A Finite Element Propagation Model For Extracting Normal Incidence Impedance In Nonprogressive Acoustic Wave Fields

Transcription

1 NASA Technical Meorandu A Finite Eleent Propagation Model For Extracting Noral Incidence Ipedance In Nonprogressive Acoustic Wave Fields Willie R. Watson Langley Research Center, Hapton, Virginia Michael G. Jones Lockheed Engineering & Sciences Copany Sharon E. Tanner and Tony L. Parrott Langley Research Center, Hapton, Virginia April 1995 National Aeronautics and Space Adinistration Langley Research Center Hapton, Virginia

2 A Finite Eleent Propagation Model For Extracting Noral Incidence Ipedance In Nonprogressive Acoustic Wave Fields Willie R. Watson, Michael G. Jones z, Sharon E. Tanner, Tony L. Parrott NASA Langley Research Center, Hapton, Virginia z LOCKHEED Engineering & Sciences Copany, Hapton Virginia Abstract A propagation odel ethod for extracting the noral incidence ipedance of an acoustic aterial installed as a nite length segent in a wall of a duct carrying a nonprogressive wave eld is presented. The ethod recasts the deterination of the unknown ipedance as the iniization of the noralized wall pressure error function. A nite eleent propagation odel is cobined with a coarse/ne grid ipedance plane search technique to extract the ipedance of the aterial. Results are presented for three dierent aterials for which the ipedance is known. For each aterial, the input data required for the prediction schee was coputed fro odal theory and then containated by rando error. The nite eleent ethod reproduces the known ipedance of each aterial alost exactly for rando errors typical of those found is any easureent environents. Thus, the ethod developed here provides a eans for deterining the ipedance of aterials in a nonprogressive wave environent such as that usually encountered in a coercial aircraft engine and ost laboratory settings. 1 Introduction The design of increasingly eective and ecient duct treatents for acoustic noise suppression continues to be a critical consideration in the design of environentally acceptable aircraft propulsion systes. To achieve the full potential of duct treatents in future aircraft engines, it will be necessary to aintain the target ipedances of acoustic treatents near their optiu values. A continuing easureent proble in treatent technology is the accurate deterination of noral incidence ipedance of acoustic aterial in grazing ow environents. Methods for deterining the noral incidence ipedance in this environent fall into three categories, \T-tube" ethod (ref. [1]), in-situ ethod (refs. [2, 3]), and the propagation odel ethod. The \T-tube" and \in-situ" ethods have several drawbacks that are discussed at length in ref. [4]. These two easureent ethods do, however, serve as useful copleents to the \propagation odel " ethod, which is the subject of this paper. Propagation odel ethods for evaluating the acoustic ipedance of a aterial are popular because of their convenience. The conventional ethod involves easuring the sound attenuation properties in a waveguide lined with the acoustic aterial over a sucient length to be eectively innite. This data is then used with the solution to the wave equation in an innite waveguide to establish the ipedance of the aterial. The evolution of waveguide 1

3 odels for this purpose began over 20 years ago with a unifor ean ow odel(ref. [4]). For this case, an analytical expression for the ipedance of the aterial was derived using known transcendental functions and the easured axial wavenuber. Validation of the odel in zero ow (i.e., grazing incidence sound only) was achieved by deonstrating that it reproduced the easured noral incidence ipedance of a test panel. Waveguide ethods were later extended to rectangular ducts with shearing ean ows in one cross sectional direction (refs. [4, 5]). The ethod presented in ref. [5] was extended to include ean ow shear in two cross sectional directions in ref. [6]. Both the one and two-diensional sheared ow odels developed in refs. [5] and [6] were validated with easured data in ref. [7]. Innite waveguide odels are applicable, in a very straightforward anner, to situations for which a single progressive ode propagates within the waveguide containing the unknown aterial. However, any conventional liner concepts generate ore coplex acoustic elds. Thus, easured data ust now be interpreted as the superposition of any propagating odes (i.e., ulti-odal eects generated by installation of the test specien and anufacturing tolerances). Broadband liners currently under study contain variable ipedance properties and produce ultiple odes in the waveguide. The current research eort was otivated by the shortcoings of the current ethodology for deterining the noral incidence ipedance in these ore realistic situations. The ethod developed here uses a propagation odel based upon a nite eleent technique for deterining noral incidence ipedance fro easured wall pressure data. This allows a deterination of the ipedance of aterials in nonprogressive acoustic wave elds containated with ulti-odal eects and reections. Although the analysis of this paper assues a two-diensional duct without ean ow, it ay be extended to three diensions and to ean ows with shear. The reainder of this paper is organized into seven sections. The following section 2 describes the physical proble and coordinate syste used in the study. Section 3 presents the governing equation and boundary conditions that are solved to obtained the unknown ipedance of the acoustic aterial. Section 4 describes the propagation odel (i.e., a linear nite eleent ethod). Measured data was not available as input to the odel. Therefore, ulti-odal analysis was used to siulate the necessary input. This is discussed in section 5. The unknown ipedance of the aterial is obtained by iniizing the dierence between the known and nuerically coputed wall pressure. The iniization is achieved by a coarse/ne grid search technique in the coplex ipedance plane. This is the subject of section 6. Results of ipedance predictions for known aterials are presented for wave elds containing nonprogressive waves in section 7. Conclusions relevant to this paper are presented in section 8. 2 Description of the Physical Proble Figure 1 shows a scheatic of the two-diensional duct used in this study. The aplitudes of right and left oving acoustic waves decay as shown scheatically in the gure. The axial and transverse directions are denoted by x and y, respectively. The duct is L units long with the source and exit planes located at x = 0 and x = L, respectively. Inputs at the source and exit planes are the source pressure, p s (y), and the noralized exit ipedance, exit (y), respectively. Throughout this work all ipedances are noralized with respect to 2

