lagrangian density exactly as in deriving the Hamiltonian density in mechanics. In the acoustic case the total energy density and the Hamiltonian density are not the same which accounts for two different “energy” fluxes. When the duct has acoustically absorptive walls neither of the two flux expressions gives correct results. A re-evaluation of the basis of derivation of the energy density and energy flux provides forms which yield consistent results for soft walled ducts.
1. INTRODUCTION
In order to assess the acoustic performance of duct non-uniformities and duct linings it is often convenient to compare incident acoustic power and transmitted acoustic power. This is a straightforward procedure in ducts with no flow. When a mean flow is present the problem is more complicated and it is by no means clear how acoustic energy density and energy flux are to be defined in the general case when the flow is non-uniform. At the present time it does not seem possible to give unique definitions which lead to practical calculations except in special cases. Partly because of this difficulty the rigorous application of the concept of power insertion loss or transmission loss is usually restricted in practical applications to calculations in uniform ducts. For example, this is done by comparing acoustic powers in uniform ducts on either side of a duct non-uniformity or by comparing acoustic powers at various axial locations in a uniform lined duct using an appropriately extended definition of power. Even in these cases with axial uniformity the presence of a flow which is non-uniform across the duct presents fundamental problems. Morfey’s [l] definition of energy density and flux in hard walled ducts appears to have gained the widest acceptance because it satisfies a conservation law in any irrotational constant entropy flow. However the extension of its use to rotational flows depends on the separation of rotational and irrotational parts of the acoustic field. Miihring’s [2] approach to an energy equation with no source terms is applicable in simple terms to sheared flows in uniform ducts with hard or soft walls. Perhaps because Miihring’s energy equation is based on Clebsch potentials. a concept somewhat removed from the usual duct acoustics theory, or perhaps because the types of flows in which the particularly simple results are useful have not been dominant in the literature, it does not appear to have replaced the more elementary type of energy expressions. Hence, in the literature energy flux calculations are generally restricted to uniform flows in uniform ducts. When this is the case the definition of energy density and tNow Missouri
at the Department
of Mechanical and Aerospace Engineering.
University
of Missouri-Rolla.
Rolla.
65401. U.S.A. 517
0022-460X/79104053
7 + 16 SO2.00/0
@ 1979 Academic
Press Inc. (London)
Limited
518
W. EVERSMAN
energy flux originally put forward by Ryshov and Shefter [3] for hard walled ducts, and used subsequently by Eversman [4], becomes equally valid. This being the case, the definitions of Morfey and Ryshov and Shefter are often used interchangeably as a basis for simple calculations of insertion loss or transmission loss. Both definitions of energy satisfy a conservation law for uniform flow and the relationship between them is discussed here. The use of the Morfey and Ryshov and Shefter energy definitions for lined ducts is tempting and has appeared in the literature [5]. As noted by Tester [6] anomalous results can occur, particularly in calculations of the energy flux for heavily attenuated modes. Numerical experimentation has confirmed this observation and has led to an extension of the hard walled duct energy definitions to yield relationships which satisfy energy equations with source terms. These energy equations introduce an acoustic flux which correctly relates to the direction of propagation for all modes.
2. DEFINITIONS
OF ACOUSTIC
ENERGY
DENSITY
AND ENERGY
FLUX
Candel [7] has reviewed much of the literature dealing with acoustic energy principles in general and their application in ducts in particular. He divides the energy densityenergy flux approaches into two types. Type I are typified by the Morfey relationships, E, = (1/2pc2)p2 + $P
I$, = pP + p(Vo.rq
+ (l/c2)&
+ (l/pc2)Vop2
I$,
(1)
+ (l/C2)Vcl(V0~Qp,
and Type II by the Ryshov and Shefter relationships,
fin = PV + V0[(1/2pc2)p2 + $V=2]. In these equations the energy density E and energy flux fi are functions of the acoustic perturbations in velocity and pressure v and p. The mean flow velocity is denoted by v0 and the ambient density and speed of sound by p and c. The Type I expressions satisfy a conservation law of the type dE/at + divfi
= 0
in the irrotational isentropic case. The Type II expressions have been shown to satisfy the conservation law when the mean flow is uniform. In the case of irrotational perturbations on a uniform flow both types are valid. In this case, it is possible to show that the two definitions of acoustic energy density and energy flux are related. When uniform flow is present in a uniform duct the isentropic acoustic problem is governed by the linearized continuity and momentum equations (l/c’)[a/&
+ v,, . grad]p + p div v = 0,
[a/& + p,,.grad]v
= -(l/p)gradp.
When the acoustic perturbations are irrotational a single equation in the velocity potential 4, ~24 =
(ip)[ajat
(5)
equations (5) and (6) can be replaced by
+
uajaij24,
(7)
ACOUSTIC
ENERGY
519
IN DUCTS
where ‘/ = grad 4,
p = -p[~3/?t
(8.9)
+ U6/i;x]+.
In equations (7H9) the mean flow is in the direction of the x axis, and p,, = C’z In terms of the velocity potential the Type I and Type II energy relations become VdJ + W2M,
E, = +/GV. RI = -pcp,vcj
+ U&J2 - ww2)~,~~,
+ pU:[(l/c2)(&
+ U&)2
(Ia)
+ w,xl.
- (U/~“)~_X(~,
-t U&l]. (2Zi)
E,,= +P[w~v4 + W2)(4,+ wJ”l. $,
= - p[&
+ U&]V$
(3a) + ( l/cZ)(&
+ p(U/2)Z[Q474
+ r/&Y.
(4a)
The difference in the two formulations lies in the acoustic energy density (this difference establishes the difference in the energy flux). E, may have the most intuitive appeal because what appears to be a kinetic energy density, in equation (1) the terms .F = $W are a reduction
to quadratic
+ (l/c2)(IQ?)p,
terms of the complete CJT= $(p + p/2)(&
kinetic
+ P)‘(‘/,
energy density + P)
less the steady flow and first order terms. However, a detailed derivation of the thermo-fluid dynamic energy equation accurate to second order terms [8] reveals the fully consistent definition of E,,. E,, as a consistent definition of total acoustic energy can be supported by defining the potential energy density and kinetic energy density as Y = (1/2pcZ)p2 = $p/c’)(& and the Lagrangian
density 2
+ U&)2,
.P = +p1/2 = $V4’
as
= J‘--.F
= $[(l/c2)(&
+ UQ
where equations (8) and (9) have been used to introduce Principle [9] in the form 6
V$.
- V#).V&]. the velocity
(10) potential.
Hamilton’s
2’(4,, &,, dx,, &,, $1 dx, dx, dx, dt = 0 ssss
yields the Lagrange
equation
Cartesian co-ordinates xi = x, x2, x? have been used. Equation (11) and equation (10) lead to equation (7). Since E,, is consistent with the thermo-fluid dynamic energy equation and since it can be used in Hamilton’s Principle to derive the convected wave equation it seems appropriate to accept it as the definition of acoustic energy density. The role of E, and its conservation equation is now explained via the alternate form of Hamilton’s Principle in terms of the Hamiltonian density. The Hamiltonian density can be defined as
(12) where 4 =
~~/O, = (PlC2)(d+ + Ud,,.
(13)
520
W. EVERSMAN
The modified Hamilton’s Principle is [&, - &‘]dxl dx,dx,dt
6
= 0
ssss and this leads to Hamilton’s equations (14.15) Equation (12) expanded yields 2
= (P/2C2)(& + WJ2
+ (P/2)V#. v4 - Wc’vJ4,(4,
+ U#J
(16)
and this can be related to SF = (1/2pc2)pZ + $W2 + (l/c2)&.
v)p
which is the Type I energy density E,. Equation (16) can be written explicitly in terms of 4 as z@ = (c2/2p)q2 + (P/2)V4. v4 - U$,q.
(17)
Equation (14) yields W/at = (c2/p)q - u$X
or
4 = (P/c’)(~, + u+.J
(18)
and equation (15) yields
aqlat= ~~24- uaqjax.
(19)
If q is eliminated between equations (18) and (19), equation (7) results. It is thus found that the Hamiltonian density, and thus E,, is also compatible with the governing equation of motion via the modified form of Hamilton’s Principle. The conservation‘law is found by considering X’(q, $,,, $x,, 4._, 4) and forming
This can be written as
Hamilton’s Equations, equations (14) and (15), reduce this to (21) Upon introduction
of the vector
3 aiwa4+
fi=-C--ii, i=l
wxzat
(22)
where < are unit vectors in the Cartesian system, equation (21) becomes ax/at
+ div 3 = 0.
(23)
This approach to the definition of fi is apparently well known as the components in equation (22) are used by Morse and Ingard [lo] without further reference in considerations of energy flux in strings, plates, and acoustic transmission in the case with no flow. In these
ACOUSTIC
ENERGY
situations the Hamiltonian density and the total energy density now investigated in more detail. The components are
These components
can be combined
fi = p3 + (l/pc2)&.
521
IN DUCTS
coincide.
