A finite element scheme for acoustic transmission through the walls of rectangular ducts: Comparison with experiment

Journal of Sound and Vibration (1984) 92(3),

A FINITE FOR ACOUSTIC THE

WALLS

387-409

ELEMENT

SCHEME

TRANSMISSION

THROUGH

OF RECTANGULAR

DUCTS:

COMPARISON

WITH

EXPERIMENT

R. J. ASTLEV Department of Mechanical Engineering, University of Canterbury. Christchurch, New Zealand AND

A. CUMMINGS Department of Mechanical and Aerospace Engineering, University of Missouri-Rolla, Rolla, Missouri 6540 1, U.S. A. (Received 25 October 1982, and in revised form 14 April 1983)

Previous work on modelling acoustic transmission through the walls of rectangular ducts has left some open questions about structural damping, radiation damping and the way in which the acoustical radiation should be treated. In an attempt to provide answers to these questions a new numerical theory for the problem is described in this paper. It is both complementary to, and more accurate than, the simpler models described elsewhere. The results are gratifying, being in good agreement with measurements, both in detailed and in overall descriptions of the transmission phenomena. The numerical results are generally in close agreement with those of the previous theoretical approaches, providing some justification for several of the approximations implicit in those models.

1. INTRODUCTION In an attempt to gain a quantitative understanding of the mechanism of sound transmission through the walls of rectangular ducts (the main application being in air moving systems), one of the authors (A.C.) has published a series of articles [l-9] in which various aspects of this phenomenon have been investigated. Both theoretical and experimental data have been presented in references [l-9]. It has been shown that “wave solutions” for both the fundamental and higher order acoustic modes propagating inside the duct are in generally very good agreement with measurements of the duct wall transmission loss (TL). Additionally, the far field acoustic radiation patterns have been modelled with acceptable accuracy for the fundamental acoustic mode (see reference [4]). Simplified “approximate asymptotic solutions”, valid at sufficiently high frequencies, have also been described in reference [9]. In the “wave” theories, both for the fundamental mode and for higher order modes, closed form solutions for the duct wall response to the internal sound field were obtained, but the radiation load on the outer surface of the duct was neglected. Thus both the sound field inside the duct and the structural wave motion were uncoupled to the external sound field. In the fundamental mode theory presented in reference [l], coupling between the internal sound field and the structural motion was included in the form of a quasi-one dimensional approximate acoustic wave equation which incorporated the effects of slightly 387 @ 1984 Academic Press Inc. (London) Limited 0022-460X/84/030387+23 $03.00/O

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yielding walls. In the higher order mode theory described in references [7] and [8], this coupling was neglected on the basis that it was not of such consequence as to affect the accuracy of the predictions significantly. The radiation models for the fundamental and higher order modes were respectively a finite length line source and an “equivalent” cylindrical radiator (equivalent in the sense that the actual duct wall displacement was extended around the surface of a cylinder with the same perimeter as the duct). These simplified models were utilized because of the rather awkward geometry of the actual duct surface. In both the fundamental and higher order mode models, it was found necessary to incorporate a rather large structural loss factor (0.1 to 0.2) in order to limit the duct wall response at frequencies near wall resonances. The need for this unusually high degree of damping was assumed to arise from the effects of the external radiation load, though this assertion was pure surmise. Although these relatively simple theories have been successful in giving fairly accurate predictions of the duct wall TL, several open questions still exist, and are as follows. First, the validity of the radiation models for ducts of large cross sectional aspect ratio is perhaps questionable, in view of the great dissimilarity in geometry between both the line source model (for the fundamental mode) and the cylindrical radiator model (for higher order modes), and the actual duct. A more accurate modelling of the radiation from such ducts is desirable, at least as a check on the accuracy of the simple models. Secondly, the effects of radiation damping in determining the structural response, particularly at and near duct wall resonances, are unknown. Whether the inclusion of coupling to the external acoustic field would account for the apparently large structural damping is not clear and this feature also merits more detailed investigation. The work described in this article represents an attempt to answer both of the above questions. Finite Element (FE) methods are ideally suited to cope with the difficult radiation geometry of a rectangular duct, which cannot be modelled in full by any reasonably simple analytical scheme. Additionally, coupling between the duct wall motion and the external sound field can readily be included in an FE model incorporating both structural and acoustic elements. Accordingly, such a numerical scheme was formulated with the object of investigating in detail the roles of radiation geometry and external coupling. The FE treatment of the structural-acoustic coupling described in this paper is not unlike similar treatments in applications involving enclosed systems (most notably the coupled oscillation and response of automobile compartments [lo]) although the analysis in the present case is modified somewhat by the presence of a prescribed axial harmonic dependence in the acoustical field which is removed from the system prior to discretization. The inclusion of the effects of radiative acoustic boundaries is a more unusual feature of the current formulation. Several alternative approaches present themselves for the treatment of such effects in FE models. These include a number of boundary integral methods [ll, 121 and more recently infinite element and wave envelope techniques [13]. For the current problem where a cylindrical outer boundary may conveniently be chosen for the numerical acoustical region and where mean flow effects are not significant the methods already referred to, although applicable, are probably unnecessarily complicated. A more straightforward modal expansion in terms of cylindrical eigenmodes with the correct radiative properties appears to offer a simple alternative and is the method adopted in the present analysis. This type of approach has been used in analogous FE water wave studies [14] but does not appear to have been applied before to problems involving acoustical radiation.

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2. THEORY

2.1. GEOMETRY

AND

EQUATIONS

The geometry of the physical problem is shown in Figure 1. Some important simplifying assumptions may, however, be made before an attempt is made to cast the problem in mathematical form. The most important of these is the assumption of discrete frequency motion for the whole system and of a single axial phase speed both for the acoustic waves (inside and outside the duct) and for the structural waves in the duct walls. In addition the duct is assumed to be of infinite length although an adjustment to the calculated transmission loss will subsequently be made to account for the finite extent of the radiating segment. It is also assumed that the structural response and the external acoustical field are “driven” by a specified internal acoustical pressure. The coupling between the internal acoustical field and the rest of the system is therefore neglected, although for the case of a fundamental internal mode this assumption must be reconsidered and will be discussed further when results are presented for that case. Although the coupling with the internal field is ignored it is important to note that the coupling between the structure and the external acoustical field (which includes the damping effect of radiated acoustical power) is included in the analysis which follows. As a consequence of the assumptions already mentioned and of a further assumption that the internal acoustical field is composed of pure rigid-walled duct modes which are individually either symmetric or asymmetric about the y and z axes it is possible to reduce the solution region to the positive quadrant in the y-zplane shown in Figure 2. Appropriate boundary conditions resulting from the symmetry or asymmetry of the particular internal mode under consideration are then applied on the y and z axes.

Figure

1. Geometry

of the physical

problem.

Figure

2. Geometry

of the solution

region.

