Calculation of perforated plate liner parameters from specified acoustic resistance and reactance

Journal o f Sound and Vibration (1975) 40(I), 119-137

CALCULATION OF PERFORATED PLATE LINER PARAMETERS FROM SPECIFIED ACOUSTIC RESISTANCE AND REACTANCE A. W. GuEss General Structures Corporation, 2141 l~Iiramar Drire, Balboa, California 92661, U.S.A. (Received 3 June 1974, and in revisedform 30 October 1974) A mathematical procedure is developed for calculating the parameters of a s{nglelayer perforated-plate acoustic liner in order to achieve a specified acoustic resistance and reactance for single-frequency excitation. The method includes the standard impedance terms, due to viscosity, radiation, and backing effects, and also the terms due to high sound amplitude and steady tangential airflow. Emphasis in the analysis is on these latter non-linear effects. Specification of acoustic resistance and reactance (at one frequency and set of environmental conditions) means that two quantities are used to determine the four liner parameters: fractional open area of plate (porosity of plate), thickness of plate, diameter of hole perforations, and depth of backing cells. The procedure has the following sequence of steps: (a) convenient values are assumed for plate thickness and hole diameter, (b) the resistance equation is solved for fractional open area, (c) the reactance equation is solved for backing depth. The solution indicates that higher values of sound amplitude and/or steady airflow necessitate larger values of fractional open area and backing depth for the fixed acoustic resistance and reactance. Various special cases give simplifications in the solution procedure and a numerical example is given. I. INTRODUCTION The computation of noise reduction in acoustically lined ducts often results in desired values for the acoustic resistance and reactance to be used for the material lining the walls of the duct. These desired values can be obtained from plots of sound power attenuation in the "wall-impedance plane" and depend on a number of factors, such as sound frequency, modes present, duct geometry and dimensions [1-3], and presence or absence of steady airflow through the duct [4]. Even the simpler problem of specification of resonance for the normal incidence absorption coefficient of a liner involves setting the acoustic reactance equal to zero (with this definition of resonance). Once it has been decided what acoustic resistance and reactance are to be used for a lining or surface, it then remains to design the lining material so as to achieve this specified impedance. A commonly used type of acoustic lining is perforated plate backed by a perpendicular, partitioned, layer of cells (often formed by honeycomb sheet) and an impervious wall. Such an arrangement represents an array of Helmholtz type resonators; and the relevant parameters are fractional open area, hole diameter (with circular holes), plate thickness, backing depth, and perhaps cell cross dimensions and cell wall thickness. The understanding of acoustic processes occurring within Helmholtz type resonators has increased greatly within the past several years, especially with regard to the non-linear effects resulting from high sound amplitude and steady airflow past the resonator orifices. There are a number of investigations into these effects, and formulas exist for the calculation of acoustic resistance and reactance resulting from the various physical processes, given the acoustic, airflow, and perforated plate parameters [5-7]. The inverse problem, important in the design of perforated 119

120

A.W. GUESS

plate liners, has been less well studied. The inverse problem consists of the calculation of the perforated plate parameters from a specified desired acoustic resistance and reactance, for a given acoustic and steady-airflow environment. It is the purpose of this paper to develop a mathematical procedure for the calculation of the parameters of a perforated plate liner, given a specified acoustic resistance and reactance for a specific frequency ofexcitation. Emphasis is on the region of high sound amplitude with steady airflow, and special cases should give simplifications to the formulas. Preliminary to formulating the procedure, expressions for the acoustic resistance and reactance for single frequency excitation of a cell-backed perforated plate are set down. The expressions treat each expected effect (high sound amplitude, stead~' airflow, viscous and mass effects, radiation impedance, and backing reactance), the regions of validity of various approximations are noted, and constants that appear are estimated from empirical data. A major limitation to the analysis is that it applies strictly to single-frequency excitation of the liner. This restriction is important with non-linear effects, when acoustic excitations at all frequencies will influence the response of the liner at a particular frequency. However, for a periodic acoustic excitation (with a Fourier series representation), usually the excitation at one frequency is dominant, and this could be used in the analysis, or a root-mean-square value of excitation ov'er all frequencies could be used. A second restriction on the analysis is that it applies to "single-degree-of-freedom" (single cell-layer) perforated-plate liners. It is probably possible to extend the formulas for calculation of acoustic resistance and reactance to "double-" or "multiple-degree-of-freedom" liners, with inclusion of non-linear effects (already done for linear effects alone, with a.c. electrical circuit analogs). However, theinversion of such formulas, to calculate the parameters and dimensions of a "multiple-degree-offreedom" liner, given specified values for the resistance and reactance at the multiple frequencies, is certain to be more difficult mathematically and iteration or numerical schemes would probably be necessary. 2. EQUATIONS FOR ACOUSTIC RESISTANCE AND REACTANCE In this section, expressions are given for the acoustic impedance of a perforated plate liner resulting from the various physical effects. The specific non-dimensional acoustic impedance z,, resistance 0,, and reactance Z, for the plate are defined locally by P z , = 0, + i x , = ~ , pCUn

(1)

where p is the acoustic pressure and un is the normal acoustic particle velocity at a position on the plate surface (z,, p, and u, complex), at frequencyf, and pc is the characteristic impedance Co = gas density, c = speed of sound). The single-frequency acoustic field is further defined by wavelength 2, = c,/f,, angular frequency co = 2:~f, acoustic pressure amplitude IPl, normal acoustic velocity amplitude lull, and angle of incidence ~ of the acoustic wave onto the plate surface.t The perforated plate liner is considered to consist of a plate of thickness t uniformly perforated with circular holes of diameter d so as to produce a fractional open area a of the total area of the plate (hence the number of holes per unit area of the plate is 4tT/nd"; the spacing, a, between centers of closest holes is x/-nd/2V'a for a square array, and a/nnd/~/2 x 3uza for a triangular array). The plate is backed with a regular partitioned single-layer cellular structure (such as metal honeycomb) with walls perpendicular to the perforated plate. The "["The impedance z. includes the effect of tangential flow of Mach number m . With continuity of particle displacement, z. is (l - M'tsin ~) times the impedance z of the liner alone, where Af~ = component of A[ in the incidence-reflection plane of sound wave. See reference [2].