4 the characteristic ipedance of the ediu in the duct. The upper wall of the duct is rigid. There are points located at x = x 1 ; x 2 ; x 3... x along the upper wall, at which the acoustic pressures are known. The sound absorbing aterial is assued to be a perforate over honeycob and constitutes the botto wall. This aterial is L units long and is assued to be point (locally) reacting (i.e. acoustic waves propagate through it noral to the faceplate). The sound absorbing aterial has an unknown noralized ipedance (x), as shown. The proble at hand is to deterine the ipedance of the aterial fro the known data. It should be noted, as suggested by gure 1, that the ath odel discussed here is liited to a 2-D description which approxiates a three-diensional ow ipedance tube. Such ow ipedance tube apparatuses can be used to obtain the unknown noral incidence boundary condition fro a knowledge of the source pressure, p s (y), exit ipedance, exit (y), and upper wall pressures. This ethod of easureent has been traditionally called the \waveguide ethod". It should be noted that this paper will use analytically based input data to deterine the noral incidence ipedance, since easured data was not available. 3 Governing Equation and Boundary Conditions Steady-state acoustic pressure waves, propagating within the duct shown in gure 1 satisfy the Helholtz 2 p(x; 2 p(x; 2 + k 2 p(x; y) = 0 (1) where k is the free space wavenuber, k = 2f, f is the frequency in Hertz, and c is the c sound speed in the duct. Before a solution to the acoustic eld can be obtained and the unknown ipedance extracted, boundary conditions ust be prescribed. Along the source plane of the duct, x = 0, the acoustic pressure is known p(0; y) = p s (y) (2) The boundary condition along the rigid upper wall is equivalent to the requireent that the gradient of acoustic pressure noral to the wall = 0 (3) At the duct terination, x = L, the ratio of acoustic pressure to the axial velocity ust equal the known exit ipedance, exit = ikp(l; y) exit (y) Finally, the lower wall boundary is assued locally reacting, so = ikp(x; 0) (x) When the ipedance (x) is known, equations (1)-(5) constitute a well posed boundary value proble that can be solved to deterine the sound eld within the duct. Exact solutions to this proble are not available for a general set of input data; therefore, a coputational ethod is required to obtain the solution to equations (1)-(5). 3 (4) (5)