Suitable equations and boundary conditions are now presented to describe the acoustical and structural components of the problem. In the internal region (R, of Figure 2) the acoustic pressure pt (x, t) is specified and assumed to be of the form pz (x, t) = Re [Ap,,,“(y, z) eiwrPikmnx],

(1)

where A denotes the complex amplitude of the internal mode, pmn the transverse mode shape, w the radian frequency of excitation, k,, the axial wavenumber, x is a position vector and t is time. For a rectangular duct of width a and depth b, pmn and k,, are given explicitly by P,,(Y,z)=COS(m~(y+~/2)lu)cos(~~(z+~/2)/~), k,, =[k;

-(mrr/a)2-(n~/b)2]“2,

(2) (3)

where k. = OJ/co (co denoting the local sound speed) and m and n are the transverse mode numbers (5 0) in the y and z directions.

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The external acoustical field, which must satisfy the linearized acoustical wave equation at all points external to the duct, will be denoted by pf (x, t) and pz (x, t) in the regions RI and RI, respectively (see Figure 2); R, is a finite region surrounding the duct and R2 extends to infinity. Since pT and pT are assumed to have the same axial wavenumber as the internal acoustical field they may be written as

The complex

pressure

pT (x, t) = Re [p, (y, z) el”‘-ik,mnx],

(44

p$ (x, t) = Re [ p2( y, z) e’w’-‘k,~~x].

(4b)

amplitudes

p1 and p2 are then solutions

V2pi + k’*p, = 0,

of the Helmholtz

i=1,2,

equation (5)

where k”- - (ki - k&,) and V2 denotes the two dimensional Laplacian operator in the y-z plane. On the boundary C2 the solutions in R, and R2 must yield the same acoustical pressure and normal particle velocity. This condition may be written as p1 =

where d/an denotes are then determined for example,

p2

on

dp,/dn = dp2/an

C2,

on C2,

(6a, b)

a derivative normal to C2. Boundary conditions on the y and z axes by the symmetry or asymmetry of the internal mode. On the z axis,

either or

pi=O,

(if m is even),

i = 1,2

api/ay = 0, i = 1,2

(if m is odd).

Analogous conditions hold on the y axis. At the inner boundary of the outer region the normal the velocity of the vibrating duct wall. If the outward 7*(x, s, t) is written as 7*(x, s, t) = Re [n(s) where s denotes a perimetral boundary condition becomes

co-ordinate

(7b) particle velocity must equate to displacement of the duct wall

eiwrPikg,,nx],

(as indicated

Vp, . n, = -pow27

(7a)

on

in Figure

C,,

2), the above kinematic

(8)

where n, is a unit outward normal from the region R, on Ci, V = (a/ay, a/at) and p. is the ambient density of air. At the outer limit of the external acoustical region the pressure field must consist entirely of radiated (rather than reflected) components. The appropriate mathematical statement of the above condition is given by the two-dimensional Sommerfeld radiation condition &[a&ar+ik’pJ-+O

as r-+03,

(9)

where r is a cylindrical polar radius as shown in Figure 2. The description of the coupled structural-acoustic system is completed by the specification of an appropriate structural equation relating the displacement of the duct wall to the applied internal and external acoustical pressures. In the present case the duct wall is assumed to behave as a thin elastic plate which is governed by the dynamic plate equation gV;n*+q(a2n*/at2) where g represents Laplacian operator

the flexural rigidity in the duct surface

=po* -p:,

of the plate, q its mass per unit area and V: the (given by VT = a2/as2+if2/ax2). The assumption

FINITE

of harmonic equation

ELEMENT

time dependence

DUCT

WALL

TRANSMISSION

k,,

and an axial wavenumber

g(D4-2k2,,D2+k4,,)77-q0277

391

then yields the structural

=po-pl,

(10)

where D = d/ds. The boundary conditions on 77 at the y and z axes are then deduced from the symmetry or asymmetry of the exciting mode. At point A for example (see Figure 2) the boundary conditions are D3~=0

and

Dr]=O

D2v=0

and

~=0

ifmiseven,

(1 la)

ifmisodd.

(1 lb)

Physically these conditions correspond to zero shear force and slope (equations (1 la)) or zero bending moment and displacement (equation (llb)). Analogous conditions hold of both structural and acoustical components at B with m replaced by n. Specification of the problem is now complete. 2.2.

A VARIATIONAL

STATEMENT

OF THE

PROBLEM

Several techniques now present themselves as the basis for a finite element formulation of this problem. The approach selected for the current analysis is based on a variational statement which implicitly satisfies the structural and acoustical field equations and all but one of the boundary conditions. The appropriate functional x is defined by

x(77,PJ =

:g[(D2n)2+2k2,,(Dn)2+

k4,,n2] ds

Cl

+

I &[VPI.

VP1- %:I

R

dy dz

1

+

--

1 2P1 POW

(12)

It is relatively easy to show (see the Appendix) that this functional yields equations (5) and (10) as Euler equations arising from variations in pI and 17.The kinematic boundary condition at the surface of the duct (equation (8)) emerges as a natural boundary condition if p1 is not constrained on C1. The symmetrical pressure boundary conditions at the y or z axes (e.g., equation (7a) on the z axis) result as natural boundary conditions on p, unless a boundary condition of type (7b) is explicitly applied. At the outer boundary of RI the particle velocity constraint (equation (6b)) occurs as a natural condition if p1 is unconstrained. Variations of n then yield natural boundary conditions which correspond to those of equations (lla) or (llb), depending upon whether the displacement or its slope is explicitly constrained. The determination of the stationary values of x with respect to all permissible functions n and pl-permissible in this context implying that they satisfy any explicit boundary conditions, are continuous and in the case of r] have continuous first derivatives-then yields an implicit solution not only of the structural and acoustical governing equations but also of all the required boundary conditions except for equation (6a) (continuity of pressure at C,) which will be satisfied independently in the course of the numerical formulation.

392

R. J. ASTLEY

AND

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It is of interest to note that some of the terms in the functional x have recognizable physical origins. The first term for example represents the mean strain energy of the duct per unit length. The second and third terms are easily recognizable as the mean kinetic energy of the duct per unit length and the potential energy per unit length of the applied loads, respectively. Clearly the functional could be modified to include more complex structural effects (the presence of longitudinal stiffeners for example) by simple adjustment of the first two terms in the expression for x. 2.3. FE DISCRETIZATION AND SOLUTION An FE discretization is now used to establish a class of trial functions with respect to which the stationary values of the functional x may be determined. The duct wall and the adjoining acoustical region are accordingly divided into one-dimensional and twodimensional elements, respectively. A typical element subdivision is shown in Figure 3.

\ Structural element

Figure node.

3. Finite element

discretization

of the duct and surrounding

region.