PERFORATED PLATE LINER PARAMETERS

121

Acouslic irflow

Figure 1. Pictorial display of parameters of a single-layer perforated-plate acoustic liner: ~r = fractional open area, t = plate thickness, d = hole diameter, L = backing depth. F o r the steady tangential airflow: M ' = Mach number, d = boundary layer thickness.

cellular structure in turn is backed by a hard wall at a distance L behind, and parallel to, the perforated plate. See Figure 1 for a pictorial display of the parameters. The plate thickness, t, hole diameter, d, and spacing, a, between closest holes are all assumed small compared with the sound wavelength, q'hat is, t,d,a,~ 2.. (2) Hence the plate is effectively thin, and with uniform properties with respect to the incident sound. For the cellular backing structure, the walls of the partitions are assumed thin compared with the cross dimensions of the cells. Also, the cross dimensions ofthe cells are assumed small compared with the wavelength, 2, although this assumption is not necessary for acoustic waves of normal incidence, ~b = 0. The cross-sectional area, S, of a cell is chosen such that the mean number of entering holes per cell, 4aS[rid 2, is somewhat greater than unity so that at least one opening per cell exists. There is no restriction on the backing depth, L, compared with the wavelength, 2. The acoustic particle velocity u, normal to the perforated plate is related to the orifice gas velocity Uo, averaged over a hole diameter, by u. = truo

(3)

from continuity considerations (with use of incompressible approximation). The specific acoustic impedance Z for an individual orifice can be calculated and converted to the nondimensional specific acoustic impedance z for the perforated plate by use of equations (1) and (3), with possible additional corrections due to end effects and interaction effects. The total acoustic pressure p at the plate is the sum of the pressure Pi of the acoustic wave incident on the plate and the pressure Pr of the acoustic wave reflected from the plate: i.e., p = Pl + P,The incident and reflected wave pressures are related to the resistance and reactance of the plate locally by P' = 8 9

l/cosq~)+89

pCtln

and

Pr = ~ ( 0 , - l/cosq~)+89

(4)

pCU n

which follow from relations derived in reference [8], with use of equation (l). The amplitude of the orifice velocity (used later) can then be given in the forms

IPl

2lp, I

luol ape[O~,+x~,W2 crpc[(0, + l/cos$)Z+X,2] t/2' from equations (1), (3) and (4).

(5)

122

A.w.

GUESS

2.1. EXPRESSIONSFOR VISCOUSAND MASSIMPEDANCE The specific acoustic impedance Z~, of a single tube (with diameter d) of short length t (<~).) resulting from viscous and mass effects, is derived by Kinslcr and Frey [9] and is 4 J t(Kd[2) Z~ = -ipcot KdJo(Kd[2)

] 1 ,

K2 =

io9 v '

(6)

where v is the gas kinematic viscosity, and Jo and J1 are the zeroth and first order Bessel functions, respectively, of the first kind (of complex argument). The non-dimensional specific acoustic impedance z~. for the perforated plate of thickness t (i.e., an array of tubes) is obtained by multiplying equation (6) by 1/ape and accounting for end effects. The approximate (limiting) values for zv are

8V/ '8-v-~t ' (cot+ ~c~t') c--7

,

for

I~_1 d (colI/2 > 1 0 ,

(7)

and

z~,=z~,e=a--b-~+i-~-~ac,

for

=2kv]

< 1,

(8)

where z,.n and z,,e are, respectively, the Helmholtz and Poiseuille values for specific acoustic impedance. (Thereal part of equation (8) is the Poiseuille flow resistance.) Equation (7) results from the use of the asymptotic value, -i, for Jl/Jo in equation (6), and equation (8) results from the use of series expansions for Jo and Jl. In the intermediate range of 1 < [Kd/21 < 10, an approximation to z~, can be obtained by a suitable combination of equations (7) and (8), and addition ofthe real parts of these equations gives an approximation for the acoustic resistance good for all values of]K/2]. (For the imaginary parts ofequations (7) and (8), the term cot/ticshould be multiplied by a factor which approaches the values 1 and 4/3 for large and small values of IKd/2], respectively.) For perforated plate liners, usually the values of the parameters are such that equation (7) is valid. In equation (7) (and also in equation (8)), the quantity t' is a corrected length (thickness) to account for viscous effects just outside the entrances to the orifices. The end correction to the remaining term wt/ac (mass inertance) results from radiation effects (given later). The use of t' = t + d

(9)

gives good agreement with experiments, see reference [10], and includes the viscous effects at both ends ofthe orifices. A general discussion of the validity and experimental verification of equations (7) and (8) is given in reference [9]. For tubes of short length, it is not necessary to consider corrections resulting from heat conduction to the w~ills, see reference [10], which also presents the results of experiments validating the use of equation (7). 2.2.

EXPRESSIONS FOR RADIATION IMPEDANCE

An expression for the specific acoustic impedance of a single circular opening of diameter d in a flat surface (i.e., a flanged open-ended tube), due to radiation effects, is derived by Morse and Ingard [2] and is

PERFORATED P L A T E L I N E R P A R A M E T E R S

123

where y = 2nd]). = o~d[c, and x/2

Ml(y) = 4 f sin (~cos c0sin2 c~d~; o

z,~-pc

[.~(d) 2

zR~pc

i 4 cod] +-~-~--~-],

l + i n - ~2).d ]],

for

y
for y >), 19

(11)

02)

Equation (11) results from the use of series expansions for Jl and MI, and equation (12) results from the use of asymptotic values of Jx and M~. The condition for validity of equation (11), y < 1/2, is satisfied for virtually all parameter ranges of interest with perforated plate liners: i.e., 2 > 12.6d for values of 2 and d of interest (in agreement with expression (2)). In order to calculate the non-dimensional radiation specific acoustic impedance za for the perforated plate, the imaginary part of equation (11), representing "mass inertance just outside the tube", should be multiplied by two (to account for the two ends of the orifice) and the entire expression multiplied by l/apc. In addition, interaction effects between adjacent holes, along with airflow and sound amplitude effects, give further corrections to the reactance part (determined mainly by experiment). These multiplications and corrections can be incorporated with equation (11) to give

7~2[ dy z. = ~ 1

0)6 + i ac

(13)

for the specific radiation impedance of the perforated plate, where 6 is the orifice "end correction" and is

8 a(l-o.Tv 6=

)(l +5

10 Mo ,

1+ 10'Mg ]

8 =0.85.