The vector
f$ it;
in vector form to yield Po)p’ + p(Vo V)V + (l/C2)&.
which is recognized from equation (2) as fi,. By comparing equations (1) and (3) and equations types of formulations can be written as E, - E,, = LIE = (l/?)(Ti,
‘v)“v, p
(2) and (4) the difference
(24)
in the two
v, p.
perturbations on a uniform It can be deduced that for irrotational conservation equation ^ g dE + div di? = 0.
flow there must be a
This can be proven by direct manipulation by making use of the continuity and momentum equations and the irrotationality consequences V x p = 0 and ‘v = grad 4. From this development it is found that when both definitions of the acoustic energy are applicable, namely to isentropic irrotational perturbations on isentropic uniform mean flow, they are both consistent with the governing equations of motion. The acoustic energy density E,, and associated flux fi,, can properly be related to the energy terms in the second order thermo-fluid dynamic energy equation and E,, can be used to construct the Lagrangian density for the derivation of the equation of motion via Hamilton’s Principle. EIis then found to be the Hamiltonian density from which the equation of motion can be derived from a modified Hamilton’s Principle. The Hamiltonian density satisfies a continuity equation in which the flux is coincident with fl,. The differences in the two formulations appear when the restrictions are relaxed. E,, and ,q,,, are applicable in the form of equations (3) and(4) to rotational disturbances on uniform mean flow (hydrodynamic modes [ll]). E, and N, are applicable in the form of equations (1) and (2) to irrotational disturbances on irrotational non-uniform flows. Finally, E, and ti, have been used to derive an energy equation with energy production terms in the general case [l] and no comparable result has been shown for E,, and g,,. However, the use of E, and fi, in the general case requires the separation of rotational and irrotational parts of the acoustic perturbation. 3. INTENSITY
AND POWER IN LINED DUCTS
The derivation of Type I and Type II energy relationships [ 1,3,8] does not include ducts with soft walls. In spite of this, both types of relationships have been used in the assessment of attenuation and results have appeared in the literature [5]. Tester [6] has pointed out that Type I relationships are generally useful for well cut-on modes but that they can be misleading in certain cases. During the course of the present research program numerous
l_ 2345-
1.288 -0.271 - 1.708 0.109 0.288
-0.605 0.257 0.762 0.859 0.864
12345-
2+ 3+ 4+ 5+
+ iO.193 + i2.422 + i5.906 + i9.390 + i12.920
- i0.532 - i4.619 - i4.824 - i8.975 - i12.685
constant
+ iO126 + i2.226 + i5.636 + i9.025 + i12.593
kxlk
1+
Mode
-0.772 0.897 1.291 1.178 1.042
3f 4+ 5+
constant
- il~OO1 - i4.078 - i4.538 - i8.501 - i12.317
kJk
Axial propagation
1.524 -5.918 0.679 0.465 0.455
Mode
Axial propagation
-0.9196 - 0.0240 0.0015 0.0015 OJIOO8
0.0572 - 0.0453 - 0.5227 - 0.0057 -0.0018
p,
Calculations
- 1.5200 0.0104 0.0125 0.0047 0.0018
0.0638 - 0.275 0.0004 - om21 -0.0011
PI
Calculations
TABLE 1
TABLE 2
- 1.7654 - 0.0339 - 0.0125 - 00041 -00X6
0.0552 0.0441 0.0132 o+IO39 0.0015
k
powers
-0.6185 - 0.0862 - 0.0249 -0.0121 - 0.0067
0.1114 -0.0192 -0.6199 - 0.0077 - om41
PI,
- 0.8204 - 0.0470 - O+.IO80 - 0.0029 -0.0013
0.0748 0.0398 0.4917 0.0049 0.0016
k
Acoustic
powers
A = 0.72 + $342
0.0796 0.3681 4.7439 0.0886 0.0406
di,Jdx
- 0.8204 - 0.0470 - O+IO80 - 0.0029 -0~0013
0.0748 0.0398 0.4917 oGO49 0.0016
x 1lpc3 - units of area
-0.3170 - 0.2276 - 0.0946 - 0.0545 -0.0331
-
- 1.7654 - 0.0339 -0.0125 - om41 -0.0016
0.0552 0.0441 0.0132 0.0039 0@015
PI,
A = 0.72 - iO.42
- 04441 -0.1509 -0.1411 - 0.0745 - 0.0392
-0.1115 - 0.3600 - 0.1202 - 0.0658 -0.0370
d@dx
x 1Jpc3 - units of area
of acoustic power; kb = 1.0, M = -0.5,
- 1.0195 - 0.0271 - 0~0000 0.0043 0.0034
0.1135 - 0.0340 0.0353 0.0144 0.0072
PII
Acoustic
of acoustic power; kb = 1.0, M = -0.5,
0.0796 0.3681 4.7439 0.0886 OG406 -0.3170 - 0.2276 - 0.0946 - 0.0545 - 0.0331
-
- 0.4441 -0.1509 -0.1411 - 0.0745 - 0.0392
-0.1115 - 0.3600 -0.1202 - 0.0658 - 0.0370
d&dx
-0.3170 - 0.2276 - 0.0946 - 0.0545 -0.0331
- 0.0796 -0.3681 - 4.7439 - 0.0886 - 0.0406
-04441 -0.1509 -0.1411 - 0.0745 - 0.0382
-0.1115 - 0.3600 -0-1202 - 0.0658 - 0.0370
-
-
-0.751 2.584 1.755 1.112 0.930
12345-
constant
0,655 t in.045 0.592 *- DO73 O.il2 i io.152 WX9 + iO.971 ,j.(>X>.J. ! i .x20 ____~
I
? .( -J 5
iO.003 iO.051 iO.736 iO.500 i2.219
1.964 1,622 0.979 0.831 0.753
1 2’ 3’ 4’ 5’
~
kJk
Mode
constant
+ iO.002 + i2.178 + i4.017 + i8.135 + ill.981
0205 i2.008 i4.032 - i8.130 - ill.975
Axial propagation
- 12.319 2.595 1.655 1.074 0.902
kxlk
1+ 2+ 3+ 4+ 5+
Mode
Axial propagation
TABLE 4
- 1.6424 - 0.0055 -0.0011 - 0.0002 -0aOO1
0.0122 0.0057 0~0010 0.0002 0~0001
PI
powers
0.0050 0.0229 oaO79 0.0028 0.0016
- 0.0059 - 0.0238 - oaO9o - 0.0029 -0@017
-
dij,/dx
04Of,
if~OOi
..__.. ~. ~.____ _...._~.__ _____ ._
-- 0.790 0.562 0.187 0.0 15
0~120 0.100 0.055 0.028 0.015
PI,
-- 1.197 -- 0.824 ~.-0.208 --0.014 0.007 .--..-
~...
0,06 1 0.055 0.032 0.015 0.008 ~ 0.643 - 1.197 -- 0.732 -- 0.824 - 0.377 - 0.208 -- 0.168 0.014 0,161 - 0.007 .-- -----------.- ~-------
oaO2 0.033 0.282 0.277 0.201
-
0.061 0.055 0.032 0.015 0.008
k,
x 1/pc3 -- units of area d?,Jd.w
powers
4
Acoustic
0.72 + iO.42
- 1.6424 - 0.0055 -0aOll - 0.0002 - oaOo1
k ~~~ 0.0122 oaO57 0~0010 0.0002 0~0001
x 1/pc3 - units of area
A = 0.01 + iO.42
of acoustic power; kh = 6.0, M = -0.5, A =
-0.9638 0.2372 0.0926 0.0302 0.0173
- 0.0103 0.2536 0.0843 0.0297 0.0172
pu
Acoustic
power; kh = 1.0, M = -0.5.
1,148 0,769 m~O.186 (‘1~005
0,061 0.056 0.037 0.016 0.006
p,
Calculation
- 1.4435 0.1743 0.047 1 0.005 1 0.0014
-0-0082 0.1796 0.0407 0.0047 0.0012
PI
C’alcu1ation.s qfucoustic
TABLF~
- 0,643 -- 0.732 0,377 0.168 0.161
- oGO2 -0.033 -- 0.282 - 0.277 - O-201
dl;,,/dx
- 0.0059 -0.0238 - oaO90 - 00029 -0aO17
-0-0050 - 0.0299 - om79 - 0.0028 - 0.0016
df’,,/dx
0.0050 0.0299 0.0079 0.0028 OTlOl6
-- 0.643 0.732 0.377 --0,168 0~161
- oaO2 - 0.033 - 0.282 - 0.277 .- 0,201
-
- 0.0059 - 0.0238 - oaO9o - 0.0029 -0.0017
-
-
524
W. EVERSMAN
I IFigure
I
I
b
>
I Duct centreline
1. Geometry
M
and co-ordinate
or hard
wail
system for two dimensional
> x
duct.
calculations of duct attenuation with use of both Type I and Type II energy definitions support Tester’s observation. Tester’s development shows that Type I axial intensity in individual modes will have a sign determined by E, = sgn(Re[kX(l - kP) + /CM]}, where k, is the axial propagation constant, M is the Mach number of the uniform mean flow and k is o/c, o being the driving frequency and c the speed of sound. Then E, = + 1. If the sign of the intensity obtained in this way is used to judge the direction of propagation it is frequently found to be in contradiction with considerations of attenuation. If Type II energy relationships are used the sign of the intensity is not so easily specified as it depends locally on the transverse velocity as well as the pressure and axial velocity. However it is found that contradictions in the sign of the intensity and the propagation direction occur although not necessarily in the same way as the contradictions based on E,. Tables 1 through 4 show the results of calculations of axial power in individual modes for four distinct cases of duct wall admittance and frequency in the geometry of Figure 1. In each case the axial propagation constant for live modes in each direction (e.g., mode l+ is the least damped mode which attenuates in the positive x direction) are shown with the corresponding value of P, and Pi,, the respective axial powers from Type I and Type II relationships (evaluated at x = 0.0). The calculations in these tables are generally power due to a unit modal amplitude, the amplitude being defined as the pressure at Y = 0. However, modes of propagation can occur in which the pressure at the wall (4’= b) is much higher than at the centre line. In these cases the acoustic power per unit amplitude is large. To avoid large entries in the tables when this occurs the power is divided by the amplitude squared of the pressure at !: = b. This has been done in Table 1, mode 2+, and Table 3, mode 1 +. It is apparent that many instances occur where the direction in which attenuation occurs does not agree with the sign of the acoustic power. Tester points out that since the modes involved are higher order, useful acoustic power arguments can still be made about the lower order “propagating” modes. This observation is usually true but Table 3, mode l+, is an example in which the lowest order upstream mode has P, and P,, of the wrong sign. This is a fairly pathological case, but illustrates the problem. Further difficulty arises in multimodal analyses in which coupling occurs. Then it is not convenient to examine each mode individually and the total power is the sum of many contributions of individual modes as well as modal coupling. Table 1 illustrates another extreme case where considerable error can be introduced as the second right running mode (positively travelling) has a Type I power which is negative and of nearly half the magnitude of the first mode Type I power. A similar problem is noted if Type II power is used. A total power calculation might well be misleading in this instance.