0, Acoustical

node; 0, structural

The two noded structural elements have Hermitian shape functions identical to those of a standard beam element. The structural degrees of freedom therefore comprise nodal values of n itself and also nodal values of its derivative with respect to s. The resulting trial function, ;i say, may conveniently be written as (13) where p denotes the number of unconstrained degrees of freedom, Si the ith degree of freedom and Si the corresponding global shape function (defined inside each element by an explicit element shape function). The discretization of the acoustical region R 1proceeds in a similar manner: nine noded isoparametric rectangles are used with Lagrangian shape functions which vary quadratically in the parent element. A trial function pI1is then implicitly defined throughout RI and may be written as

b = iilNPi

=

[Nl{Pl7 1xq qx1

where 4 is the number of unconstrained acoustical value of pressure and Ni the corresponding global At this point it is convenient to specify the form is chosen so that it exactly satisfies the appropriate

(14)

degrees of freedom, pi the ith nodal shape function. of the outer acoustical field p2. This field equation in R2 (equation (5))

FINITE

ELEMENT

and is written as an expansion radiation condition (equation metrical boundary conditions eigensolutions that individually

DUCT

WALL

393

TRANSMISSION

in outward traveling orthogonal eigensolutions so that the (9)) is also explicitly satisfied. The symmetrical or asym(of type (7a) or (7b)) may then be satisfied by selecting satisfy these conditions. Then p2 may finally be written as I p2 = C AiF( aiO)Hg)( kir). I=1

(1%

Here 2i-2, F(X)

=

cos X

(n even)

1 sin X

(n odd)

(m and II even)

(y_= 2i- 1, 2i-1, ’

’

1 2i,

}

(m even, n odd) (m odd, n even)

’

(n and m odd)

1 ki = (k” - kkn)1’2, and H’*‘( ) denotes a Hankel function of the second kind. Equation (15) may be rewritten in matrix form as ~2

=

[

‘4 {A). IXl IX1

(1Sa)

where ‘Pi( r, 0) = F( aiO)HLt’( kir) and {A} is the column vector of the modal coefficients Ai. It is convenient to write the radial deviative of p2 in the same form: i.e., @Jar =

(16)

PWAL

where @i(r+3)= k&‘( aJ)Hi:)‘( kir). The trial functions ;i and i1 and the modal expansion for ap2/ar(equation (16)), are now substituted into the expression for x to give

The matrices [K], [Ml, [B], [Cl, [D] and the vector {Fo} in the above expression are given by

[Kl=g

I

Cl

[[~‘[S11’[D2[S11+k2,nPISllT[~Pll

[Ml=q

[D]=-

I Cl

[Sl=l-SldS,

--&Wl’Pl

I c* POW

(19,20)

USI = 1 P~=P,,(Y, z) ds.

(22723)

Cl

ds,

B

(18)

LB1= j- [Nl=[Sl dS,

Cl

Minimization of x with respect to the components linear equations of the form K-o*M

+ k2,,[Sl’[Sll ds,

0

of (6) and {p} then yields a set of

(24)

K.

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J. ASTLEY

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CUMMINGS

Since the modal coefficients {A} (as well as the structural and acoustical degrees of freedom (6) and {p}) must be regarded as unknown parameters of the problem the above set of equations is incomplete, a result of the pressure continuity condition at C, (equation (6a)) having not yet been included in the discretized problem. This is now done (in an integrated sense) by weighting the residual pressure difference ( c1 - p2) with appropriate weight functions and integrating over C,. If the weight functions are chosen to be the component functions of [@I this process yields

[Dl? where [D] is as previously

defined

PI

(25)

+CElIAI = I()),

and [E] is given by

[El=-j[@]‘[P]ds. c‘L Equations equations

(24) and (25) may now be combined for {a}, {p} and {A} of the form

to yield a complete

(26) system

of linear

(27)

The submatrices [K], [M] and [C] are clearly symmetric (from equations (18), (19) and (21)) and since the submatrix [E] can be shown to be diagonal (a consequence at the orthogonality of [ @] and [q] on C,) the entire coefficient matrix on the left-hand side of equation (27) is symmetric. The various submatrices within the coefficient matrix are not without some physical significance: [K] and [M] for example are the stiffness and mass matrices for the duct, [B] provides the coupling between the structure and the external acoustical field and {F,,} contains the equivalent structural loads caused by the internal acoustical field. The numerical portion of the external acoustical field (with a combined mass-stiffness matrix [Cl) is then coupled to the far field eigenexpansion by the matrix [D]. This general form of the discretized structural-acoustic problem is not unlike that found in several other applications. If the coupling with the radiative outer solution is deleted the resulting subset of the equations is similar to that encountered in models for the structural-acoustic interaction of closed compartments [lo]. The equations as they stand closely resemble those presented in reference [ 121 for the response of loudspeaker systems though in this case the outer radiative field is modelled by an integrated source distribution (rather than an eigenfunction expansion) and an additional enclosed reverberant region is coupled to the system at our inner boundary. The solution of equation (27) was achieved by means of a banded Gaussian reduction scheme (the equations are strongly banded if the degrees of freedom are appropriately ordered), the element contributions to the coefficient submatrices having been first assembled in the usual way. The discretizations used to produce the results presented in this paper involved typically 200-400 degrees of freedom (a relatively coarse mesh with 253 acoustical degrees of freedom and 24 structural degrees of freedom is shown in Figure 3) and 10 terms in the eigenfunction expansion. The accuracy of the solutions was regularly checked by refining the mesh and extending the eigenfunction expansion to ascertain whether significant changes occurred in the resulting solution. It was generally found that the resolution of the structural deformation (rather than the acoustical field) placed the greatest demands upon the mesh.

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CALCULATION

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395

A convenient (and experimentally verifiable) way in which to present results from the current numerical scheme is through the calculation of an appropriate transmission loss representing an integrated measure of the ratio of the acoustic power transmitted through the duct walls to the incident power of the internal acoustical mode. The acoustic power radiated per unit length by an infinitely long duct, WT say, is given by ws”=4 I C2 where

1, is the radial

component

I, ds,

of the acoustic 1, =p*(x,

(28)

intensity

on C,, defined

by

t)uT (x, t).

(29)

The overbar in this expresssion denotes averaging over one period variable UT is the radial velocity on CZ, given by UT = Re [(i/wp,)(ap/ar)

of oscillation

and the

eiwt~ik~~~n~].

The effect of a finite length radiating source may then be approximated by the theory reference [4] which predicts a radiation efficiency C, that is given by the expression C,=l

I-cosK_Si(K)+z 7T [ K

K

of

, I

in which K = L( k,, - k), L is the length of the duct and Si( ) denotes the sine integral. The sound power radiated per unit length from a finite length duct, W,, is then given by w, = c, w;. The internal acoustical power, Wi,,, is simply calculated intensity of the internal mode over a control surface within to the axis of propagation. This gives W,,, = IA]2k,,ab12wp,,~,~,,

(30) by integrating the acoustic the duct and perpendicular

(31)

of the internal mode (see equation (1)) and E, is a Neumann where A is the amplitude symbol (equal to 1 if the subscript is zero, otherwise equal to 2). The transmission loss is then defined by TL = IO log,,, ( W,,J W,).