(14)

The factor (I - 0 - 7 ~ ) accounts for orifice interactions (sum of interior and exterior effects, see reference [10]). The factor 1/(1 + 305M 3) accounts for steady airflow effects (given in reference [11] and see also data in reference [5]) with M = V/c being the steady airflow Mach number and V the airflow velocity tangentially along plate. The factor (I + 5 x 103Mo2)] (1 + 104Mo2) is given here as an attempt to account for sound amplitude effects, with Mo = luol/c being the orifice Mach number, and results from a fit to the data in reference [5]. The corrections due to steady flow and high sound amplitude should be further verified by experiment, and a dependence on the Reynolds number luol d/v and Strouhal number [uol[fd (for oscillatory flow through the orifices) might be expected. Corrections to the resistance part of equation (15) are usually not considered (the radiation resistance is often quite small for perforated plate liners). Non-linear effects on the acoustic reactance, which may be attributed to effects on the orifice end correction 6, are known to occur experimentally. Experimental results presented in reference [5] indicate that, as the sound pressure is increased, the end correction 6 approaches one-half of its low-amplitude value. This can be attributed to turbulent jet formation on the exit side of the orifice (for the oscillatory flow) which "washes out" the end

124

A.W. GUESS

correction (mass inertance) on that side [5, 12]. Also, experimental results presented in reference [5] indicate that as the steady airflow increases (M increases), the end correction approaches zero. If the steady flow effects are attributed to existence of a turbulent boundary layer [7], then the turbulence could remove the end correction on the boundary layer side, while air from velocity fluctuations passes through the orifice to form local turbulence on the opposite side (and remove the end correction on that side also). The factors involving Mo and M in the expression for 8, equation (14), are fits to experimental data, without theoretical derivation. When equation (13) is added to equation (7), the term is seen to be the mass end correction to the mass inertance term The end correction 8 results from the radiation reactance and may be comparable to t even though the radiation resistance may be negligibly small compared with the viscous resistance. References [9] and [10] present discussions of the comparison of experiments with theoretical formulas for radiation impedance (and also viscous and mass impedance), for Helmholtz resonators, without non-linear effects. For perforated plate liners, the discrepancies between theory and experiment are accounted for quite well by the corrective terms.

co3/ac

cot/ac.

2.3. EXPRESSXONSFOR NON-LIN.EARRESISTANCE Expressions for th~ non-linear resistance ofan orifice, resulting from high sound amplitude and steady airflow through the orifice, have been developed by Ingard and Ising [6] using both theory and experiment. For a perforated plate, the non-linear resistance due to high sound amplitude and steady flow tangential to the plate may be written, according to Ingard [7], as

~

]

g [u~ c+ [vl

(15)

where ]vIis an estimate ofthe magnitude of the turbulent velocity fluctuation due tO a turbulent boundary layer. The factor (I - a 2) can be important for larger values oftr. The [uol/c term, from the effect of high sound amplitude, is the result of turbulent dissipation due to jet formation on the exit side of the orifices (with the oscillatory acoustic motion) and comes from equation (5). (See also the papers by Zinn [13] and by Ingard and Labate [14].) The term, resulting from steady flow along the plate, is based on Ingard's concept that the velocity fluctuations in the boundary layer are at a characteristic frequency much less than the acoustic frequency and hence act like a steady flow through the orifices to enhance turbulent jet formation (with acoustic oscillations) and resulting turbulence dissipation [7]. Other mechanisms have been proposed for the non-linear resistance resulting from steady flow tangential to the plate [15, 16] (see also the review by Groeneweg [5]). The value of Ivl in equation (15) can be related to the steady flow Mach number M (along the plate) by means of turbulent boundary layer relations.-[" For the present, the empirical result that the acoustic resistance due to flow is approximately proportional to the steady airflow Mach number will be used. This means that the relation

Ivl/c

Ivl

"kM

(16)

C

is to be used in equation (15), where M is the steady airflow Mach number, and k is a constant determined from acoustic resistance measurements with flow (use of boundary layer theory should given an increase o f k with increase of boundary layer thickness A, and perhaps further t Either smooth-wall or rough-wall turbulent boundary layer theory may apply; however it is not intended to undertake here the application of these theories (this is under investigation).

PERFORATED PLATE LINER PARAMETERS

125

dependence on 3t). For perforated plates, data [17] indicate that k = 0.3, and this value has been used by Rice [18]. A review of experiments on non-linear resistance effects connected with the use of equations (15) and (16) is given in reference [5]. In particular, the data of Feder and Dean [17] indicate that the effects of high sound amplitude and steady grazing airflow act independently, are additive, and that the acoustic resistance due to grazing airflow is proportional to the Mach number and frequency independent. Recent experimental and theoretical work by Melling [19] demonstrates the frequency independence of the high amplitude sound term (luol/c term) in equation (15), and also indicates that the coefficient of this term should be modified to take account of dependence on the discharge coefficient (and hence dependence on Reynolds number in the orifice; see also reference [13]). The coefficient is still near unity; however, use of the r.m.s, value (rather than the amplitude value) of the orifice particle velocity in equation (15) would give better agreement with Melling's experimental results. 2.4.

EXPRESSIONS FOR BACKING IMPEDANCE

The backing impedance depends on the nature of the backing to the perforated plate. For the single-layer cellular, structure under consideration (with cell walls perpendicular to perforated face sheet and to hard wall backing, and with cell wall thickness,~ cell cross dimensions ,~ 2), the backing impedance is purely reactive, and the specific backing reactance is

Hence 7.n depends only on backing depth L and frequency f = co/2n. Special values and limits are as follows: 7.8 = 0 when toL/c is an odd integer multiple of n/2, and ZB "-+ 7-/-ooas coLic approaches an integer multiple ofn from above or below. The low frequency approximation is Xn "" -

,

for

(18)

,~ I.