ACOUSTIC
ENERGY
525
IN DUCTS
It is obviously desirable to have flux expressions for uniform flow in uniform ducts with soft walls. The following development produces such expressions in dimensional form. Figure 1 shows the geometry appropriate for the development. Two dimensional ducts will be considered explicitly, but only as a matter of convenience. The duct is considered to have a hard wall at y = 0 and a soft wall at y = b. The flow in the duct is uniform. The duct lining is treated as one of point reaction with a relation between applied pressure and particle displacement characteristic of a single degree of freedom oscillator . (25) P& + r& + &,i = P. This is consistent with the usual impedance model, p/t = Z, with 2 = rb + i(wp, - k&o),
Gm
where pb, rb and k, are respectively the effective mass, damping coefficient, and spring constant per unit area. In a particular lining specified only in terms of numerical impedance values, only rb would. be identifiable, the reactance being a combination of “mass” and “spring” effects. 5 is the particle displacement of the lining and is a function of time and position on the wall. Of course the impedance relationship may actually be due to a higher order system which can be viewed as a multiple degree of freedom oscillator. It can be shown that the results for energy flux obtained by the following argument based on the single degree of freedom oscillator are unchanged if the lining behaviour is in fact characteristic of a multiple degree of freedom system. Energy expressions of Type II are considered first since the implication of total energy density lends a physical motivation to the development. The total energy contained in a volume element of the duct is made up of the acoustic energy plus the lining energy ,$, = jj$p~.
iidz + jjjs
d* +Jji’J;
ds + Ij$kJ’ds.
(27)
The volume and surface integrals are taken over a fixed region. ds is an area element on the soft wall boundary. The local rate of change of total energy is
a&at =
b”v.
? + (~/Pc~) PP,] dz +
it[pbitt + k,[] ds.
A volume element of fixed size is taken even though one boundary is flexible. This is consistent with the retention of quadratic terms in the energies. This can be rewritten as
a&/at
=
sss
[pv. (c + PO * grad P + (l/p) grad pj
+ (l/pc2) p(p, + q. . grad p + pc2 div ?)I dz -
[pp . (qO . grad ii + (l/p) grad p) + ( l/pc2) p( go . grad p + pc2 div t’)] d7 U1
+
i,l&i,,
+ rbit + k& - PJ ds -
i,[r,C, - PJ ds.
By making use of acoustic field equations. equations (5) and (6) and the lining equation of motion, equation (25), it is found that with Q0 = U?and V x v = 0
a&/at
= -
sss
div[pv + (?J2)(pP*
The divergence theorem can be
used
to
p + p2/pc2)] dz + [,,pJ,ds
- {lS,r,,C: ds.
change the volume integral into a surface integral
526
W. EVERSMAN
a&at = -
JJ
[pV + ‘&/2)(p+
“v + p2/pc2)] .3 ds +
S
In equation(28), S denotes the entire boundary of the duct volume and S, denotes the lined wall. Note that the first surface integral contains fil,. Consider now an element of duct of length Ax, for the two dimensional case with duct width b. Then, with pb and ub denoting the values of p and u assumed on the soft boundary,
a4
at -
Jb
[N,&) - N,& + A41 d_v- P,f+,Ax+ p,‘&Ax - ‘-,ifAx.
o
Time averaging and taking the limit Ax --, 0 yields “’ 0&J s 0 dx
dy = - (&,‘,,) +>
(29)
where ( ) denotes time averaging over one period of harmonic motion. N,, denotes the x component of the definition of g,,. The condition of continuity of particle gsplacement at the boundary y = b takes the form vb = { + UC,, where < = c (x, t) is the wall displacement field. Equation (29) can be written as
Jbd 0
dx
W,,J dy = -
u- &,rf>*
(30)
One can define a new axial power, which includes the hardwall axial power P,, =
J
b (N,,J
dy,
0
as
&,x= p,,=+ u
J
P&,xdx,
(31)
and then equation (30) becomes -d dx
+uJdr) =(s(Nnx)dy b
(32)
,
0
which is a time averaged continuity equation with a source term (actually a power sink). @,,is defined as the axial power consistent with the Type II energy. In the case of a Type I energy formulation the additional energy related to the lining is appended in the same way: I?, =
JJJ
$pv+ddz
+
JJJ$p2/pcz) JJJ dz +
(l/~“)(~~ * 7) p dz +
JJ JJ$
$p,[; ds
+
kbc2 ds.
(33)
This is justified because E, was identified as the Hamiltonian density and it would be desired to add addition%1 terms to the Hamiltonian. In particle mechanics the Hamiltonian coincides with the total energy and as equation (25) is in effect a particle mechanics equation of motion it is appropriate to append total energy terms to the Hamiltonian. When this is done, the development of a new conservation equation and the definition of a new axial
ACOUSTIC
ENERGY
527
IN DUCTS
power proceeds as above and yields
k,* = P,x +
U
(pbSx) dx + PU
(ubub) dx. s
s
It can be shown by use of the axial momentum equation that this is equivalent to
p,x =PIx + w$>
+ PW,i”)l.
134)
The axial power balance is then
P, is defined as the axial power consistent with the Type I energy. Equation (34) for p, is perhaps more appealing than equation (31) for $,, in that the integration required in the extra terms can be carried out explicitly and results in an expression in the local pressure, velocity and boundary displacement. Equations (32) and (35) show that if uniform mean flow is present there are additional terms to be added to the definitions of flux and power. When these terms are included the * axial rate of change of power is negative definite pb > 0) and terms with p, or P,, positive decay in the positive x direction and those with P, or f’,, negative decay in the negative Y direction. From equation (32) and equation (35) it follows that d&dx
= d&,,;dx:
that is, 6, and @,,could at most differ by a constant. independent of x, which would be irrelevant. Whether p, and p,, actually are equal depends on the constants which are included in the extra integral terms which add to P, and P,,. An appropriate choice, adopted here, is to set the constants equal to zero, or alternatively, view 8,X and P,,= as those parts of the definitions which vary with x. For any useful application it is only this part which is of any significance. 4. COMPUTATIONAL
FORMULAE
Computational formulae for intensity and power in the two dimensional case have appeared in various forms in the literature [2, 4-61. In this section formulae for P, and P,, * * are given in the present notation as well as the modified P, and P,,. While in previous sections dimensional equations were used, the computational formulae given here are in terms of non-dimensional acoustic perturbations. In two dimensional ducts the solution for acoustic propagation in terms of pressure and axial and transverse particle velocity is, in vector-matrix form. p(x. J‘,t) = e’“”Ien cos 7cn4.]{p}, u(x, y, t) = e”” [ena,, cos K~JJ jpl, u(x, y, t) = e”‘l [e,fl,I sin
K~J‘]
{p).
(M--38)
where row matrices are defined by fP)?‘=
p,.
pp.
.
&j.
[en COS K,,.v]
=
[e,
COS Xl)‘.
, 02y
COS K2,v~j
and e,, = exp( - ik_x), cl,, = (kJk)/(l
- MkJk),
/j,, = - i(ic,Jk)/(1 - Mkx,,: kj
The modes of propagation could conveniently be ordered so that modes 1 to N are decaying for positive x and modes N + 1 to 2N are decaying for negative x. In equations
528
W. EVERSMAN
(36H38) the acoustic perturbations are non-dimensional, p with respect to pc2 and the velocities with respect to c. The mean flow Mach number is denoted by M. From the boundary kinematic relation (ur,is the non-dimensional fluid transverse particle velocity at L’= b) cub = [* + ciV& It foil ows that in each mode the boundary displacement and velocity are related to the fluid velocity at the boundary by i,,(x, t) = (llik)~bn/(l - MkJk), Note that [, the boundary lead to
i,,(x, t) = -@,,v~,,,i,,(x, 0 = c~J1
- MkJk).
deflection remains dimensional, as does time. These relations
i(x. t) = eiof[enmPn sin ic,bJ {P>,5,(x, t) = eiorr - enaJn sin ic,bJ {p}, i,(x, t) = c eiwf[e,dJn sin icnbJ {p},
(3941)
where cS~= l/(1 - MkX,/k), y, = G,,/ik. Expressions for the wall displacement are also available through the impedance condition itn = c&b,,? where pb is the fluid pressure at the wall. This yields the alternate expressions 5(x, t) = (A/ik) eiot[e,, cos ~,b] {p}, c,(x, t) = A eiotr - (kX,/k) e,, cos IC,,~](p}, [,(x,
t)
CA eiot[en cos IC,~] {p}.
=
(42-44)
The eigenvalues K,,and kXnare defined by Kb tan rcb = ikbA(l
- MkX/k)2, k,/k = [l/(1 - M’)] [ - M + 41 - (1 - M2)(rc/k)2]. (45746)
A is the wall admittance, non-dimensional with respect to l/pc. Equations (42)-(44) are consistent with equations (39)-(41) through the eigenvalue equation equation (45). By using the real parts of the acoustic disturbances in equations (2) and (4) there results, in terms of non-dimensional perturbations,
4(N,J/pc3
= (1 + M2)(pu*
4(NIIX)/pc3
+ p*u) + 2M[uu* + pp*],
= pu* + p*u + M [w*
(47)
+ vu* + pp*].
(48)
The superscripted asterisk denotes the complex conjugate. If equations (36H38) are used in equations (47) and (48) the following relationships result: 4P,lpc3 = {pjT[(l 4P,,lpc3
+ M’)[Al
+ 2M PI]
(P*>,
(49)
= {P)~[CAI + M Ccl] {P*>,
(50)
where b An,,,
=
(a,,
+
a:)&,
exp
(-@,,
-
k:Jx),
Inm
cos
=
xny
cos
FC;L’
dy,
s0 Bm
=
(1
+
a,&)Z,, exp( - i(kX, - kzm)x),
[C] = [B] + [D], D,,,,, = /?np*J m nmexp( -i(kX, - kfQx), J,,, =
b sin K,J sin lczq’dy. s0 It can be verified that Tester’s expression [6] for the individual powers of Type I in each mode, P,,,, results from equation (49): that is,
k12
“n = 2 11 - MkX,/k12
k M + (1 - M2) Re 2
(
)I
I,,,.