(32)

In the current numerical model the evaluation of Wg” (via equations (28) and (29)) can be performed either inside the numerical region R, (in which case p* and U: are deduced from the FE trial functions) or in the outer region R2 (in which case p* and u: are obtained from the modal expansion). Both calculations were programmed into the numerical scheme as a check on the implicit matching procedure at the boundary C’*. The two sets of values obtained were in close agreement (a 5% discrepancy in W, was the largest noted; typically the difference was less than 2%). In all results presented later in this paper the modal expansion values of W, are used.

3. EXPERIMENTS Three sets of experimental data were obtained in selected cases for galvanized steel ducts with cross section 457 X 229 mm, wall thickness 0.64 mm and length 4.3 m. First, the duct wall TL was measured in l/10 octave bands of noise for an internally propagating (1,0) mode, a continuous frequency analysis being used to obtain fine

396

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CUMMINGS

frequency resolution. By this means, the duct wall resonance frequencies could be found, and the extent of damping determined. Secondly, measurements of the amplitude and phase of the duct wall displacement (at a given axial position) for the (1,O) mode were made by using pure tone excitation. A displacement profile could thus be plotted and compared with theoretical patterns. Thirdly, measurements of the radiated acoustic pressure in the near field were made for noise filtered in a l/ 10 octave band, once again for the (1,O) mode propagating inside the duct. In all cases, commercially made ductwork was used, and the test ducts were made up from three sections which were connected axially by “standing S slip” joints across the longer sides and “drive” joints across the shorter sides (see reference [15] for further details of the joints). Two opposite duct corners incorporated “Pittsburg lock” seams (see again reference [15]) while the other two corners were continuous, and formed simply by bending the metal sheet. The TL and displacement measurements were made on a duct with flat walls, whilst the near field pressure measurements were taken on a duct with “cross breaking” on all walls (two diagonal creases, making each wall into four sides of a very flat rectangular pyramid). By the use of ordinary commercial ducts, some idea of the applicability of the theoretical methods to practical systems could be obtained, and the accuracy of the predictions-both in local details and in overall behaviour-could be assessed. 3.1.

TRANSMISSION

LOSS

TESTS

The (1,O) acoustic mode was excited in the test duct by two loudspeakers mounted in one wall of a massive acoustic enclosure; one end of the duct abutted against the enclosure wall in which the speakers were mounted and was sealed to it with modelling clay. The speakers were fed with random electrical noise, from a General Radio (GR) type 1390-B random noise generator and a power amplifier, and were connected so that they vibrated in antiphase. Measurements of the acoustic field within the duct revealed that a sufficiently pure (1,0) mode could be excited over the entire frequency range of the test data. The duct was first fitted with an “anechoic” termination-consisting of 0.6 m long glass fibre wedges-at the end opposite the speakers, closed off with a 16 mm sheet of chipboard and sealed. The duct and loudspeaker enclosure were both situated in a reverberant chamber of volume 161 m’, and the sound pressure level (L,) was measured at five positions within the chamber by using a Briiel and Kjaer (B & K) type 4133 13 mm condenser microphone, a B & K type 2609 measuring amplifier and a GR type 1521-B graphic level recorder. A continuous frequency analysis in a l/ 10 octave bandwidth was plotted for each microphone position. The recorder pen writing speed was 25 mm/s and the chart paper speed was 38 mm/s. In these tests, the space averaged Lp at each frequency was related to the sound power radiated from the duct walls. The voltage fed to the loudspeakers was kept constant throughout the tests. Next, the anechoic termination and chipboard cover were removed and the space averaged L, was measured, a priori, with the same voltage being fed to the loudspeakers. The L, in this case was representative of the internal sound power entering the duct. The difference between the two space averaged Lp values is equal to the difference between the internal and radiated sound power levels. The L, difference at each frequency was corrected for the duct dimensions to yield a TL figure consistent with the aforementioned definition. At frequencies approaching cut-on for the (1,0) mode, the modal reflection coefficient at the open duct termination was not insignificant, and so measurements were made of this quantity (by using a pure tone excitation for a range of

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frequencies), the data being derived from the standing wave ratio. These measurements were then used to correct the 7Z figures. In this process one neglects the effects of axial resonances in the duct, but since the corrections were relatively small (6.5 dB at most, and elsewhere less than 3 dB), the resonances were fairly heavily damped, and little loss of accuracy was incurred. 3.2. WALL VIBRATION MEASUREMENTS With the duct situated in the reverberant chamber and fitted with the anechoic termination, the (1,0) acoustic mode was again excited in the duct as previously described, except that a pure tone was fed to the loudspeakers. With a signal frequency of 650 Hz, the vibration amplitude and phase were measured around the duct walls on a cross section approximately midway along one of the duct sections. A B & K type 4374 accelerometer (of mass 0.65 g) was used as the vibration transducer, and its output was fed to a B & K type 2635 charge amplifier and thence to a GR type 1564 sound and vibration analyzer. Calibration was effected by means of a B & K type 4291 accelerometer calibrator. The phase of the accelerometer signal was measured using an AD-YU type 406L precision phase meter, the loudspeaker input acting as the reference signal. The acoustic pressure amplitude inside the duct was also measured by using a type 4133 13 mm B & K microphone, calibrated by means of a B & K type 4220 piston-phone. PRESSURE MEASUREMENTS 3.3. NEAR FIELD ACOUSTIC These measurements were taken outdoors, with the duct placed vertically on the ground. Figure 4 shows the arrangement. A pair of loudspeakers was mounted in an enclosure inside the lower end of the duct, and an anechoic termination was fitted at the upper end. The ground surface surrounding the duct was covered with about 150 mm of glass fibre blanket, which acted as an acoustic absorbent, and any other reflecting surfaces were sufficiently distant so as not to affect the acoustic near field of the duct significantly.

Anecholc termmotlon

Acoustlcoily

/ 150 mm thickness of gloss flbre on ground surface

Figure

4. Experimental

arrangement

for near field acoustic

pressure

measurements.

R. J. ASTLEY

398

AND

A. CUMMINGS

The loudspeakers were fed, in antiphase, by a random noise signal from a GR type 1390-B random noise generator and a power amplifier. The voltage was kept constant and was monitored by using a B & K type 2409 electronic voltmeter. The near field acoustic pressure was detected by means of a B & K type 4133 13 mm condenser microphone, the output of which was fed to a B &K type 2609 measuring amplifier, and the signal was filtered in a l/10 octave frequency band (centred on 650 Hz in these tests) by a GR type 1564 sound and vibration analyzer. The microphone was fitted with a small hook, and was placed in a known position relative to the duct by means of a wire grid of 13 mm mesh, the hook being placed at an interstice of the grid; the grid itself was surrounded by a metal frame which was held in place by supports attached to the duct. These supports were covered with glass hbre blanket to minimize acoustic reflections. Sound pressure readings were taken at alternate grid points within the test area adjacent to the duct wall. These values were plotted at each grid point and contours of pressure amplitude drawn by interpolation between the grid points.