2.5. COLLECTED IMPEDANCE EXPRESSIONS

The full specific acoustic impedance z for the cell-backed perforated plate is obtained from (19)

z = z v + zR + 0~,L + iT.n = 0 + ix,

where the individual terms result strictly from equations (6), (10), (15) and (17), with factors l[trpc for equations (6) and (I0) and with corrections for end effects. With the supposition that > I0 and

2n-=--< ). c

!/2,

(20)

equations (7) and (11) can be used for Zv and za in equation (19). The further use ofequations (15), (16) and (17) gives

O-X/g~

i+

+

+0

+kM

(21)

126

A. W. GUESS

and 7. = - - a c

+

ac

c ,) \1+~

-cot

(22)

as the expressions for specific acoustic resistance and reactance of the single-layer perforated plate liner, with I"ol from equation (5) and ~i from equation (14), and often ihe value k _~ 0.3 can be used. The importance ofthe various terms in equations (21) and (22) depend upon their magnitudes in comparison with those of the other terms. A specific comparison 0fmagnitudes can be done only for specific values of liner parameters and environmental (acoustic and grazing airflow) parameters. In general, it may be noted that the terms involving viscosity will be larger for liners with thick face sheets and small diameter orifices, while the radiation resistance term will be larger for large diameter orifices. The non-linear resistance terms, of course, will become important with high amplitude sound and grazing airflow. In equation (22), usually the term involving viscosity will be negligible compared with the first term (mass inertance term with end correction). Equation (21) is actually an implicit relation for 0 since luol is to be expressed in terms of 0 and Z by means of equation (5). It might seem that the inequalities of relations (20) could be contradictory since the first inequality holds for large values ofo9 and d, while the second inequality holds for small values of o9 and d. However, the usual situation with perforated plate liners is that both the inequalities ofexpressions (20) hold. The first inequality compares the orifice diameter dwith the viscous boundary layer thickness (2v/o9)'/2, while the second inequality compares the orifice diameter d with ihe acoustic wavelength 2 and is a basic assumption (see relations (2)). The Poiseuille fl0w resistance, 32v(t + d ) / a c d z from equations (8) and (9), can be added to equation (21) to give an expression for acoustic resistance good even when the first inequality in (20) does not hold. (In equation (22), the term ogt/ac would need a factor which goes to 4/3 when d < (2v/o9)'/2.) The Poiseuille flow resistance may be important at low frequencies since it is frequency independent, while the Helmholtz resistance becomes small at low frequencies. 2.6. ABSORPTION COEFFICIENT

The acoustic absorption coefficient c~for the liner may be derived as 40, cos q5 ct = (0. cos q~ + 1)2 + 7.2 cos 2 4,"

(23)

While equations (21) and (22) could be substituted into equation (23) to express ~ in terms of the liner and environmental parameters, the resulting expression would still implicitly involve 0. The derivative of~ with respect to o9 is d~ --

4(Z,z - O,2 + I / c o s ' ~b) 0,~ - 80,Z,7.~ =

=' =

do9

(24)

[(0, + l/cos 402 + Z~12cos 4~

and for ~ to be a maximum, ~' = 0 (along with ~" < 0). This requirement gives

(z,~-o,~+ 1/cos2~)o; =2o, z,z, (for-'=o).

(25)

as a necessary condition for c~to be a maximum. The values =l,a'=O

for

O,=l/cos~b

and Z , - - O = 7 .

(26)

PERFORATEDPLATELINERPARAMETERS

127

may be noted, so that the absorption coefficient is unity (and is a maximum) for these special values o f 0. and 7.. together. In general, the resonance condition, defined as 7.. = 0, will not satisfy equation (25) unless the left-hand side is zero. Besides the case noted in equation (26-), this will occur only i f 0 ' . = 0 for Z. = 0 (i.e., acoustic resistance independent of frequency at resonance). It can be shown that this condition ( 0 ' . = 0 for 7., = 0) will be satisfied by equation (21) for 0 ifthe viscous and radiation terms are neglected, leaving only the non-linear resistance terms. The solution o f equation (25) with the most general expression for 0 is expected to occur offresonance (i.e., with ;(, ~ 0). That is, maximum absorption does not in general occur at liner resonance. Ifthe backing reactance )~Bcan be approximated by equation (i 8), and the viscous reactance in equation (22) is neglected, then the liner reactance can be written as

Z=

co(t + ,~)

c

trc

toL"

(27)

The resonance condition will give the resonance frequencyfo or,

A•L o"

7.. = 0 = Z gives

too = 2nfo = c

(t + 6)

(28)

from equation (27). With use of equations (27) and (28), equation (23) can be written as

" =

40,/cos q5 ,too

where

Q = too(t + 3)(1 + ?,It sin ~)

(29)

ac(O, + 1/cos 4~)

cos

The quantity Q is a measure of the absorption bandwidth when an absorption maximum occurs at (or near) resonance. If the acoustic resistance is independent of frequency (0 independent of to) then Q is the"q value" defined by resonance frequency divided by difference o f frequencies for half resonance absorption. A small value of Q gives a broad frequency absorption band. From equation (29) with specified values of too, 0. and q~, smaller values of Q are achieved by smaller values of(t + 6)/a (see also reference [20]). These results indicating an increase in absorption bandwidth with a decrease in (t + (5) and/or an increase in a are still expected to apply when equation (22) for Z must be used in full generality and when 0 is not independent of to. 3. SOLUTION OF EQUATIONS When values of acoustic resistance and reactance have been specified for a set of environmental conditions (acoustic and airflow) and for a given frequency, equations (21) and (22) give two independent equations for the determination of the four liner parameters a, t, d a n d L. The solution for these four quantities is thus "underdetermined" and two of the quantities can be chosen arbitrarily. The most reasonable procedure is to assume convenient values for plate thickness t and hole diameter d according to manufacturing, structural strength, and weight requirements, and then derive values for fractional open area a and backing depth L. This procedure is developed here. The associated, more difficult, problem of determining values for tr, t, d, and L for specified values ofresistance and reactance at two separate sets of

128

A.W. GUESS

environmental conditions is not undertaken (a physically acceptable solution is not always possible). The problem can hence be approached as follows. The perforated plate liner is supposed to achieve acoustic resistance and reactance values 0 and Z, respectively, for frequency f = o~/2rr, in the presence o f environmental conditions specified by incident sound pressure amplitude IP, I, angle of incidence ~b of the acoustic wave onto the plate surface, and steady flow Mach number M. Plate thickness t and hole diameter d are constants determined by extrinsic considerations, perhaps in conjunction with frequency bandwidth considerations, and it is desired to solve for fractional open area 0- and backing depth L. Equation (21), not involving L, can be solved for 0-, and then this value of o" can be used in equation (22) to solve for L. Equation (21) with inclusion of the Poiseuille flow resistance, can be written as

A

0 = - - + ~

0-

+C

0-

],

(30)

where

lz2(d~2 ~ (

A=7@ + c B = a

t)