(51)
ACOUSTIC
ENERGY
529
IN DUCTS
The equivalent Type II result is, from equation (50)
+ M
II2 ’ 1. J,!,,!.
(52)
I The exponential term is suppressed in equations (51) and (52) or, alternatively. included in the definition of the amplitude, p,. Note that P, and P,,. the acoustic powers, when divided by pc3/4 are not wholly non-dimensional as there is no scaling of lengths in the analysis. The resulting dimension is that of duct area (a unit depth is assumed). The additional terms in the power flux are now expanded. Define in non-dimensional terms
s
k,, = % +PC3M@,i,>dx = P,,x + $ &x = P,
+ pc3M
(p,(,)
M
dx + pc3M
s =Plx + f
(p,(:
(u~II,,)
+ p,*[,) d.y.
dx
s
M
(p&I,* + p,*[,)dx +$ M (u,,u; + u,*u,) dx. s s From equations (36) and (40) it can be determined that
I
+ p,*i,)dx = (P&x*
where Hnm =
--l-
kx, - kx*,
{p}'[H]{p)
(M,/?,sin rc,,bcos KEb + cczfl: cos i$,b sin IcEb)exp ( - i(kX,,- k,*,,)u).
(53)
If equation (43) is used this also can be written as Hmn =
--I A* % k& - k:_ (
cos icnbcos rczb + A $
cos Knbcos rczb exp( -i(k,,
- k:,,).x)
j (54)
It is apparent that if A = 0, the hard wall case, Hnm = 0. When real eigenvalues occur Hnm becomes singular for n = wt.A form of equation (53) or (54) can be derived to circumvent this by returning to the original integral definitions. These formulae are not given explicitly here. Alternatively, by defining the integral to have a lower limit x = 0, a constant can be introduced into equation (54) which makes evaluation of Hnm for real eigenvalues a straightforward limiting process. This would have the interesting effect of making Hqm(x = 0) = 0 and thus p,(.u = 0) = P,(x = 0), although the decay rate dp,/dx(x = 0) is not equal to dP,/dx (x = 0). The present choice of constant (zero) amounts to taking a lower limit x = f co (kX, assumed to have at least a small imaginary part) and has the salutary effect of producing a power which is of the proper sign for a given direction of propagation. This choice and a similar one in constructing the p,, terms also ensures that p, = I’,, (equation (55) below). By proceeding in the same way it is established that
s
(PJ:
+ p,*i,) dx +
s
by,*
+ ++,,
dx = {P)’ [f’l{p*l, .-
W. EVERSMAN
530
where pnnI = Hnm + k
.x.
: kzm(cc*P:cos rC,b sin Kzb + ~:a, sin rC,bcos $b)
x
x exp (- i(kX, - kzm)x).
Equation (45) allows this to be written as i p,,,,, = i
A* (, _ Mkx,,,)
A ~0s K,b ~0s C,b - c1 _ Mkznlk,
~0s ~,b ~0s Gb
1 x
x exp ( - i(kX, - kz”)x).
(55)
In the hard walled case P,,,,, = 0. Finally, the computational formulae are 4@pc3 = (P>’ [(l + M’)[A]
+ 2M[B] + M[P]] {P*},
43,J~c~ = {PI’ [L-AI f MCCI +
(56)
Wffl] @*I.
(57)
The individual powers of Type I and Type II. including the additional terms to account for wall impedance now follow from equations (56) and (57) with n = m:
4P,12
M + (1 - M’) Re
Inn
nn = (1 - MkJk(’ & Mlp,Y Re{(a,, - cr,*)/?Xcos lc,b sin rc:b), fm(kXn) lP,12
(58)
I II k
M + 2(1 - M2)Re
- M(l - M2) 2
nn = 11 - Mk,Jk12
2
Inn
MlpI2Re{G[~P~cos K,,b sin K,*b}. Wkxn)
- n
(59)
Again, the exponential term is suppressed in equations (58) and (59) or included in the definition of the amplitude p,. The axial rate of change of power dfi,/d.x or d?,,/dx is easily computed from equations (56) and (57) by noting that the x dependence is contained in the exponential factors in the coefficient matrices. Thus, for example, & [A] = [A’], A;, = -i(kXn - kzm)A,,,. Similar relations obtain for the other coefficient matrices. The dissipation term, W, is defined by W = (r,C:> = (rJ2) This leads to the vector-matrix
[i,C:l.
formulation 4WlP3
= {PI ‘cwP*L
where R,,,,, = 2?$ nmnm S*p fi* exp( - i(k,, - kzm)x) sin rC,b sin Iczb,
where P,,is the real part of Z/PC (non-dimensional).
(60)
The dissipation in the individual modes
ACOUSTIC
ENERGY
IN DUCTS
531
is thus
with the exponential suppressed or included in p,. Tables 1 through 4 show, in addition to calculations of power based on the Morfey and Ryshov and Shefter relations, the calculations of axial power in individual modes based on the new relations developed here (equations (58) and (59)). The power gradients and the dissipation term are also shown. It is apparent that PI and p,, coincide and that the sign of the axial power is consistent with the propagation direction: that is, positively propagating modes (attenuated for positive x) have positive power and the reverse for negatively propagating modes. It is also noted that the axial gradient of power equals the dissipation term. as stipulated by the time averaged continuity equation. Because of the factor k ‘, it would appear from equation (55) that at higher frequencies the correction P,,,, to P, is likely to be small. For well cut-on modes, ones for which P, is relatively large, Pnmis expected to be of only minor importance. This does in fact happen as shown in Table 4 where the reduced frequency kb = 6.0. Note, however, that for well cut-off modes moving with the flow the correction still dominates and the sign of p, and P, are different. 5. OBSERVATIONS Tester [6] used Mohring’s [2] definition of intensity, specialized to sheared flow in a two dimensional duct, to explain the deficiencies of the Morfey and Ryshov and Shefter definitions of intensity when the duct wall is soft. This was accomplished by treating the uniform flow as one with a vanishingly thin shear layer at the wall. When suitable modification is made to account for the fact that Tester’s result is dimensional, the additional power
in equation (56) with [P] defined by equation (55) can be shown to yield the identical diagonal terms defined by his equation (22). This is done by using the conjugate of the expression in brackets in equation (55) and by introducing the magnitude squared of the term (1 - MkJk) in the denominator. Equation (31) with the extra power term appearing in integral form, may be open to criticism as depending on an acoustic fluctuation product integrated with respect to the axial co-ordinate. The extra terms in equation (34) also were originally arrived at in the same way but were representable in terms of an explicit integration yielding results in terms of local pressure, velocity, and lining displacement. In either case, when the computational expressions of equations (56) and (57) are obtained the power is in terms of local pressure fluctuations. The acoustic power considerations addressed here verify the results of Tester for the diagonal terms and extend the implications to include power conservation laws which show the effect of the wall as a power sink. The results in this analysis can be viewed as having been derived from relatively conventional energy conservation concepts with no necessity to consider a wall shear layer. In the Tester approach, via Mohring’s intensity definition, the shear layer is essential. Miihring’s [2] equation (15) does not reduce to the uniform flow result (with soft walls) unless the proper limiting behaviour across the shear layer is accounted for. 6. CONCLUSION It
has been shown that both of the common acoustic energy density-energy
flu\- relation-
532
W. EVERSMAN
ships, of which Morfey [l] and Ryshov and Shefter [3] are the principal proponents, are related to the governing field equation for irrotational acoustic disturbances on a uniform duct flow by a variational principle. The Ryshov and Shefter energy density can be viewed as the sum of a potential energy density and a kinetic energy density from which the Lagrangian density can be formed. Hamilton’s Principle then yields the convected wave equation. The Morfey form of the energy density has been shown to be the Hamiltonian density, related to the Lagrangian density by a Legendre transformation. The modified form of Hamilton’s Principle then yields the convected wave equation. The concept of energy flux in the Morfey representation follows naturally from the definition of the Hamiltonian density. The problem asociated with the use of either the Morfey or Ryshov and Shefter energy formulations in the case of ducts with soft walls has been addressed and a new energy equation including an energy sink due to the wall dissipation has been derived. The acoustic flux, and hence the acoustic power, associated with the modified energy equation is consistent in sign with the direction of propagation of acoustic disturbances as deduced from attenuation arguments. ACKNOWLEDGMENT
The research reported here was supported by the National Aeronautics and Space Administration Lewis Research Center under Grant NSG 7192. The author wishes to thank the reviewers for particularly useful comments. Several significant shortcomings in presentation and content were brought to light and these have been modified, in certain cases by using ideas proposed by the reviewers. REFERENCES
1. C. L. MORFEY1971 Journal of Sound and Vibration 14, 1.59-170. Acoustic energy in non-uniform flows. 2. W. M~HRING 1971 Journal of Sound and Vibration 18, 101-109. Energy flux in duct flow. 3. 0. S. RYSHOVand G. M. SHEFTJZR1962 Journal of Applied Mathematics and Mechanics 26, 1293-1309. On the energy of acoustic waves propagating in moving media. 4. W. EVERSMAN1970 Journal of the Acoustical Society of America 49, 1717-1721. Energy flow criteria for acoustic propagation in ducts with flow. 5. S. H. Ko 1971 Journal of the Acoustical Society of America SO, 14181432. Sound attenuation in lined rectangular ducts with flow and its application to the reduction of aircraft noise. 6. B. J. TESTER1973 Journal of Soundand Vibration 28,205-215. Acoustic energy Bow in lined ducts containing uniform or “plug” flow. 7. S. M. CANDEL 1975 Journal of Sound and Vibration 41, 207-232. Acoustic conservation principles and an application to plane and modal propagation in nozzles and diffusers. 8. W. EVERSMAN1970 Boeing Company Document 03-8152. The propagation of acoustic energy in a flow duct. 9. H. GOLDSTEIN1959 Classical Mechanics. New York: Addison Wesley Publishing Company. Chapters 7 and 11. 10. P. M. MORSEand K. U. INGARD1968 Theoretical Acoustics. New York: McGraw-Hill Book Co. See pages 100-106, 196-200 and 248-250. 11. W. M~~HRING1976 Fortschritte der Akustik, DAGA 76, VDI-Verlag, Dusseldorf, 543-546. Uber Schallwellen in Scherstromungen.
qf Sound
and Vibration
ACOUSTIC
(1979) 62(4), 517-532
ENERGY IN DUCTS:
FURTHER
OBSERVATIONS
W. EVERSMAN~ Department
of’Mechanical Engineering, (Received
University qf Canterhurv,
22 April 1978, and in revised,form
Christchurch.