5 4 Duct Propagation Model The coputational ethod chosen to solve equation (1), coupled with the boundary condition equations ((2)-(5)), is a Galerkin nite-eleent ethod. Details on the ethod are given in several texts (refs. [8, 9]), and only sucient detail is presented here for continuity. When applied to the current acoustic proble, the nite-eleent ethod ay be interpreted as an approxiation of the continuous acoustic eld as an asseblage of rectangular eleents as illustrated in gure 2. Here it is assued that there are N nodes in the axial and M nodes in the transverse directions of the duct. A typical rectangular eleent, [I; J], is shown in gure 3. Each eleent consists of four local node nubers labeled 1, 2, 3 and 4, respectively. Each eleent is considered to have width a = (x I+1 x I ) and height b = (y J+1 y J ) as shown. The objective of the ethod is to obtain the unknown acoustic pressure at the nodes of each of the (M 1)(N 1) eleents. Galerkin's nite eleent ethod is eployed to iniize the eld error. It should be noted that the eld error is distinct fro the wall error function, which is used later to extract the unknown ipedance. Dene the eld error function as E(x; y) p(x; 2 p(x; 2 + k 2 p(x; y) (6) Within each eleent p(x; y) is represented as linear cobination of four functions, N 1 ; N 2 ; N 3 and N 4 which coprise a coplete set of basis functions p(x; y) = N 1 (x; y)p 1 + N 2 (x; y)p 2 + N 3 (x; y)p 3 + N 4 (x; y)p 4 (7) (x x N 1 (x; y) = [1 I ) (y y ][1 J ) ]; N a b 2 (x; y) = [ (x x I) (y y ][1 J ) ] a b N 3 (x; y) = [ (x x I)(y y J ) (x x ]; N ab 4 (x; y) = [1 I ) (8) ][ y y J ] a b in which p is the values of p(x; y) at local node. The variable ipedances exit (y) and (x) are represented in a siilar anner along each boundary eleent " # 1 (y yj ) y yj exit (y) = exit (y J ) exit (y J+1 ) b b (x) = [ 1 (x x I) a ](x I ) + (x x I) (x I+1 ) (9) a In an ideal sense, the solution to the sound eld is obtained when the eld error, E(x; y), is identically zero at each point of the doain. This is approxiately achieved by requiring that the eld error function be orthogonal to each basis function N (x; y). Contributions to the iniization of the eld error function fro a typical eleent are Z xi+1 Z yj+1 Z xi+1 Z yj+1 " p [I;J] EN I dydx = p [I;J] + k 2 p [I;J] N I dydx (10) y 2 R xi+1 x I x I x I y J The second derivative ters in equation (10) are integrated by parts in order that the linear basis functions can be used R yj+1 h y J EN I dydx I R xi+1 x I + R y J+1 y J + R x I+1 x I R yj+1 y I (L; y) (x;h) I (x; k 2 p [I;J] N I i dydx N I (0; y)]dy N I (x; 0)]dx (11)

6 Substituting the wall and exit boundary condition into the line integrals in (11) gives R xi+1 x I R yj+1 y J EN I dydx = R xi+1 x I + R y J+1 y J ik R x I+1 x I R yj+1 y J [ ikp[i;j] (L;y) exit (y) [ p[i;j] (x;0) I i k 2 p [I;J] I dydx N I (L; y) (0;y) I (0; y)]dy N I (x; 0)]dx (12) where the line integrals in equation (12) are evaluated only for eleents which lie along the boundary of the duct. The contribution to the iniization of the eld error for each eleent is expressed in atrix for as Z xi+1 Z yj+1 x I y J EN I dydx = [A [I;J] ]f [I;J] g (13) where [A [I;J] ] is a 4x4 coplex atrix for each eleent [I; J], and f [I;J] g is a 4x1 colun vector containing the unknown acoustic pressure at the four nodes of the eleent. The coecients in the local stiness atrix, [A [I;J] ], were coputed in closed for. Assebly of the global equations for the coputational doain is a basic procedure in the nite eleent ethod. Appropriate shifting of rows and coluns is all that is required to add the local eleent atrix, [A [I;J] ], directly into the global atrix, [A]. Assebling the eleents for the entire doain results in a atrix equation of the for: [A]fg = ff g (14) where [A] is a coplex atrix whose order is MN, and fg and ff g are MNx1 colun vectors. The vector fg contains the nodal values of the unknown acoustic pressure and ff g is the zero vector. It is necessary to apply the source pressure condition to this syste of equations before a solution can be obtained. Satisfying the noise source boundary condition consists siply of setting all nodal values of acoustic pressure at the source plane (x = 0) to the known value of source pressure, p s (y). Thus inserting these conditions into the assebled global atrix equation (14), introduces nonzero eleents into the rst 2M coponents of ff g. Further details on iposing source conditions are described elsewhere (refs. [8, 9]). The global atrix [A] generated by Galerkin's Method following application of the source conditions is a coplex atrix. Fortunately, owing to the discretization schee used, it will also be block tridiagonal. The structure of atrix [A] prior to iposing boundary conditions is shown in gure 4, where the superscript T denotes atrix transpose. Note that [A] is a square syetric block tridiagonal atrix whose order is MN. This global atrix contains a nuber of ajor blocks (A I ; B I ) which are theselves square and tridiagonal as shown in the gure. The diagonal ajor blocks, A I are also syetric. Much practical iportance arises fro this syetric structure as it is convenient for iniizing storage and axiizing coputational eciency. Special atrix techniques exist for a solution of this structure following application of source conditions 1. All coputation and storage is perfored only on the lower triangular portion of the atrix [A]. 1 Gaussian eliination with partial pivoting and equivalent row innity nor scaling is used to reduce the rectangular syste to upper triangular for. Back substitution is then eployed to obtain the solution for the acoustic pressure at the NM node points 5