32V(1

l+-d

luol

t)

(31)

+d'

21p, I

C = 0-Mo = pC2[(O.-I- 1/COS ~b)2 -t- 7.,z]1/2'

(32)

and (33)

C = kM,

with use of equation (5). Equation (30) is an algebraic equation in a, and the physically acceptable root lies between zero and one. With multiplication by o z, equation (30) can be put into the standard cubic form, Ca 3 + (0 + B) 0-2

-

-

(A + C) 0- - B = 0,

0 < 0- < 1 for physically acceptable value, (34)

with A, B, C and 0 all equal to or greater than zero. Before studying equation (34) in full generality, some special cases are examined. 3.1. SOLUTION FOR SMALL O" First note that when 0- is small, say a < 0.I, then 0-z < 0.01 and (1 - crz) -~ 1 to within 1 ~o. With use of this approximation, equation (30) can be put into the form (35)

0 a l - (A + C ) a, - B = 0.

Here a, denotes "sigma small" and is specifically the physically acceptable solution of equation (35), A + C

a,=

20

[{A

"]- CI2

B] u2

+..'-z-;"-~. [ \ zO ] +o-'j

'

and

a=a,

for

as
(36)

Consequently, a reasonable approach to the general solution is as follows. Evaluate the quantities A, B, and C and then determine a~ from equation (36). If the value of as so determined is small, say a, < 0.1, then an excellent approximation to a has been obtained. However,

PERFORATED PLATE LINER PARAMETERS

129

if the value ofcr, determined from equation (36) is large, then for good accuracy, equation (34) should be solved. Evidently, from equation (36), an increase in B and/or C (hence, from equations (32) and (33), an increase in incident sound pressure amplitude IP~l and/or steady flow Mach number M) causes an increase in ors (with 0 kept constant). 3.2.

SOLUTION FOR ff x,VITll C EQUAL ZERO

The condition C = 0 corresponds to no tangential airflow: i.e., M = 0. Equation (34) then reduces to a quadratic equation in a, with the acceptable solution

[

A

.4"

B"

a=2(0+B---~-~+ [ ~ , ( 0 ~ ) i , ) 2 t - ~ ]

]'~

, for C=0.

(37)

When A is set equal to zero in equation (37), hence implying no viscous or radiation resistance, then the solution reduces to that for high sound amplitude alone, which has been studied by Ingard [10] for unit absorption at resonance. Equation (37) gives

a

[b--+--~j

with

=

l+(iP, l/pd)cos24 '

0=l/cos~b

and

for A = 0 (and C = 0),

(38)

Z=0,

with use of equations (32) and (26). Equation (38) is Ingard's result. 3.3.

SOLUTION FOR o" WITH B SMALL

When B is neglected in comparison with other terms in equation (34), the condition corresponds to very low inciden t sou nd pressure amplitude on the plate: i.e., Ip, I small, from equation (32). Equation (34) reduces again to a quadratic equation in ~r, with the acceptable solution

[(o)2(A,12o re

3.4.

+

for

B ~ 0.

(39)

EQUATION FOR (7 WITH ,4 EQUAL ZERO

When A is set equal to zero, the condition corresponds to the neglect of the linear acoustic resistance terms (viscosity and radiation resistance) in comparison with non-linear resistance terms. With high sound pressure levels and high steady airflow conditions (such as may exist in the ducts of turbofan engines), the assumption A _~0 is often valid. In this important special case, equation (34) is still a cubic equation in a. Rather than study this special case alone, the general cubic equation (34) will be examined. 3.5.

GENERAL SOLUTION FOR O"

In order to study the solution of equation (34), first set f(o)=Caa+(0+B)aU--(A+C)a-B;

A,B, C,O >~O.

(40)

The first and second derivatives off(a) with respect to ~r are f'(a)= 3Ca2+2(O+B)a-(A+C)

and f"(a)=6Ca+2(O+B).

(41)

130

A.w. GUESS

A plot o f f ( a ) versus a will give information concerning the roots ofequation (34), or.f (a) = 0, and attention need only be directed to the interval 0 ~
and f ( l ) = 0 -- A,

(42)

and a l s o f ( a ) --+ +oo as a -+ +co. From equations (41), the values o f f ' ( a ) at the ends of the interval are f ' ( 0 ) = - (A + C)

and f'(l.) = 2(B + C ) + 0 + (0 - A).

(43)

Also from equations (41), the roots o f f ' = 0 are

aM•

= -+[ \[(o + --~-]

+

+91j

3C

-k

/o 3+,q c }'

with f ' ( a M •

(44)

These roots locate a minimum off(g) at aM+ > 0 and a maximum at aM- < 0 (also,if(aM+) > 0 andf"(aM_) < 0). Equations (40) through (44) give enough information for a schematic plot o f f ( g ) in the interval 0 ~< a ~< 1, shown in Figure 2. At a = 0 b o t h f ( a ) and its slope are equal to or less than zero, from equations (42) and (43). Also, at a = 1 b o t h f ( a ) and its slope are positive i f 0 > A. T h u s f ( a ) has a minimum at a = aM+ and a single positive root at a = a+. The condition 0 > A is necessary for a+ to be in the physically acceptable range 0 ~< a+ ~< 1. I f 0 = A, then a+ = 1 is a root. Further, a+ is in the range aM+ < a+ < I (for 0 > A). tO

/t

0-,~ "\

-B

J

a

Figure 2. Schematic plot o f f ( a ) in interval 0 ~
Finally, the general formula for a+ can be written down. The theory of cubic algebraic equations develops formulas for the roots o f such equations, and the particular formulas to be used depend on relations between the coeff• of powers in the equation. For equation (34), with the restrictions on the coefficients as given, the general formula for the positive root is

:o §

a+ = 2r

where

- cos-'(~/~3)] - \

3C }'

:o+,y ~= \--Y6-]+ 89\-YU] I+-6

2c

(45)

(46)

and

(I A (3-U/+ ~ +~)"

~,2= (0+B/2

(47)

131

PERFORATED PLATE LINER PARAMETERS

3.6.