New Zcuhd
8 August 1978)
The transmission of acoustic energy in uniform ducts carrying uniform flow is investigated with the purpose of clarifying two points of interest. The two commonly used definitions of acoustic “energy” flux are shown to be related by a Legendre transformation of the Lagrangian density exactly as in deriving the Hamiltonian density in mechanics. In the acoustic case the total energy density and the Hamiltonian density are not the same which accounts for two different “energy” fluxes. When the duct has acoustically absorptive walls neither of the two flux expressions gives correct results. A re-evaluation of the basis of derivation of the energy density and energy flux provides forms which yield consistent results for soft walled ducts.
1. INTRODUCTION
In order to assess the acoustic performance of duct non-uniformities and duct linings it is often convenient to compare incident acoustic power and transmitted acoustic power. This is a straightforward procedure in ducts with no flow. When a mean flow is present the problem is more complicated and it is by no means clear how acoustic energy density and energy flux are to be defined in the general case when the flow is non-uniform. At the present time it does not seem possible to give unique definitions which lead to practical calculations except in special cases. Partly because of this difficulty the rigorous application of the concept of power insertion loss or transmission loss is usually restricted in practical applications to calculations in uniform ducts. For example, this is done by comparing acoustic powers in uniform ducts on either side of a duct non-uniformity or by comparing acoustic powers at various axial locations in a uniform lined duct using an appropriately extended definition of power. Even in these cases with axial uniformity the presence of a flow which is non-uniform across the duct presents fundamental problems. Morfey’s [l] definition of energy density and flux in hard walled ducts appears to have gained the widest acceptance because it satisfies a conservation law in any irrotational constant entropy flow. However the extension of its use to rotational flows depends on the separation of rotational and irrotational parts of the acoustic field. Miihring’s [2] approach to an energy equation with no source terms is applicable in simple terms to sheared flows in uniform ducts with hard or soft walls. Perhaps because Miihring’s energy equation is based on Clebsch potentials. a concept somewhat removed from the usual duct acoustics theory, or perhaps because the types of flows in which the particularly simple results are useful have not been dominant in the literature, it does not appear to have replaced the more elementary type of energy expressions. Hence, in the literature energy flux calculations are generally restricted to uniform flows in uniform ducts. When this is the case the definition of energy density and tNow Missouri
at the Department
of Mechanical and Aerospace Engineering.
University
of Missouri-Rolla.
Rolla.
65401. U.S.A. 517
0022-460X/79104053
7 + 16 SO2.00/0
@ 1979 Academic
Press Inc. (London)
Limited
518
W. EVERSMAN
energy flux originally put forward by Ryshov and Shefter [3] for hard walled ducts, and used subsequently by Eversman [4], becomes equally valid. This being the case, the definitions of Morfey and Ryshov and Shefter are often used interchangeably as a basis for simple calculations of insertion loss or transmission loss. Both definitions of energy satisfy a conservation law for uniform flow and the relationship between them is discussed here. The use of the Morfey and Ryshov and Shefter energy definitions for lined ducts is tempting and has appeared in the literature [5]. As noted by Tester [6] anomalous results can occur, particularly in calculations of the energy flux for heavily attenuated modes. Numerical experimentation has confirmed this observation and has led to an extension of the hard walled duct energy definitions to yield relationships which satisfy energy equations with source terms. These energy equations introduce an acoustic flux which correctly relates to the direction of propagation for all modes.
2. DEFINITIONS
OF ACOUSTIC
ENERGY
DENSITY
AND ENERGY
FLUX
Candel [7] has reviewed much of the literature dealing with acoustic energy principles in general and their application in ducts in particular. He divides the energy densityenergy flux approaches into two types. Type I are typified by the Morfey relationships, E, = (1/2pc2)p2 + $P
I$, = pP + p(Vo.rq
+ (l/c2)&
+ (l/pc2)Vop2
I$,
(1)
+ (l/C2)Vcl(V0~Qp,
and Type II by the Ryshov and Shefter relationships,
fin = PV + V0[(1/2pc2)p2 + $V=2]. In these equations the energy density E and energy flux fi are functions of the acoustic perturbations in velocity and pressure v and p. The mean flow velocity is denoted by v0 and the ambient density and speed of sound by p and c. The Type I expressions satisfy a conservation law of the type dE/at + divfi
= 0
in the irrotational isentropic case. The Type II expressions have been shown to satisfy the conservation law when the mean flow is uniform. In the case of irrotational perturbations on a uniform flow both types are valid. In this case, it is possible to show that the two definitions of acoustic energy density and energy flux are related. When uniform flow is present in a uniform duct the isentropic acoustic problem is governed by the linearized continuity and momentum equations (l/c’)[a/&
+ v,, . grad]p + p div v = 0,
[a/& + p,,.grad]v
= -(l/p)gradp.
When the acoustic perturbations are irrotational a single equation in the velocity potential 4, ~24 =
(ip)[ajat
(5)
equations (5) and (6) can be replaced by
+
uajaij24,
(7)
ACOUSTIC
ENERGY
519
IN DUCTS
where ‘/ = grad 4,
p = -p[~3/?t
(8.9)
+ U6/i;x]+.
In equations (7H9) the mean flow is in the direction of the x axis, and p,, = C’z In terms of the velocity potential the Type I and Type II energy relations become VdJ + W2M,
E, = +/GV. RI = -pcp,vcj
+ U&J2 - ww2)~,~~,
+ pU:[(l/c2)(&
+ U&)2
(Ia)
+ w,xl.
- (U/~“)~_X(~,
-t U&l]. (2Zi)
E,,= +P[w~v4 + W2)(4,+ wJ”l. $,
= - p[&
+ U&]V$
(3a) + ( l/cZ)(&
+ p(U/2)Z[Q474
+ r/&Y.
(4a)
The difference in the two formulations lies in the acoustic energy density (this difference establishes the difference in the energy flux). E, may have the most intuitive appeal because what appears to be a kinetic energy density, in equation (1) the terms .F = $W are a reduction
to quadratic
+ (l/c2)(IQ?)p,
terms of the complete CJT= $(p + p/2)(&
kinetic
+ P)‘(‘/,
energy density + P)
less the steady flow and first order terms. However, a detailed derivation of the thermo-fluid dynamic energy equation accurate to second order terms [8] reveals the fully consistent definition of E,,. E,, as a consistent definition of total acoustic energy can be supported by defining the potential energy density and kinetic energy density as Y = (1/2pcZ)p2 = $p/c’)(& and the Lagrangian
density 2
+ U&)2,
.P = +p1/2 = $V4’
as
= J‘--.F
= $[(l/c2)(&
+ UQ
where equations (8) and (9) have been used to introduce Principle [9] in the form 6
V$.
- V#).V&]. the velocity
(10) potential.
Hamilton’s
2’(4,, &,, dx,, &,, $1 dx, dx, dx, dt = 0 ssss
yields the Lagrange
equation
Cartesian co-ordinates xi = x, x2, x? have been used. Equation (11) and equation (10) lead to equation (7). Since E,, is consistent with the thermo-fluid dynamic energy equation and since it can be used in Hamilton’s Principle to derive the convected wave equation it seems appropriate to accept it as the definition of acoustic energy density. The role of E, and its conservation equation is now explained via the alternate form of Hamilton’s Principle in terms of the Hamiltonian density. The Hamiltonian density can be defined as
(12) where 4 =
~~/O, = (PlC2)(d+ + Ud,,.
(13)
520
W. EVERSMAN
The modified Hamilton’s Principle is [&, - &‘]dxl dx,dx,dt
6
= 0
ssss and this leads to Hamilton’s equations (14.15) Equation (12) expanded yields 2
= (P/2C2)(& + WJ2
+ (P/2)V#. v4 - Wc’vJ4,(4,
+ U#J
(16)
and this can be related to SF = (1/2pc2)pZ + $W2 + (l/c2)&.
v)p
which is the Type I energy density E,. Equation (16) can be written explicitly in terms of 4 as z@ = (c2/2p)q2 + (P/2)V4. v4 - U$,q.
(17)
Equation (14) yields W/at = (c2/p)q - u$X
or
4 = (P/c’)(~, + u+.J
(18)
and equation (15) yields
aqlat= ~~24- uaqjax.
(19)
If q is eliminated between equations (18) and (19), equation (7) results. It is thus found that the Hamiltonian density, and thus E,, is also compatible with the governing equation of motion via the modified form of Hamilton’s Principle. The conservation‘law is found by considering X’(q, $,,, $x,, 4._, 4) and forming
This can be written as
Hamilton’s Equations, equations (14) and (15), reduce this to (21) Upon introduction
of the vector
3 aiwa4+
fi=-C--ii, i=l
wxzat
(22)
where < are unit vectors in the Cartesian system, equation (21) becomes ax/at
+ div 3 = 0.
(23)
This approach to the definition of fi is apparently well known as the components in equation (22) are used by Morse and Ingard [lo] without further reference in considerations of energy flux in strings, plates, and acoustic transmission in the case with no flow. In these
ACOUSTIC
ENERGY
situations the Hamiltonian density and the total energy density now investigated in more detail. The components are
These components
can be combined
fi = p3 + (l/pc2)&.