4.

RESULTS

In the discussion that follows, theoretical results for a duct of the type shown in Figure 1 are compared with experimental data obtained by using the procedures described in section 3. In addition to the numerical results produced by the FE method described in section 2, theoretical results are also given for a simpler model (involving approximation of the duct wall by a cylindrical radiating surface and denoted henceforth as the CR model) described in reference [8]. The presentation of a comprehensive set of results creates some practical difficulties since a variety of duct sizes would need to be studied, and for each of these, data should ideally be compared over an appropriate frequency range and for a complete set of incident modes, as well as for different degrees of damping. This would involve a proliferation of data quite inappropriate in a paper of this length. The authors have therefore chosen to present results only for some “typical” cases. Attention is initially confined to the propagation of the (1,0) mode in a duct with a 2: 1 aspect ratio as described in section 3. This configuration forms the basis of the comparison between theoretical and experimental data. Some conclusions are then reached regarding the effects of acoustical radiation and structural damping. These simplify the presentation of results for a more complete set of internal modes and other duct sizes, this latter set of results being presented primarily in an attempt to establish the accuracy and limits of validity of the simpler CR model. 4.1.

COMPARISON

WITH

EXPERIMENTAL

RESULTS

In this section a comparison between theoretical and experimental results is presented for a duct of width 0.457 m and depth 0.229 m subject to excitation by the (1,0) internal mode. The thickness, Young’s modulus and density of the duct wall are 0.635 mm, 181.6 GPa and 7812 kg/m3, respectively (this duct will be termed “duct 1” in all discussion to follow). Results are presented over a frequency range 400-1500 Hz (the lower end of this range being just above the cut-on frequency for the (1,O) internal mode). Duct 1 is perhaps slightly smaller than average (within the general spectrum of air moving ductwork) but has the advantage of yielding relatively well separated structural resonances in the frequency range of interest (for larger ducts they tend to be more densely packed, making comparison with data-even from l/10 octave band measurements-more difficult).

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The qualitative features of a typical FE solution for this configuration are shown in Figure 5 for a frequency, f, of 650 Hz. On the right-hand side of the figure are shown the calculated contours of sound pressure amplitude within the FE portion of the external region (the contours are plotted at 10 equal linear increments, between the maximum

Figure 5. Calculated structural response and pressure contours p, Pressure amplitude contours; --O--, displaced shape (imaginary component).

for the (1,O) mode. Duct 1, f = 650 Hz, 5 = 0. (real component); -a--, displaced shape

value and zero). On the left-hand side of the figure is plotted the displaced shape of the duct wall. The wall displacement is suitably scaled and its real and imaginary components are represented by continuous and broken lines, respectively (the location of the structural modes being indicated by solid and open circles). For frequencies far from structural resonances-and 650 Hz falls into this category-the displacement is almost in phase with the pressure yielding predominantly real values for n as seen in Figure 5. The nature of the pressure field shown in Figure 5 leads to an interesting preliminary observation. Although the wall displacement driving the external pressure functions is of a relatively complicated shape, the acoustic field itself is of a very simple two-lobed, dipole-like pattern (allowing for the planes of symmetry and asymmetry in the y and z directions, respectively). This would indicate that it is the overall integrated effect of the wall displacement, rather than the detailed perimetral variation, which dominates the external field even in regions close to the duct walls. A qualitative confirmation that the measured pressure field does correspond to the predicted solution is provided by a comparison of the pressure pattern of Figure 5 with the experimental pressure contours of Figure 6 obtained at the same frequency (see Par1 of measurement

Pittsburgh

gr!d

lock

A?,,

Figure

6. Measured

pressure

amplitude

contours

for the (1,0)

mode in duct 1. f = 650 Hz.

400

R. J. ASTLEY

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section 3.3). As with the numerical solution, the measured contours represent linear increments in sound pressure amplitude. Clearly the general nature of the field is strikingly similar in both cases although some asymmetry is evident in the real situation. A comparison of predicted and measured duct wall displacement patterns for the same frequency is shown in Figure 7. The correspondence is not exact (there being some asymmetry again in the real situation) but the general scale of the perimetral variation

Pittsburgh

Figure 7. Measured 5 = 0. -, Predicted;

lock

and calculated duct wall displacement . . . . ., measured.

patterns

for the (1,O) mode.

Duct 1, f = 6.50 Hz,

and the number of lobes on each side is qualitatively in good agreement. Quantitative comparison of the response is not easy to make since the theoretical and experimental displacement patterns do not exactly correspond, but the experimental values appear to be generally rather larger than predicted. The differences are not great, however, and presumably reflect the fact that the theoretical model does not represent all details of the real system (the “Pittsburgh lock” corner joints, for example). Pure tone measurements tend to exaggerate detailed features of geometry, while measurements in bands of noise suppress them. The necessity of obtaining phase data precluded the use of bands of noise in the wall displacement measurements. It should be emphasized that the experimental and theoretical curves of Figure 7 do not represent the actual magnitudes of the displacement, each curve being normalized with respect to its largest value. They therefore give only a comparison of the shape of the displacement patterns. For a representative comparison of the magnitudes involved one must look to the measured and predicted TL over the appropriate frequency range. Before this is done, however, it is appropriate to reach some assessment of the effects of acoustical and structural damping on the system. Both effects are included in the current numerical formulation, the first via the coupled treatment of the structural and acoustical response and the second via the inclusion of a structural loss factor, 6, in the definition of the plate rigidity (the flexural rigidity of the plate, g, is given by g = Et3 (1 + it)/ (1 - v*) where E, t and v denote Young’s modulus, plate thickness and Poisson’s ratio, respectively). The results presented in Figure 5 were calculated for 5 =0 (no structural damping) though the inclusion of quite large values of 5 (as great as 0.2) have little perceptible effect at off-resonant frequencies such as 650 Hz. To gauge the relative effects of acoustical and structural damping the response of the system must be studied near resonance. A structural resonance of the idealized model for duct 1 occurs at a frequency close to 830 Hz. A representation of the predicted solution in the region close

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resonance is presented in Figures 8 and 9. Calculated TL curves are shown in Figure 8 for structural loss factors of 0, 0.01 and 0.1 for a frequency range which includes the structural resonance. Also shown are measured (l/10 octave band) values of TL?’and an “uncoupled” TL curve which is obtained from the present FE model by removing the submatrix [B] from the upper partitioned row of equation (27). This has the effect of removing the external acoustical pressure as a structural load and thus decoupling the system. The “uncoupled” curve therefore represents a solution with no damping at all (structural damping having also been removed by the specification of t=O) and consequently predicts infinite displacements (and a TL of -co) at resonance.

to

-10.