VARIATION OF a+

It is ofinterest to determine how a+ varies with the variation of the quantities A, B and C, with the specified 0 and 7. values held fixed. This will indicate how the fractional open area should be changed in order to account for changing acoustic and flow environmental conditions. For this purpose, the differential of equation (34) is taken. The differential is then solved for da and the result can be put into the form o"

dr7 = f - - - ~ d A + (l - a 2) dB + a(l - a 2) dC -

f'(a)

f'(a)

a2 dO,

(48)

f'(a)

wheref'(a) comes from equation (41), and with a = tr+. The differentials of A, B and C in terms of the environmental parameters result from equations (31)-(33) and one obtains

+llogdlZ]dt~ 2 \ cJJ2"--~"

(49)

dB = B dlp+l (variations in r omitted),

(50)

dM dC = C - M"

(51)

dA=

[__~_CV~( t ) 1+7

tp, t

and

From Figure 2, the derivativef'(a) is positive at a = tr+ and also 0 < a+ < 1, so that the coefficients ofdA, dB, and dC in equation (48) are always positive. In equations (49)--(51) the differentials do), dlp~l, and d M appear separately, and their coefficients are always positive. Consequently equations (48)-(51) show generally that increasing the frequency, sound pressure amplitude, or steady airflow velocity causes an increase in the required fractional open area to achieve the specified acoustic resistance and reactance (in agreement with the discussion following equation (36) for try). Equation (48) is the expression to be used in determining the variation of tr+ with variations in the environmental parameters. + 3.7.

APPROXIMATION FOR o'+

Equation (45), for a+, often results as the difference between two nearly equal quantities. For calculation purposes, it is thus advantageous to represent tr+ as an expansion which corrects a~, given by equation (36), to account for larger fractional open area. For this purpose set a+ = a+(l + e), with I*1 < 1, ('52) which assumes that the correction to tr~ is expected to be small. The substitution of equation (52) into equation (34) and the neglect of second order terms gives

Ca](l + 3e) + (0 + B) a~(l + 2e) - (A + C) a~(l + ~) - B = 0.

(53)

The solution for e is

as(B + Ca,) _ e~" (A + C ) - 2(O + B)t~,- 3CaZs

((as) a,f'(a,)'

(54)

132

A. %V.GUESS

with use of equation (35). Errors in the value of ~r+ as determined from the approximate formulas ofequations (52) and (54) are expected to be of the order ofa, x ~2. Another method by which an approximation to a+ might be obtained is to assume a series expansion in powers ofa,. However, the expansion ofequation (52) should give good accuracy since (r+ is expected to deviate from or, by only a small amount in the region of interest. 3.8. SOLUTIONFOR L Once equation (21) has been solved for a tthe root or+ ofequation (34) that lies between zero and one), this value can be used in equation (22) to solve forL, the backing depth. The assumed values oft and dand the specified values of;( and o9are used. Besides the exact solution for(r+ as given in equations (45)-(47), any of the various approximations for ~r+ represented by equations (36), (37), (39) or (52) may be used, if applicable. The solution of equation (22) for L gives c co(t + ~) L = --arccotco [ - g ~ c +

.?( ')] I +~

- 7.

(55)

,

where the end corr6ction 6 comes from equation (14), with 3/o from equation (32). If the approximation for backing reactance ZBrepresented by equation (18) can be used in equation (22), then the solution for L is

/l/r:

L -~- --

o ~ (-t-+ ~ ) + ~ g T - ~

ko9ll L

ac

c

(D) ] 1+

-z

,

for

- - , ~ 1.

(56)

c

If it is expected that L is small (L ,~ 2), then L can be calculated from equation (56), and one can then afterwards verify that the inequality is satisfied. The change in L resulting from changes in the environmental parameters can be obtained by taking the differential of equation (55), with 7. kept constant. This operation gives dL=sin 2

(t+6)~-i----

+ N//'~ ( I + ~.) [ dsinZ o --c-" ogL - t [~-

L+

cr sin2--c-j--~--

2-=~o]'do9

(57)

Usually the terms involving viscosity are negligible in equations (55)-(57). In equation (57), the differentials da and d8 depend on the environmental parameters, through equations (48) and (14). The effect of increasing the sound pressure amplitude and/or steady airflow velocity on the value of L can then be examined. As shown previously, these increases cause an increase in o (do > 0). Also, from equation (14), these increases cause a decrease in 6 (dr < 0) since Mo and M increase. In equation (57), the coefficients of da and d5 are positive and negative, respectively, so that dL > 0 (considering increases in sound pressure amplitude and steady airflow only) and L is increased. Equation (57) can be used generally to calculate changes in L due to variations in the environmental parameters. 3.9.

SAMPLE CALCULATION

For a sample calculation assume the acoustic environmental conditions f = 2 2 0 0 Hz,

lp, I =

10s dynes/cm2 (i.e., ~171 dB),

ff = 0 ,

(58)

133

PERFORATED PLATE LINER PARAMETERS

and the steady airflow environmental conditions M =

0.5 (with k = 0 . 3 1 ) .

(59)

These conditions, while somewhat extreme, may be typical near fan blades of a turbofan engine, where high amplitude and flow effects are important. Values of constants are p = 1 - 2 1 x 10-3gm/cm 3, c = 3 " 4 3 x I0*cm/s,

v=0.150cm2/s.

(60)

Typical values for perforated plate thickness t and hole diameter d are t=0-04in=0.1016cm,

d = 0 . 1 i n =0.254cm.

(61)

The use of these values for frequency and hole diameter gives ( - ~ ) 1 / 2 ( d ) = 3 8 . 6 > I0, and

2red=0-102<89

(2=15.6cm),

(62)

so that the inequalities (20) are satisfied, and equations (21) and (22) are applicable. In this example, unit absorption at resonance will be specified so that, by equation (26), the values 0,=0=1,

Z,=Z=0

(63)

are specified as the values of acoustic resistance and reactance to be achieved. The use of all the foregoing quantities in equations (31)-(33) gives A=~7-34x 10-3 , B = 0 " 0 7 0 2 = a M o ,

C=0-155.