521
IN DUCTS
coincide.
The vector
f$ it;
in vector form to yield Po)p’ + p(Vo V)V + (l/C2)&.
which is recognized from equation (2) as fi,. By comparing equations (1) and (3) and equations types of formulations can be written as E, - E,, = LIE = (l/?)(Ti,
‘v)“v, p
(2) and (4) the difference
(24)
in the two
v, p.
perturbations on a uniform It can be deduced that for irrotational conservation equation ^ g dE + div di? = 0.
flow there must be a
This can be proven by direct manipulation by making use of the continuity and momentum equations and the irrotationality consequences V x p = 0 and ‘v = grad 4. From this development it is found that when both definitions of the acoustic energy are applicable, namely to isentropic irrotational perturbations on isentropic uniform mean flow, they are both consistent with the governing equations of motion. The acoustic energy density E,, and associated flux fi,, can properly be related to the energy terms in the second order thermo-fluid dynamic energy equation and E,, can be used to construct the Lagrangian density for the derivation of the equation of motion via Hamilton’s Principle. EIis then found to be the Hamiltonian density from which the equation of motion can be derived from a modified Hamilton’s Principle. The Hamiltonian density satisfies a continuity equation in which the flux is coincident with fl,. The differences in the two formulations appear when the restrictions are relaxed. E,, and ,q,,, are applicable in the form of equations (3) and(4) to rotational disturbances on uniform mean flow (hydrodynamic modes [ll]). E, and N, are applicable in the form of equations (1) and (2) to irrotational disturbances on irrotational non-uniform flows. Finally, E, and ti, have been used to derive an energy equation with energy production terms in the general case [l] and no comparable result has been shown for E,, and g,,. However, the use of E, and fi, in the general case requires the separation of rotational and irrotational parts of the acoustic perturbation. 3. INTENSITY
AND POWER IN LINED DUCTS
The derivation of Type I and Type II energy relationships [ 1,3,8] does not include ducts with soft walls. In spite of this, both types of relationships have been used in the assessment of attenuation and results have appeared in the literature [5]. Tester [6] has pointed out that Type I relationships are generally useful for well cut-on modes but that they can be misleading in certain cases. During the course of the present research program numerous
l_ 2345-
1.288 -0.271 - 1.708 0.109 0.288
-0.605 0.257 0.762 0.859 0.864
12345-
2+ 3+ 4+ 5+
+ iO.193 + i2.422 + i5.906 + i9.390 + i12.920
- i0.532 - i4.619 - i4.824 - i8.975 - i12.685
constant
+ iO126 + i2.226 + i5.636 + i9.025 + i12.593
kxlk
1+
Mode
-0.772 0.897 1.291 1.178 1.042
3f 4+ 5+
constant
- il~OO1 - i4.078 - i4.538 - i8.501 - i12.317
kJk
Axial propagation
1.524 -5.918 0.679 0.465 0.455
Mode
Axial propagation
-0.9196 - 0.0240 0.0015 0.0015 OJIOO8
0.0572 - 0.0453 - 0.5227 - 0.0057 -0.0018
p,
Calculations
- 1.5200 0.0104 0.0125 0.0047 0.0018
0.0638 - 0.275 0.0004 - om21 -0.0011
PI
Calculations
TABLE 1
TABLE 2
- 1.7654 - 0.0339 - 0.0125 - 00041 -00X6
0.0552 0.0441 0.0132 o+IO39 0.0015
k
powers
-0.6185 - 0.0862 - 0.0249 -0.0121 - 0.0067
0.1114 -0.0192 -0.6199 - 0.0077 - om41
PI,
- 0.8204 - 0.0470 - O+.IO80 - 0.0029 -0.0013
0.0748 0.0398 0.4917 0.0049 0.0016
k
Acoustic
powers
A = 0.72 + $342
0.0796 0.3681 4.7439 0.0886 0.0406
di,Jdx
- 0.8204 - 0.0470 - O+IO80 - 0.0029 -0~0013
0.0748 0.0398 0.4917 oGO49 0.0016
x 1lpc3 - units of area
-0.3170 - 0.2276 - 0.0946 - 0.0545 -0.0331
-
- 1.7654 - 0.0339 -0.0125 - om41 -0.0016
0.0552 0.0441 0.0132 0.0039 0@015
PI,
A = 0.72 - iO.42
- 04441 -0.1509 -0.1411 - 0.0745 - 0.0392
-0.1115 - 0.3600 - 0.1202 - 0.0658 -0.0370
d@dx
x 1Jpc3 - units of area
of acoustic power; kb = 1.0, M = -0.5,
- 1.0195 - 0.0271 - 0~0000 0.0043 0.0034
0.1135 - 0.0340 0.0353 0.0144 0.0072
PII
Acoustic
of acoustic power; kb = 1.0, M = -0.5,
0.0796 0.3681 4.7439 0.0886 OG406 -0.3170 - 0.2276 - 0.0946 - 0.0545 - 0.0331
-
- 0.4441 -0.1509 -0.1411 - 0.0745 - 0.0392
-0.1115 - 0.3600 -0.1202 - 0.0658 - 0.0370
d&dx
-0.3170 - 0.2276 - 0.0946 - 0.0545 -0.0331
- 0.0796 -0.3681 - 4.7439 - 0.0886 - 0.0406
-04441 -0.1509 -0.1411 - 0.0745 - 0.0382
-0.1115 - 0.3600 -0-1202 - 0.0658 - 0.0370
-
-
-0.751 2.584 1.755 1.112 0.930
12345-
constant
0,655 t in.045 0.592 *- DO73 O.il2 i io.152 WX9 + iO.971 ,j.(>X>.J. ! i .x20 ____~
I
? .( -J 5
iO.003 iO.051 iO.736 iO.500 i2.219
1.964 1,622 0.979 0.831 0.753
1 2’ 3’ 4’ 5’
~
kJk
Mode
constant
+ iO.002 + i2.178 + i4.017 + i8.135 + ill.981
0205 i2.008 i4.032 - i8.130 - ill.975
Axial propagation
- 12.319 2.595 1.655 1.074 0.902
kxlk
1+ 2+ 3+ 4+ 5+
Mode
Axial propagation
TABLE 4
- 1.6424 - 0.0055 -0.0011 - 0.0002 -0aOO1
0.0122 0.0057 0~0010 0.0002 0~0001
PI
powers
0.0050 0.0229 oaO79 0.0028 0.0016
- 0.0059 - 0.0238 - oaO9o - 0.0029 -0@017
-
dij,/dx
04Of,
if~OOi
..__.. ~. ~.____ _...._~.__ _____ ._
-- 0.790 0.562 0.187 0.0 15
0~120 0.100 0.055 0.028 0.015
PI,
-- 1.197 -- 0.824 ~.-0.208 --0.014 0.007 .--..-
~...
0,06 1 0.055 0.032 0.015 0.008 ~ 0.643 - 1.197 -- 0.732 -- 0.824 - 0.377 - 0.208 -- 0.168 0.014 0,161 - 0.007 .-- -----------.- ~-------
oaO2 0.033 0.282 0.277 0.201
-
0.061 0.055 0.032 0.015 0.008
k,
x 1/pc3 -- units of area d?,Jd.w
powers
4
Acoustic
0.72 + iO.42
- 1.6424 - 0.0055 -0aOll - 0.0002 - oaOo1
k ~~~ 0.0122 oaO57 0~0010 0.0002 0~0001
x 1/pc3 - units of area
A = 0.01 + iO.42
of acoustic power; kh = 6.0, M = -0.5, A =
-0.9638 0.2372 0.0926 0.0302 0.0173
- 0.0103 0.2536 0.0843 0.0297 0.0172
pu
Acoustic
power; kh = 1.0, M = -0.5.
1,148 0,769 m~O.186 (‘1~005
0,061 0.056 0.037 0.016 0.006
p,
Calculation
- 1.4435 0.1743 0.047 1 0.005 1 0.0014
-0-0082 0.1796 0.0407 0.0047 0.0012
PI
C’alcu1ation.s qfucoustic
TABLF~
- 0,643 -- 0.732 0,377 0.168 0.161
- oGO2 -0.033 -- 0.282 - 0.277 - O-201
dl;,,/dx
- 0.0059 -0.0238 - oaO90 - 00029 -0aO17
-0-0050 - 0.0299 - om79 - 0.0028 - 0.0016
df’,,/dx
0.0050 0.0299 0.0079 0.0028 OTlOl6
-- 0.643 0.732 0.377 --0,168 0~161
- oaO2 - 0.033 - 0.282 - 0.277 .- 0,201
-
- 0.0059 - 0.0238 - oaO9o - 0.0029 -0.0017
-
-
524
W. EVERSMAN
I IFigure
I
I
b
>
I Duct centreline
1. Geometry
M
and co-ordinate
or hard
wail
system for two dimensional
> x
duct.
calculations of duct attenuation with use of both Type I and Type II energy definitions support Tester’s observation. Tester’s development shows that Type I axial intensity in individual modes will have a sign determined by E, = sgn(Re[kX(l - kP) + /CM]}, where k, is the axial propagation constant, M is the Mach number of the uniform mean flow and k is o/c, o being the driving frequency and c the speed of sound. Then E, = + 1. If the sign of the intensity obtained in this way is used to judge the direction of propagation it is frequently found to be in contradiction with considerations of attenuation. If Type II energy relationships are used the sign of the intensity is not so easily specified as it depends locally on the transverse velocity as well as the pressure and axial velocity. However it is found that contradictions in the sign of the intensity and the propagation direction occur although not necessarily in the same way as the contradictions based on E,. Tables 1 through 4 show the results of calculations of axial power in individual modes for four distinct cases of duct wall admittance and frequency in the geometry of Figure 1. In each case the axial propagation constant for live modes in each direction (e.g., mode l+ is the least damped mode which attenuates in the positive x direction) are shown with the corresponding value of P, and Pi,, the respective axial powers from Type I and Type II relationships (evaluated at x = 0.0). The calculations in these tables are generally power due to a unit modal amplitude, the amplitude being defined as the pressure at Y = 0. However, modes of propagation can occur in which the pressure at the wall (4’= b) is much higher than at the centre line. In these cases the acoustic power per unit amplitude is large. To avoid large entries in the tables when this occurs the power is divided by the amplitude squared of the pressure at !: = b. This has been done in Table 1, mode 2+, and Table 3, mode 1 +. It is apparent that many instances occur where the direction in which attenuation occurs does not agree with the sign of the acoustic power. Tester points out that since the modes involved are higher order, useful acoustic power arguments can still be made about the lower order “propagating” modes. This observation is usually true but Table 3, mode l+, is an example in which the lowest order upstream mode has P, and P,, of the wrong sign. This is a fairly pathological case, but illustrates the problem. Further difficulty arises in multimodal analyses in which coupling occurs. Then it is not convenient to examine each mode individually and the total power is the sum of many contributions of individual modes as well as modal coupling. Table 1 illustrates another extreme case where considerable error can be introduced as the second right running mode (positively travelling) has a Type I power which is negative and of nearly half the magnitude of the first mode Type I power. A similar problem is noted if Type II power is used. A total power calculation might well be misleading in this instance.