-EOi-_750

J 800 Frequency

Figure 8. Calculated and measured transmission ---. theory (uncoupled); 0, measurements.

I-,

850

(Hz)

loss for the (1,0)

mode in duct 1. -,

Theory

(coupled);

Referring to Figure 8 note first that as resonance is approached from below all solutions show an initial increase in TL. The reason for this behaviour is demonstrated by the sequence of predicted wall displacement and pressure contour patterns shown in Figure 9 for f = 800, 814 and 833 Hz. The maximum in the TL (which occurs near the second of these values) is clearly attributable to “cancellation” in that the net integrated effect of the wall displacement becomes small and all that remains of the pressure field are some strongly attenuated local solutions. It has always been assumed that TL behaviour of this type was due to cancellation but the current results provide somewhat vivid graphical confirmation of this. Of interest also in Figure 9 is the rapidity with which the pronounced dipole radiation pattern reasserts itself on either side of the cancellation frequency (within about 10 Hz.). These solutions were calculated for zero structural ? Parenthetically one can note that a l/ 10 octave frequency bandwidth is sufficiently resonances exhibited in the TL curves shown. Had the resonances been much sharper, bandwidth would have been required.

narrow to resolve the however, a narrower

402

R. J. ASTLEY

(0)

AND

A.

CUMMINGS

(b)

Figure 9. Calculated structural response and pressure field for the (1,O) mode in duct 1 with .$= 0 (coupled). Pressure amplitude contours, --O--, displaced shape (real (a) f = 800 Hz; (b) f = 814 Hz; (c) f = 833 Hz. -, component); --0--, displaced shape (imaginary component).

damping (5 = 0) but similar general behaviour (i.e., a cancellation “peak” followed by a resonance “dip”) is found for all values of 5. It can be noted in passing that even though no structural damping is included in the solutions of Figure 9 the effects of acoustical damping are detectable in the solution for 833 Hz, a frequency which is very close to the structural resonance. These effects are indicated by the large out-of-phase component in the wall displacement. Returning to the TL curves of Figure 8 and a comparison of the predicted and measured values one can see that the measured results correspond most closely to the predicted response of a system with significant damping, being close to the predicted TL curve for 5 = 0.1 (the correspondence is in fact a good deal more apparent when a larger frequency range is considered, as in Figure 10 to follow). This value of the effective loss factor (i.e.. 5 =O*l) is in general agreement with similar observations for the fundamental mode behaviour of such systems [6], where a loss factor of 0.2 was deemed appropriate. What is clearly revealed by the present analysis, however, is the very small contribution of radiation coupling to the observed levels of damping. A comparison of the “uncoupled” response and that for 5 = 0 and 0.01 shows that the damping effect of acoustical radiation (represented by the difference between the 5 = 0 and uncoupled curves), although detectable, is small compared with the effect of even a minimal structural loss factor of 0.01. Since the actual damping in the system appears to be in the region of ten times this amount (i.e., t= Oal), one may conclude that whatever the origin of the damping, acoustical wave radiation is not the primary mechanism. This is a useful result in the context of further theoretical studies since the inclusion of coupling between the structure and the external acoustical field occasions considerable additional complexity in the formulation of either analytical or numerical models. A confirmation that these deductions are not merely a consequence of sampling the TL solutions over an “untypical” frequency range is given by the TL curves of Figure 10. Calculated values of TL (for [=O.O and t=O.l) are presented over an extended frequency range (400-1500 Hz) and comparison is again made with measured values. The predicted curve for .$ = 0.1 is seen to correspond closely to the observed TL over

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,

I._._-

IO

I

-5 )O

I

I

1

I

Ik

Frequency (Hz) Figure 10. Calculated measured.

and measured

transmission

loss for the (1,0)

mode in duct 1. -,

Theoretical;

0,

the entire frequency range though there is an interesting shift of the actual resonant frequency in the vicinity of 800 Hz. It seems probable that this results from an inadequate description of the “Pittsburgh lock” joints, assumed to be rigid in the numerical model. That they are in all probability somewhat less than rigid is already indicated by the small asymmetries in the observed wall displacements and pressure field of Figures 6 and 7. A decrease in the actual stiffness of the system would tend to lower the structural eigenfrequencies and consequently produce the type of frequency shift observed in Figure 10. The structural model could easily be modified to include some type of flexible corner condition but since there is no clear indication of what precisely this should be, such modification has not been considered worthwhile at this stage. 4.2. COMPARISON

WITH

PREVIOUS

THEORETICAL

RESULTS

Although the comparison between the present numerical results and measured data, as presented in the preceding section, is generally good it must be frankly admitted that good correspondence has also been reported [8] between the simpler CR model and analogous measured data. One of the main objectives therefore in developing the FE scheme was to check the validity of the CR approximation by including those effects not modelled by that approach (viz. the structural-acoustic coupling in the outer region and a correct geometric representation of the external field) and thereby to assess the importance of their exclusion in practical calculations. In this section a comparison of results obtained from the FE and CR models is presented for a comprehensive set of modes. The comparison is extended to a study of the effects of varying the aspect ratio of the duct. In the light of the results already presented a few tentative predictions may be made regarding the comparisons to follow. First it was suggested in section 4.1 that the

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R. J. ASTLEY

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contribution of radiation damping to the overall response of the system, certainly for realistic values of the structural loss factor, was extremely small. One may therefore anticipate that the inclusion of this effect in the FE model and its exclusion from the CR model will result in few significant differences. Secondly one may note that the simple nature of the predicted acoustical field for the (1,0) mode as indicated by the pressure contours of Figure 5 might be interpreted (in the context of other modes) as a general indication that the model of a cylindrical radiating surface for the external pressure calculation may not be a bad approximation since it would appear that the net integrated displacement (rather than its detailed perimetral variation) dominates the external field. Both of these suppositions are in fact substantially borne out by the results that follow. All results are calculated for a structural loss factor of 0.1. Predicted TL curves for duct 1 are shown in Figure 11 for a comprehensive set of internal modes, the cut-on frequency of each mode being indicated by the starting point TL curve. Both FE and CR results are presented for the frequency of the corresponding range 100-1500 Hz. For all modes, the FE and CR theories give values of TL which differ

30 -

I

0. 100

Frequency (Hz)

Figure 11. Calculated transmission loss of duct 1 with [=O.l. FE results: 3---, mode; -Cl-, (2,O) mode; 4(0, 1) mode; -/I--, (3,0) mode. CR results: High frequency approximation: --_ (0,O) mode; ---, (3,0) mode.