(64)

Neither ofthe special cases of equations (37) or (39) apply, and substitution into equation (36) then gives as = 0.358 > 0-1. (65) Consequently, a, is not particularly small, and the general solution to equation (34) should be considered. The use of equations (64) and (65) in equations (54) and (52) yields -~-0"0678,

~+ = 0-334

(66)

as the approximate solution of equation (34), and further substitution into equations (45)(47) produces ~=13.17, ~p2=5.65; cr+=2.635-2"302=0.333 (67) for the exact solution. Hence, a, may be corrected to give better accuracy, and the approximate value for a+ is probably sufficient. The use of a+ in equations (64), (14) and (55) gives finally 34o = 0-211, ~ = 1.65 x 10-3 cm, L = 3.55 cm = 1.40 in (68) for the backing depth of the liner (and the liner approaches being a quarter wave resonator). A comparison of magnitudes of the viscous, radiation, and non-linear terms in the expressions for acoustic resistance and reactance can be made for this example, and will also provide

134

A.W. GUESS

a check on the calculation. Substitution of the numerical values into equation (21) gives 0 = 0-0158 + 0-00394 + [0-564 + 0-414] = 0-998.

(69)

The radiation resistance (second term) is small compared with the Helmholtz viscous resistance (first term) which in turn is small compared with the non-linear resistance terms due to sound pressure amplitude (third term) and steady airflow (fourth term). At lower sound pressure levels and lower steady airflow velocities (and with smaller hole diameters), the viscous resistance will be relatively more important. The Poiseuille flow resistance is 32v(t + d)[acd 2 = 2.32 x 10-3 and is small compared with the Helmholtz resistance in this example. Substitution of the numerical values into equation (22) gives ;( = 0"125 + 0"0158 - 0"141 = 0"00

(70)

and the reactance depending on viscosity (second term) is small compared with the mass inertance (first term). Finally, the effect of variations in the environmental parameters on the answers for a and L can be considered~ through use of equations (48) and (57). Suppose the values for f , Ip, I, and M in equations (58) and (59) were increased by 10Yo (omit variations in ~b): i.e., let d~o

to

_

dlP, I _ IP, I

dM

---0"I. M

(71)

The use of these values with the previous numerical values in equations (49)2(51), and (41) gives d A = 5 . 2 5 x l0 -4, dB=0.00702, dC=0.0155, f'(tr)=0.601. (72) Substitution of these quantities (and the value for a+) into equation (48) then yields da = 2.91 x 10-* + 0.0104 + 0.00764 = 0.0183,

(73)

and the first term (due to change in frequency) is negligible compared with the second and third terms (due to changes in sound Pressure amplitude and steady airflow, respectively). The result in this example is that the fractional open area is not especially sensitive to changes in the environmental parameters (i.e., da[a is about 6 ~o). The determination of the variation in L results from substitution of the numerical values into equation (57), with use ofd~r from equation (73),and equation (14). The result is that L is increased by less than I Yo due to the increases in IP, I and M (da and d~5 contributions) while L is decreased by about 11 Yo due to the increase in co (as might be surmised directly from equation (55)). The assumed environmental conditions for this sample calculation were chosen so as to make the non-linear effects dominant. It may be remarked that perforated face sheets of fractional open area as large as 20~o to 25 ~o have found some use in lining ducts of aircraft turbofan engines, and also as acoustic and shock absorbing liners in wind tunnels. 4. DISCOSSION AND COMMENTS The procedure developed here for calculation of the parameters of a single-layer perforated plate liner in order to achieve a specified acoustic resistance and reactance can be summarized as follows. The total environment under which the liner is supposed to operate is assumed

PERFORATED PLATE LINER PARAMETERS

135

known, and results from the acoustic environment (given by sound frequency, incident sound pressure amplitude, and angle of incidence of acoustic wave), plus the flow environment (given by steady airflow Mach number and boundary layer thickness). The two independent equations for resistance and reactance involve all of the four liner parameters, and convenient values are chosen for perforated plate thickness t and hole diameter d. The equation for acoustic resistance, which does not involve backing depth, is solved for the required fractional open area tr. If neither of the special cases represented by equations (37) or (39) applies, then equation (36)is solved for cq. Iftr~ is not small, then the fractional open area may be obtained from equation (52) for better accuracy (or even from the exact solution of equation (45)). This derived fractional open area is then used in the' equation for acoustic reactance to derive the required backing depth L, from equation (55), or equation (56) if applicable. The entire procedure is for a single acoustic frequency only, since it is for this frequency that the acoustic resistance and reactance are specified. For example, in the propagation of noise through ducts of turbofan engines, the frequency under consideration for which the acoustic resistance and reactance are specified might be the fan blade or turbine blade passing frequency (for acoustic liners in the fan ducts or turbine exhaust duct, respectively). As is known, t h e attenuation of sound propagating through acoustically lined ducts depends not only upon the absorption characteristics of the liner but also upon how the geometry of the duct, with liner, affects the overall propagation of sound through the duct, and consequent attenuation and absorption of sound energy. If the acoustic resistance and reactance are chosen so as to produce an attenuation maximum, the question then arises as to what frequency bandwidth may be expected. The discussion of frequency bandwidth following equation (29) was specifically for absorption. It was found that small values of (t + ~5) and larger values of tr produced a broader absorption bandwidth. By equation (14), t5 is proportional to d. Consequently smaller values of plate thickness t and/or hole diameter d will produce a broader absorption frequency bandwidth. While it is not proper to carry over directly these results for absorption bandwidth (a property of the liner only) to attenuation bandwidth (a property of duct geometry combined with liner), it may be surmised that the results will be similar: both may depend primarily on the change of reactance with frequency, and equations (22) or (27) indicate that the effects will be similar. These considerations imply the use of smaller values of t and d to achieve wider bandwidth (however, note that smaller values of t and d may change the value of A given by equation (31) and hence may change the derived value of tr). While equations (21) and (22) include the main and essential contributions to acoustic resistance and reactance, respectively, a number of additions and corrections have been proposed by various authors. In particular, it has been proposed [18] that the non-linear resistance term in equation (21) depending on high sound amplitude (namely the luol/c term) should be multiplied by the factor exp [-(d/XM) 2] where XM is the maximum net acoustic fluid displacement in the orifice. This factor approaches zero or unity as d/XMbecomes large or small, respectively, and represents physically the concept that the acoustic fluid displacement must be large compared with the orifice diameter in order to have turbulent jet formation. The factor could be included in equation (21) and in the solution procedure for a. Also, from the same source, data indicate that the constant k in equation (21) should be a weak function of or. This correction is not so easily included in the solution procedure for or, but is expected to have little effect on the accuracy. All in all, the additions and corrections can either be accounted for in the solution procedure for tr, or should not significantly affect the accuracy. In addition, see the recent paper by Melling [19] where a comprehensive discussion is given of sound pressure level effects on acoustic impedance (including both linear and non-linear regimes). The analysis presented in this paper strictly applies only to single-frequency sinusoidal