ACOUSTIC
ENERGY
525
IN DUCTS
It is obviously desirable to have flux expressions for uniform flow in uniform ducts with soft walls. The following development produces such expressions in dimensional form. Figure 1 shows the geometry appropriate for the development. Two dimensional ducts will be considered explicitly, but only as a matter of convenience. The duct is considered to have a hard wall at y = 0 and a soft wall at y = b. The flow in the duct is uniform. The duct lining is treated as one of point reaction with a relation between applied pressure and particle displacement characteristic of a single degree of freedom oscillator . (25) P& + r& + &,i = P. This is consistent with the usual impedance model, p/t = Z, with 2 = rb + i(wp, - k&o),
Gm
where pb, rb and k, are respectively the effective mass, damping coefficient, and spring constant per unit area. In a particular lining specified only in terms of numerical impedance values, only rb would. be identifiable, the reactance being a combination of “mass” and “spring” effects. 5 is the particle displacement of the lining and is a function of time and position on the wall. Of course the impedance relationship may actually be due to a higher order system which can be viewed as a multiple degree of freedom oscillator. It can be shown that the results for energy flux obtained by the following argument based on the single degree of freedom oscillator are unchanged if the lining behaviour is in fact characteristic of a multiple degree of freedom system. Energy expressions of Type II are considered first since the implication of total energy density lends a physical motivation to the development. The total energy contained in a volume element of the duct is made up of the acoustic energy plus the lining energy ,$, = jj$p~.
iidz + jjjs
d* +Jji’J;
ds + Ij$kJ’ds.
(27)
The volume and surface integrals are taken over a fixed region. ds is an area element on the soft wall boundary. The local rate of change of total energy is
a&at =
b”v.
? + (~/Pc~) PP,] dz +
it[pbitt + k,[] ds.
A volume element of fixed size is taken even though one boundary is flexible. This is consistent with the retention of quadratic terms in the energies. This can be rewritten as
a&/at
=
sss
[pv. (c + PO * grad P + (l/p) grad pj
+ (l/pc2) p(p, + q. . grad p + pc2 div ?)I dz -
[pp . (qO . grad ii + (l/p) grad p) + ( l/pc2) p( go . grad p + pc2 div t’)] d7 U1
+
i,l&i,,
+ rbit + k& - PJ ds -
i,[r,C, - PJ ds.
By making use of acoustic field equations. equations (5) and (6) and the lining equation of motion, equation (25), it is found that with Q0 = U?and V x v = 0
a&/at
= -
sss
div[pv + (?J2)(pP*
The divergence theorem can be
used
to
p + p2/pc2)] dz + [,,pJ,ds
- {lS,r,,C: ds.
change the volume integral into a surface integral
526
W. EVERSMAN
a&at = -
JJ
[pV + ‘&/2)(p+
“v + p2/pc2)] .3 ds +
S
In equation(28), S denotes the entire boundary of the duct volume and S, denotes the lined wall. Note that the first surface integral contains fil,. Consider now an element of duct of length Ax, for the two dimensional case with duct width b. Then, with pb and ub denoting the values of p and u assumed on the soft boundary,
a4
at -
Jb
[N,&) - N,& + A41 d_v- P,f+,Ax+ p,‘&Ax - ‘-,ifAx.
o
Time averaging and taking the limit Ax --, 0 yields “’ 0&J s 0 dx
dy = - (&,‘,,) +
(29)
where ( ) denotes time averaging over one period of harmonic motion. N,, denotes the x component of the definition of g,,. The condition of continuity of particle gsplacement at the boundary y = b takes the form vb = { + UC,, where < = c (x, t) is the wall displacement field. Equation (29) can be written as
Jbd 0
dx
W,,J dy = -
u
(30)
One can define a new axial power, which includes the hardwall axial power P,, =
J
b (N,,J
dy,
0
as
&,x= p,,=+ u
J
P&,xdx,
(31)
and then equation (30) becomes -d dx
+uJ
(32)
0
which is a time averaged continuity equation with a source term (actually a power sink). @,,is defined as the axial power consistent with the Type II energy. In the case of a Type I energy formulation the additional energy related to the lining is appended in the same way: I?, =
JJJ
$pv+ddz
+
JJJ$p2/pcz) JJJ dz +
(l/~“)(~~ * 7) p dz +
JJ JJ$
$p,[; ds
+
kbc2 ds.
(33)
This is justified because E, was identified as the Hamiltonian density and it would be desired to add addition%1 terms to the Hamiltonian. In particle mechanics the Hamiltonian coincides with the total energy and as equation (25) is in effect a particle mechanics equation of motion it is appropriate to append total energy terms to the Hamiltonian. When this is done, the development of a new conservation equation and the definition of a new axial
ACOUSTIC
ENERGY
527
IN DUCTS
power proceeds as above and yields
k,* = P,x +
U
(pbSx) dx + PU
(ubub) dx. s
s
It can be shown by use of the axial momentum equation that this is equivalent to
p,x =PIx + w$>
+ PW,i”)l.
134)
The axial power balance is then
P, is defined as the axial power consistent with the Type I energy. Equation (34) for p, is perhaps more appealing than equation (31) for $,, in that the integration required in the extra terms can be carried out explicitly and results in an expression in the local pressure, velocity and boundary displacement. Equations (32) and (35) show that if uniform mean flow is present there are additional terms to be added to the definitions of flux and power. When these terms are included the * axial rate of change of power is negative definite pb > 0) and terms with p, or P,, positive decay in the positive x direction and those with P, or f’,, negative decay in the negative Y direction. From equation (32) and equation (35) it follows that d&dx
= d&,,;dx:
that is, 6, and @,,could at most differ by a constant. independent of x, which would be irrelevant. Whether p, and p,, actually are equal depends on the constants which are included in the extra integral terms which add to P, and P,,. An appropriate choice, adopted here, is to set the constants equal to zero, or alternatively, view 8,X and P,,= as those parts of the definitions which vary with x. For any useful application it is only this part which is of any significance. 4. COMPUTATIONAL
FORMULAE
Computational formulae for intensity and power in the two dimensional case have appeared in various forms in the literature [2, 4-61. In this section formulae for P, and P,, * * are given in the present notation as well as the modified P, and P,,. While in previous sections dimensional equations were used, the computational formulae given here are in terms of non-dimensional acoustic perturbations. In two dimensional ducts the solution for acoustic propagation in terms of pressure and axial and transverse particle velocity is, in vector-matrix form. p(x. J‘,t) = e’“”Ien cos 7cn4.]{p}, u(x, y, t) = e”” [ena,, cos K~JJ jpl, u(x, y, t) = e”‘l [e,fl,I sin
K~J‘]
{p).
(M--38)
where row matrices are defined by fP)?‘=
p,.
pp.
.
&j.
[en COS K,,.v]
=
[e,
COS Xl)‘.
, 02y
COS K2,v~j
and e,, = exp( - ik_x), cl,, = (kJk)/(l
- MkJk),
/j,, = - i(ic,Jk)/(1 - Mkx,,: kj
The modes of propagation could conveniently be ordered so that modes 1 to N are decaying for positive x and modes N + 1 to 2N are decaying for negative x. In equations
528
W. EVERSMAN
(36H38) the acoustic perturbations are non-dimensional, p with respect to pc2 and the velocities with respect to c. The mean flow Mach number is denoted by M. From the boundary kinematic relation (ur,is the non-dimensional fluid transverse particle velocity at L’= b) cub = [* + ciV& It foil ows that in each mode the boundary displacement and velocity are related to the fluid velocity at the boundary by i,,(x, t) = (llik)~bn/(l - MkJk), Note that [, the boundary lead to
i,,(x, t) = -@,,v~,,,i,,(x, 0 = c~J1
- MkJk).
deflection remains dimensional, as does time. These relations
i(x. t) = eiof[enmPn sin ic,bJ {P>,5,(x, t) = eiorr - enaJn sin ic,bJ {p}, i,(x, t) = c eiwf[e,dJn sin icnbJ {p},
(3941)
where cS~= l/(1 - MkX,/k), y, = G,,/ik. Expressions for the wall displacement are also available through the impedance condition itn = c&b,,? where pb is the fluid pressure at the wall. This yields the alternate expressions 5(x, t) = (A/ik) eiot[e,, cos ~,b] {p}, c,(x, t) = A eiotr - (kX,/k) e,, cos IC,,~](p}, [,(x,
t)
CA eiot[en cos IC,~] {p}.