-,

(0,O) mode; --O---, (I, 0) for the above modes.

only by a small amount (seldom more than 1 dB) though the FE model does give consistently higher values. The fundamental ((0,O)) mode FE and CR results deserve k’ of equation (5), is zero some explanation since the external effective wave number, for the fundamental mode if the internal acoustical field and the structural response are assumed to be uncoupled (as proposed in the FE model). This of course excludes any acoustical radiation from the duct and corresponds to a sonic axial phase speed of the disturbance (for higher modes the axial phase speed is always supersonic). In order to apply the FE (or the CR) formulations to the fundamental mode case the assumption of an uncoupled internal field must therefore be abandoned. Fortunately a relatively simple analysis, reported in reference [l], may be used in the calculation of the coupled axial wave number and phase speed in the fundamental mode case. The results of this analysis involving the (0,O) mode; for other modes k,, were used to specify k,, in calculations is simply given by the rigid-walled value of equation (3). The same analysis is incorporated in the CR model (for the fundamental mode only) so the comparison between FE and CR results remains consistent. It is also worth noting at this stage that since the axial wavelength in the case of fundamental mode propagation, especially at low frequencies, is relatively large compared with the overall length of the duct the TL correction factor C, can be significant. For the higher modes C, is always small-of the order of a fraction of 1 dB-and the adjustment to the calculated TL values is almost imperceptible.

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The close correspondence between FE and CR results demonstrated in Figure 11 is to some extent explained by the nature of the predicted external fields. In section 4.1 it was noted (see the discussion of Figures 4 and 5) that a surprisingly orderly external field resulted from (1,O) excitation even though the wall displacement itself was of a rather complex form. Similar deductions may now be made regarding higher and lower modes. In Figure 12 for example typical wall displacement and pressure contour plots are shown for the (0,O) and (2,O) modes at 191 Hz and 800 Hz, respectively. In the first case a

(a)

(b)

Figure 12. Calculated structural response and pressure field for duct 1 with t=O.l. (a) (0,O) mode, f= 191 HZ; (b) (2,0) mode, f=800Hz. -, Pressure amplitude contours; -O--, displaced shape (real component); -a-, displaced shape (imaginary component).

clearly monopole-like, line source field is generated despite the rather complex displaced shape (which contains a significant out of phase contribution because the frequency is near that of a structural resonance). In the second case the external field is seen to adopt a regular four lobed pattern (allowing for the vertical and horizontal axes of symmetry) although once again the displaced wall shape is rather complicated (with eight stationary values of displacement in each quadrant). In the light of such results it is perhaps not surprising that the CR model produces good values for the TL. Clearly the fine geometrical detail of the wall displacement does not contribute significantly to the radiated acoustical field and its inclusion in the FE model may therefore be expected to result in only minor modification of calculated TL values. Two further TL curves are shown in Figure 11. These are asymptotic “high frequency” solutions (reported in reference [9]) for the (0,O) and (3,O) modes. They lack the detailed, structural resonance related aspects of the response-which are in fact almost entirely absent in the (3,O) mode because of the level of structural damping-but are in good agreement in an average sense with the FE and CR curves. The asymptotic theory is discussed more fully in reference [9] and the TL curves of Figure 11 are included here merely as a further comparison of general (though somewhat peripheral) interest in the context of the present paper. Returning to the principal comparison-between FE and CR values-one might anticipate that any differences between the two theories might manifest themselves in ducts with high aspect ratios; that is, insofar as the actual rectangular duct surface can be approximated by a cylinder, the approximation is likely to be most satisfactory if the cross-section is not too flat. Results were therefore calculated for two additional duct cross-sections of larger aspect ratio. The first of these, “duct 2”, has an aspect ratio of 4: 1 being the same width as duct 1 but one half as deep. The second, “duct 3”, is both half as deep and twice as wide as duct 1 giving an aspect ratio of 8: 1. Duct 3 was considered to be more or less an upper limit for aspect ratios of practical interest. FE and CR results were calculated for a comprehensive set of internal modes within the frequency ranges 100-1500 Hz (for duct 2) and 40-800 Hz (for duct 3). These values are plotted in Figures 13 and 14. The differences between the two sets of results are

406

R.

0

J.

ASTLEY

AND

A.

CUMMINGS

I I I I Subsom phase speed

too

i

Frequency

1

1

I

,

I

I

/I1118. Ik

(Hz)

Figure 13. Calculated transmission loss of duct 2 with 5 = 0.1. FE results: mode; --O--, (2,O) mode. CR results: -, for the above modes.

-20

/

I

----,

/

(0,O) mode; --C--,

(1,O)

1

500

100

Frequency

(Hz)

Figure 14. Calculated transmission loss of duct 3 with 5 = 0.1. FE results: --O--, (2,O) mode. CR results: -, for the above modes. mode; -• --,

(0,O) mode; --U-,

(I, 0)

more noticeable than in the case of the original duct but not dramatically so, the most pronounced discrepancies (for duct 3) being only of the order of 2 or 3 dB. Some typical wall displacement-pressure contour plots for ducts 2 and 3 are shown in Figure 15. Figure 15(a) shows a solution for duct 2 with a (1,O) internal mode at 800 Hz. Figures 15(b) and (c) are for duct 3 and show (2,O) and (0,O) solutions at 500 Hz and 125 Hz, respectively. The general feature which again emerges from all three plots is the regular nature of the radiation pattern in each case despite the rather large aspect ratios of the ducts and the complexity of the wall displacements, the radiation patterns being easily recognizable as dipole, quadrupole and monopole line sources, respectivelyt. Even for the ducts with large aspect ratios, therefore, the regular nature of the radiated field appears to explain to some degree the success of the CR approximation. One feature of the (0,O) TL curves for duct 2 (Figure 13) which deserves brief additional comment is the behaviour near 200 Hz. The coupled solution which is used to predict t At

first sight, the (2,0) mode in duct 3 might be expected to give a wall displacement pattern tantamount quadrupole, in view of the large aspect ratio of the duct, with four sources disposed along the greater dimension of the duct cross section. The radiation pattern would, however, appear more nearly characteristic of a lateral quadrupole. Clearly, scattering effects by the duct surface cause an equivalent geometric separation of the radiation from the central “phase cells” along each of the longer duct sides, thereby producing a kind of hybrid quadrupole. to a longitudinal

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(b)

(c) Figure 15. Calculated

structural response and pressure contours with 5= 0.1 (a) duct 2, (1,O) mode, f = 800 Hz; (b) duct 3, (2,0) mode, f = 500 Hz; (c) duct 3, (0,O) mode, f = 125 Hz. -, Pressure amplitude contours; --4--, displaced shape (real component); -a-, displaced shape (imaginary component).

the fundamental mode value of k ,,,” gives a subsonic axial phase speed for the narrow frequency band indicated in Figure 13. This precludes any radiation at all according to sense for this the FE model and an FE value of TL does not exist in any meaningful frequency band. The CR analysis, however, includes a finite length assumption in its initial calculation of TL for the fundamental mode (rather than modifying afterwards the result for an infinitely long duct) and hence gives finite TL values even for subsonic phase speeds. The result is the rather sharp peak in TL shown in Figure 13 in the region of subsonic phase speed, which is no doubt caused by a sudden decrease in C, (see the earlier expression for C, in terms of K).