136

A.W. GUESS

excitation ofthe perforated plate liner. It is known that the acousticimpedance of a perforated plate liner at one frequency is influenced by the acoustic excitation at other frequencies. This is especially pronounced at high overall sound pressure levels when the acoustic resistance goes. over into the non-linear regime. Hence the acoustic impedance of a perforated plate liner, for any given frequency, depends on the noise frequency spectrum itself, and this fact is not taken into account in the analysis. However, with a periodic acoustic excitation at a precise repetition rate, so that the excitation may be represented as a Fourier series expansion in time, this consideration may not be too important. For example, consider an incident acoustic wave with a sawtooth pressure trace, which can be considered as the ultimate form of a periodic high amplitude acoustic signal due to distortion by non-linear effects. The Fourier expansion of a sawtooth pressure trace gives the amplitude of the second harmonic as one-half the amplitude of the fundamental (first harmonic). Hence the second harmonic of sound pressure is 6 dB down compared with the first harmonic. Experimental work (see reference [5]) has shown that the second harmonic affects the acoustic resistance of the liner to the fundamental only when the sound pressure level o f the second harmonic is equal to or greater than that of the first harmonic. Consequently, with the situation of a highly repetitive sound pressure trace, the fundamental is dominant, and a single frequency analysis for acoustic impedance should be satisfactory (this is expected to be the situation in the very near field of fan or compressor blades). For broadband, high-amplitude noise, composed of a continuous spectrum of sound frequencies, certainly the single frequency impedance model should be supplemented by taking into account the effect of pressure levels at other frequencies (see reference [18]). In this situation, the root-mean-square value of pressure (over all frequencies) mightbe used in the acoustic impedance expressions for the non-linear regime. As mentioned at the beginning of section 3, an associated problem to the one considered in this paper is the problem of solving for all four of the liner parameters, a, t, d a n d L, given specified values o f acoustic resistance and reactance at two separate sets of environmental conditions. The four values of resistance and reactance then completely determine the four liner parameters, with no leeway for choice of t and d. This solution procedure has been carried out, and a unique solution results if there is sufficient difference between the two sets of environmental conditions. However, (i) a physically acceptable solution does not always result (e.g., tr, t, d or L may be negative), and (ii) even if a physically acceptable solution results, the answers may be so unreasonable as to be impractical in any application.

ACKNOWLEDGMENT Most of this work was done at Douglas Aircraft Company and supported by the Douglas Independent Research and Development Program. The author wishes to acknowledge helpful discussions with Dr G. M. Schindler. REFERENCES 1. L. CREMER1953 Acustica 3, 249-263. Theoretical study of propagation of sound between infinite parallel walls and the resulting wall impedance for maximum noise reduction (in German). 2. P. M. MORSEand K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw Hill Book Company, Inc. See pp. 383-385 and 705-715. 3. E. J. RICE 1969 in Aerodynamic Noise, Proceedings of AFOSR-UTIAS Symposhtm, Toronto, 20-21 May 1968 (H. S. Ribner, editor) 229-249. Attenuation of sound in soft-walled circular ducts. University of Toronto Press. (See also NASA TMS-52442, 1968). 4. E. J. RICE 1969 in Basic Aerodynamic Noise Research, NASA SP-207 (I. R. Schwartz, editor), 345-355. Propagation ofwaves in an acoustically lined duct with a mean flow. Washington, D.C.: U.S. Government Printing.Office.~

PERFORATED PLATE LINER PARAMETERS

137

5. J. F. GROENEWEG 1969 in Basic Aerodynamic Noise Research, NASA SP-207 (I. R. Schwartz, editor), 357-368. Current understanding of Helmholtz resonator arrays as duct boundary conditions. Washington, D.C.: U.S. Government Printing Office. 6. U. INWARDand H. lslt~o 1967 Journal of the Acoustical Society of America 42, 6-17. Acoustic non-linearity of an orifice. 7. U. INGARD 1968 Journal of the Acoustical Society of America 44, 1155-1156. Absorption characteristics of non-linear acoustic resonators. 8. R. W. B. STEP,tENS and A. E. BA'rE 1966 Acoustics and Vibrational Physics. New York: St. Martin's Press. See pp. 660-661. 9. L. E. KINSl.ER and A. R. FREY 1950 Fundamentals of Acoustics. New York: John Wiley and Sons. See sections 8.1-8.4 and 9.3. 10. U. INGARD 1953 Jotlrnal of the Acoustical Society of America 25, 1037-1062. On the theory and design of acoustic resonators. 11. E. J. RXCE, C. E. FEtLER and L. W. ACKER 1971 NASA TN D-6178. Acoustic and aerodynamic performance of a 6-foot-diameter fan for turbofan engines, III--performance with noise suppressors. 12. P. J. WESTERVELT 1951 Journal of the Acoustical Society o f America 23, 347-348. Acoustical impedance in terms of energy functions. 13. B. T. ZINN 1970 Journal of Sound a?td Vibration 13, 347-356. A theoretical study of non-linear damping by Helmholtz resonators. 14. U. INGARDand S. LABATE1950Journalofthe AcousticalSocietyofAmerica 22, 211-218. Acoustic circulation effects and the nonlinear impedance of orifices. 15. W. A. SIRI~NANO and T. S. TosoN 1968 NASA CR-72426. Non-linear aspects of combustion instability in liquid propellant rocket motors. 16. E. J. RICE 1973 American blstitute of Aeronautics attd Astronautics Paper No. 73-995. A model for the pressure excitation spectrum and acoustic impedance of sound absorbers in the presence o f grazing flow. 17. E. FEDER and L. W. DEAN 1969 NASA CR-1373. Analytical and experimental studies for predicting noise attenuation in acoustically treated ducts for turbofan engines. 18. E. J. RICE 1971 NASA TAt X-67950. A model for the acoustic impedance of a perforated plate with multiple frequency excitation. 19. T. H. MELLINO 1973 Jottrnal ofSotmd a?td Vibration 29, 1-65. The acoustic impedance of perforates at medium and high sound pressure levels. 20. A. W. BLACKMAN 1960 American Rocket Society Journal 30, 1022-1028. Effect of non-linear losses on the design of absorbers for combustion instabilities.