=
(42-44)
The eigenvalues K,,and kXnare defined by Kb tan rcb = ikbA(l
- MkX/k)2, k,/k = [l/(1 - M’)] [ - M + 41 - (1 - M2)(rc/k)2]. (45746)
A is the wall admittance, non-dimensional with respect to l/pc. Equations (42)-(44) are consistent with equations (39)-(41) through the eigenvalue equation equation (45). By using the real parts of the acoustic disturbances in equations (2) and (4) there results, in terms of non-dimensional perturbations,
4(N,J/pc3
= (1 + M2)(pu*
4(NIIX)/pc3
+ p*u) + 2M[uu* + pp*],
= pu* + p*u + M [w*
(47)
+ vu* + pp*].
(48)
The superscripted asterisk denotes the complex conjugate. If equations (36H38) are used in equations (47) and (48) the following relationships result: 4P,lpc3 = {pjT[(l 4P,,lpc3
+ M’)[Al
+ 2M PI]
(P*>,
(49)
= {P)~[CAI + M Ccl] {P*>,
(50)
where b An,,,
=
(a,,
+
a:)&,
exp
(-@,,
-
k:Jx),
Inm
cos
=
xny
cos
FC;L’
dy,
s0 Bm
=
(1
+
a,&)Z,, exp( - i(kX, - kzm)x),
[C] = [B] + [D], D,,,,, = /?np*J m nmexp( -i(kX, - kfQx), J,,, =
b sin K,J sin lczq’dy. s0 It can be verified that Tester’s expression [6] for the individual powers of Type I in each mode, P,,,, results from equation (49): that is,
k12
“n = 2 11 - MkX,/k12
k M + (1 - M2) Re 2
(
)I
I,,,.
(51)
ACOUSTIC
ENERGY
529
IN DUCTS
The equivalent Type II result is, from equation (50)
+ M
II2 ’ 1. J,!,,!.
(52)
I The exponential term is suppressed in equations (51) and (52) or, alternatively. included in the definition of the amplitude, p,. Note that P, and P,,. the acoustic powers, when divided by pc3/4 are not wholly non-dimensional as there is no scaling of lengths in the analysis. The resulting dimension is that of duct area (a unit depth is assumed). The additional terms in the power flux are now expanded. Define in non-dimensional terms
s
k,, = % +PC3M@,i,>dx = P,,x + $ &x = P,
+ pc3M
(p,(,)
M
dx + pc3M
s =Plx + f
(p,(:
(u~II,,)
+ p,*[,) d.y.
dx
s
M
(p&I,* + p,*[,)dx +$ M (u,,u; + u,*u,) dx. s s From equations (36) and (40) it can be determined that
I
+ p,*i,)dx = (P&x*
where Hnm =
--l-
kx, - kx*,
{p}'[H]{p)
(M,/?,sin rc,,bcos KEb + cczfl: cos i$,b sin IcEb)exp ( - i(kX,,- k,*,,)u).
(53)
If equation (43) is used this also can be written as Hmn =
--I A* % k& - k:_ (
cos icnbcos rczb + A $
cos Knbcos rczb exp( -i(k,,
- k:,,).x)
j (54)
It is apparent that if A = 0, the hard wall case, Hnm = 0. When real eigenvalues occur Hnm becomes singular for n = wt.A form of equation (53) or (54) can be derived to circumvent this by returning to the original integral definitions. These formulae are not given explicitly here. Alternatively, by defining the integral to have a lower limit x = 0, a constant can be introduced into equation (54) which makes evaluation of Hnm for real eigenvalues a straightforward limiting process. This would have the interesting effect of making Hqm(x = 0) = 0 and thus p,(.u = 0) = P,(x = 0), although the decay rate dp,/dx(x = 0) is not equal to dP,/dx (x = 0). The present choice of constant (zero) amounts to taking a lower limit x = f co (kX, assumed to have at least a small imaginary part) and has the salutary effect of producing a power which is of the proper sign for a given direction of propagation. This choice and a similar one in constructing the p,, terms also ensures that p, = I’,, (equation (55) below). By proceeding in the same way it is established that
s
(PJ:
+ p,*i,) dx +
s
by,*
+ ++,,
dx = {P)’ [f’l{p*l, .-
W. EVERSMAN
530
where pnnI = Hnm + k
.x.
: kzm(cc*P:cos rC,b sin Kzb + ~:a, sin rC,bcos $b)
x
x exp (- i(kX, - kzm)x).
Equation (45) allows this to be written as i p,,,,, = i
A* (, _ Mkx,,,)
A ~0s K,b ~0s C,b - c1 _ Mkznlk,
~0s ~,b ~0s Gb
1 x
x exp ( - i(kX, - kz”)x).
(55)
In the hard walled case P,,,,, = 0. Finally, the computational formulae are 4@pc3 = (P>’ [(l + M’)[A]
+ 2M[B] + M[P]] {P*},
43,J~c~ = {PI’ [L-AI f MCCI +
(56)
Wffl] @*I.
(57)
The individual powers of Type I and Type II. including the additional terms to account for wall impedance now follow from equations (56) and (57) with n = m:
4P,12
M + (1 - M’) Re
Inn
nn = (1 - MkJk(’ & Mlp,Y Re{(a,, - cr,*)/?Xcos lc,b sin rc:b), fm(kXn) lP,12
(58)
I II k
M + 2(1 - M2)Re
- M(l - M2) 2
nn = 11 - Mk,Jk12
2
Inn
MlpI2Re{G[~P~cos K,,b sin K,*b}. Wkxn)
- n
(59)
Again, the exponential term is suppressed in equations (58) and (59) or included in the definition of the amplitude p,. The axial rate of change of power dfi,/d.x or d?,,/dx is easily computed from equations (56) and (57) by noting that the x dependence is contained in the exponential factors in the coefficient matrices. Thus, for example, & [A] = [A’], A;, = -i(kXn - kzm)A,,,. Similar relations obtain for the other coefficient matrices. The dissipation term, W, is defined by W = (r,C:> = (rJ2) This leads to the vector-matrix
[i,C:l.
formulation 4WlP3
= {PI ‘cwP*L
where R,,,,, = 2?$ nmnm S*p fi* exp( - i(k,, - kzm)x) sin rC,b sin Iczb,
where P,,is the real part of Z/PC (non-dimensional).
(60)
The dissipation in the individual modes
ACOUSTIC
ENERGY
IN DUCTS
531
is thus
with the exponential suppressed or included in p,. Tables 1 through 4 show, in addition to calculations of power based on the Morfey and Ryshov and Shefter relations, the calculations of axial power in individual modes based on the new relations developed here (equations (58) and (59)). The power gradients and the dissipation term are also shown. It is apparent that PI and p,, coincide and that the sign of the axial power is consistent with the propagation direction: that is, positively propagating modes (attenuated for positive x) have positive power and the reverse for negatively propagating modes. It is also noted that the axial gradient of power equals the dissipation term. as stipulated by the time averaged continuity equation. Because of the factor k ‘, it would appear from equation (55) that at higher frequencies the correction P,,,, to P, is likely to be small. For well cut-on modes, ones for which P, is relatively large, Pnmis expected to be of only minor importance. This does in fact happen as shown in Table 4 where the reduced frequency kb = 6.0. Note, however, that for well cut-off modes moving with the flow the correction still dominates and the sign of p, and P, are different. 5. OBSERVATIONS Tester [6] used Mohring’s [2] definition of intensity, specialized to sheared flow in a two dimensional duct, to explain the deficiencies of the Morfey and Ryshov and Shefter definitions of intensity when the duct wall is soft. This was accomplished by treating the uniform flow as one with a vanishingly thin shear layer at the wall. When suitable modification is made to account for the fact that Tester’s result is dimensional, the additional power
in equation (56) with [P] defined by equation (55) can be shown to yield the identical diagonal terms defined by his equation (22). This is done by using the conjugate of the expression in brackets in equation (55) and by introducing the magnitude squared of the term (1 - MkJk) in the denominator. Equation (31) with the extra power term appearing in integral form, may be open to criticism as depending on an acoustic fluctuation product integrated with respect to the axial co-ordinate. The extra terms in equation (34) also were originally arrived at in the same way but were representable in terms of an explicit integration yielding results in terms of local pressure, velocity, and lining displacement. In either case, when the computational expressions of equations (56) and (57) are obtained the power is in terms of local pressure fluctuations. The acoustic power considerations addressed here verify the results of Tester for the diagonal terms and extend the implications to include power conservation laws which show the effect of the wall as a power sink. The results in this analysis can be viewed as having been derived from relatively conventional energy conservation concepts with no necessity to consider a wall shear layer. In the Tester approach, via Mohring’s intensity definition, the shear layer is essential. Miihring’s [2] equation (15) does not reduce to the uniform flow result (with soft walls) unless the proper limiting behaviour across the shear layer is accounted for. 6. CONCLUSION It
has been shown that both of the common acoustic energy density-energy
flu\- relation-
532
W. EVERSMAN
ships, of which Morfey [l] and Ryshov and Shefter [3] are the principal proponents, are related to the governing field equation for irrotational acoustic disturbances on a uniform duct flow by a variational principle. The Ryshov and Shefter energy density can be viewed as the sum of a potential energy density and a kinetic energy density from which the Lagrangian density can be formed. Hamilton’s Principle then yields the convected wave equation. The Morfey form of the energy density has been shown to be the Hamiltonian density, related to the Lagrangian density by a Legendre transformation. The modified form of Hamilton’s Principle then yields the convected wave equation. The concept of energy flux in the Morfey representation follows naturally from the definition of the Hamiltonian density. The problem asociated with the use of either the Morfey or Ryshov and Shefter energy formulations in the case of ducts with soft walls has been addressed and a new energy equation including an energy sink due to the wall dissipation has been derived. The acoustic flux, and hence the acoustic power, associated with the modified energy equation is consistent in sign with the direction of propagation of acoustic disturbances as deduced from attenuation arguments. ACKNOWLEDGMENT
The research reported here was supported by the National Aeronautics and Space Administration Lewis Research Center under Grant NSG 7192. The author wishes to thank the reviewers for particularly useful comments. Several significant shortcomings in presentation and content were brought to light and these have been modified, in certain cases by using ideas proposed by the reviewers. REFERENCES
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