5. CONCLUDING

REMARKS

The finite element model described in this paper has been shown to provide acceptably accurate quantitative predictions of acoustic transmission through the walls of rectangular ducts, both in local details of wall displacement and acoustic near field, and in overall behaviour in terms of the transmission loss, despite the “non-ideal” construction of the experimental ducts. An important result is that acoustical radiation damping has been shown to be negligible in comparison to structural damping, and it may, one feels, safely be omitted from future models for duct wall transmission loss. The accurate modelling of the acoustical radiation has provided useful insight into the interrelationship between the acoustic near field and far field. It has been shown that small scale wall displacement patterns are lost in the acoustic field away from the immediate vicinity of the duct walls, and that the far field radiation patterns may be generally expected to be quite simple.

K.

408

J. ASTLEY AND A. CUMMINGS

The “cylindrical radiator” model (reported elsewhere) has been shown to give surprisingly good results even for ducts of high cross sectional aspect ratio, by comparison to the FE model results. It appears to give a slight underestimate of the radiated power, and therefore slightly conservative transmission loss predictions. Calculated data (not shown here for want of space) on the duct wall response show that the CR theory and the FE model give virtually identical displacement patterns. One question which has not yet been answered is why there is a need for a large structural loss factor to limit the duct wall response near resonances. As yet, a physical explanation of the apparently large degree of damping has not been offered and further work on this might be enlightening.

REFERENCES 1. A. CUMMINGS 1978 Journal of Sound and Vibration 61, 327-345. Low frequency acoustic transmission through the walls of rectangular ducts. 2. A. CUMMINGS 1979 Journal of Sound and Vibration 63, 463-465. Low frequency sound transmission through the walls of rectangular ducts: further comments. 3. A CUMMINGS 1979 Journal of Sound and Vibration 67, 187-201. The effects of external lagging on low frequency sound transmission through the walls of rectangular ducts. 4. A. CUMMINGS 1980 Journal of Sound and Vibration 71, 201-226. Low frequency acoustic radiation from duct walls. 5. A. CUMMINGS 1981 Journal of Sound and Vibration 74, 351-380. Stiffness control of low frequency acoustic transmission through the walls of rectangular ducts. 6. A. CUMMINGS 1981 Journal of Sound and Vibration 78, 269-289. Design charts for low frequency acoustic transmission through the walls of rectangular ducts. 7. A. CUMMINGS 1981 American Society of Mechanical Engineers Paper Sl-WA/NCA-11. Higher order mode acoustic transmission through the walls of rectangular ducts. 8. A. CUMMINGS 1983 Journal ofSound and Vibration 90,193-209. Higher order mode acoustic transmission through the walls of rectangular ducts. 9. A. CUMMINGS 1983 Journal of Sound and Vibration 90, 211-227. Approximate asymptotic solutions for acoustic transmission through the walls of rectangular ducts. 10. D. J. NESKE, J. A. WOLF and L. J. HOWELL 1982 Journal of Sound and Vibration 80, 247-266. Structural-acoustic finite element analysis of the automobile passenger compartment: a review of current practice. 11. S. J. HOROWITZ, R. K. SIGMAN and B. T. ZINN 1981 American Institute of Aeronautics and AstronauticsPaper 81-1981. An iterative finite element-integral technique for predicting sound radiation from turbofan inlets. 12. Y. KAGAWA, T. YAMABUCHI, K. SUGIHARA and T. SHINDOU 1980 Journal of Sound and Vibration 69, 229-243. A finite element approach to a coupled structural acoustic radiation system with application to loudspeaker characteristic calculation. 13. R. J. ASTLEY and W. EVERSMAN 1983 Journal of Sound and Vibration 88, 47-64. Finite element formulations for acoustical radiation. 14. H. S. CHEN and C. C. MEI 1975 Proceedings of the Modelling Techniques Conference (Modelling 1975), Sun Francisco, 3-5 September 1975, Volume 1, pp. 63-81. Hybrid element method for water waves. 15. 1976 Low Pressure Duct Construction Standards. Sheet Metal and Air Conditioning Contractors National Association (U.S.A.).

APPENDIX

A: DERIVATION

OF THE VARIATIONAL

FUNCTIONAL

The relationship between the complex functional x of equation (12) and the preceding acoustical and structural equations (equations (5)-( 11)) may be established by the standard application of variational calculus. x is a complex functional of the complex functions q and pl. The function inside the integrand for x is a polynomial function of 77, p, and their derivatives and is consequently an analytic function of these variables. In establishing the conditions for stationary values of x two approaches may be adopted. The real and

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imaginary parts of x may be expressed as real functionals of the real and imaginary parts of n and p,. The variations of these two real functionals may then be calculated for arbitrary small increments of real functions representing the real and imaginary parts of n and pl. This process appears to involve two functionals each of four variable functions (i.e, r], vi, p,, and pr, where the subscripts r and i denote real and imaginary parts). Due to the analytic nature of the integrand, however, only one of these functionals need be considered, variations of the other then yielding redundant equations. An alternative, and much simpler, procedure is to retain the complex nature of the functional and its variable functions throughout the variational process. This approach yields the appropriate equations and boundary conditions in their full complex form, as presented in equations (5)-( 1 l), and involves a single complex functional of two complex, variable functions (r] and pr). If the real and imaginary parts of the resulting variational equations are separated and written as two real equations for each complex one the results are identical to those obtained from the variation of the real or imaginary parts of x as outlined in the preceding paragraph. The equivalence of the complex functional x to the appropriate field and boundary equations is now established by using the second of these approaches. The variation Sx of the functional x of equation (12) resulting from infinitesimal incremental functions 67 and 6p, over C1 and R, respectively, may be written as

=

&g[2D2nD’6n c-1

+ 2k2,,, 2DnD6r]

+ k4,, 276771 ds

(Al) all terms of order (Sr])‘, (6~~)’ or &$p, having been neglected. Integration by parts of the integral over C, and the use of the divergence theorem for the integral over R, then yields

sx =

{g[D47)--kZ,,D277+k4,,nl-W2477-tp"-p1)}~~ds Cl

+

7vP,.,~+~},,ds+~c~{$-z[~-~]}Sp~ds.

(A2)

If Sx = 0 for arbitrary choices of the incremental functions Sn and 6pl equation (A2) then yields the result that the terms inside the braces must be individually zero (unless 67 or 6p, is constrained at that point). The variation with 67 therefore yields the structural equation for n (equation (10)) directly from the first integral in expression (A2). Boundary conditions (lla) and (llb) arise from the second term in (A2). The variation of Sp, yields the Helmholtz equation (equation (5)) for pi-from the third term in equation (A2)-the kinematic boundary condition (equation (8)) on C,-from the fourth term of equation (A2)-and the requirement of matched pressure gradient at C, (equation (6b))-from the last term in equation (A2).