7 5 Data Input to Duct Propagation Model Three sets of boundary data are required in addition to the rigid upper wall condition, in order for the duct propagation odel to uniquely deterine the upper wall pressure. The foregoing equations ake use of this unique relationship between the upper wall pressures, p(x I ; H), and the following three sets of data 1. The source plane pressure, p s (y) 2. The exit plane ipedance, exit (y) 3. The lower wall ipedance function, (x) If any two and the upper wall pressures are known, the reaining can be deterined. Here we are seeking the unknown ipedance function of the lower wall, (x). It will be deterined by specifying the upper wall pressures, the source pressure and exit ipedance. Experiental data were not available for input to the nite eleent duct propagation odel. Thus, in this eort, we assue a unifor liner ipedance of the botto wall, and use ulti-odal analysis to deterine the upper wall pressures, the source pressure, and exit ipedance. To begin, the conventional odal solution in the duct of gure 1, for a constant ipedance at the lower wall, is p(x; y) = X nodes n=1 [A n e iknx + B n e iknx ]p n (y) (15) p n (y) = cos( n y) + tan( n H) sin( n y) (16) = k 2 n = k 2 2 n (17) ikh n H tan( n H) Here, nodes is the nuber of odes, A n and B n are the chosen ode aplitude coecients of the right and left oving acoustic waves in the duct, respectively, and the eigenvalues, n, are obtained by solving the transcendental equation (18). The source pressure and exit ipedance used as input data here, are obtained by substituting the series in equation (15) (i.e., with chosen values of A n ; B n and nodes) into equations (2) and (4), respectively, to obtain p s (y) = X nodes n=1 (18) [A n + B n ]p n (y) (19) exit (y) = k P nodes n=1 [A n e iknl + B n e iknl ]p n (y) P nodes (20) n=1 k n [B n e iknl A n e iknl ]p n (y) In order to obtain the known upper wall pressure that is required to extract the unknown ipedance, the series in (15) is evaluated at the axial locations along the upper wall p(x I ; H) = X nodes n=1 [A n e iknx I + B ne iknx I ]p n(h) (21) 6

8 6 Extraction of the Unknown Ipedance The goal of the ipedance extraction ethod described in this work is to deterine the unknown ipedance, (x), of an acoustic aterial fro the data input. The procedure is to nuerically deterine the ipedance function (x), such that the pressure along the top wall reaches its known value at each of the points. The procedure consists of repeatedly cycling through the solution to the boundary value proble (equations (1)-(5)), and obtaining a set of upper wall pressures for each ipedance function. As each new set of wall pressures is coputed, it is copared to the known values until convergence is achieved. Convergence of the procedure is guaranteed, since the boundary value proble is well-posed. The idea is best illustrated by considering a constant ipedance,. We dene the unknown ipedance as = + i (22) where is the resistance and the reactance. Resistance values are positive whereas reactance values span the real axis 0 1; 1 1 (23) It should be apparent that searching the entire upper half plane of the resistance/reactance space for the unknown ipedance is ipractical. Thus, we introduce the tranforation = cot(kd); 0 kd (24) and search for the unknown ipedance in the (; kd) plane, where is liited to 0 ax. Rules for selecting ax will be discussed later. It should be noted that equations (22) and (24) represent the ipedance odel for any perforates over honeycob used in current aircraft engines and the paraeter d is the backing depth of the perforate (ref. [10]). We now divide the coplex plane (; kd) into IMAX evenly spaced intervals in the direction and JMAX evenly spaced points in the kd direction, as shown in gure 5. The increent spacing and kd are = ax IMAX 1 ; kd = JMAX 1 Thus a point IJ in the unifor ipedance grid is (25) IJ = I + i J ; I = (I 1); J = cot (J 1)kd (26) We will establish a rule of thub for deterining IMAX, and JMAX in the next section. We now dene the global noralized wall error function at a point (I; J) in the ipedance plane. Let denote the ipedance of the unknown aterial. If the known upper wall pressures corresponding to are p(x n ; H) and those coputed fro the nite eleent solution with IJ are p(x n ; H), then a easure of the closeness of IJ to is given by the noralized wall error function, EW ( IJ ) EW ( IJ ) = EW ( IJ) E ax (27) 7

9 EW ( IJ ) = 1 X n=1 j p(x n ; H) p(x n ; H) j (28) in which j j denotes the absolute value of a coplex quantity, E ax is the axiu value of EW for all points IJ in the ipedance grid, and is the nuber of known wall pressures. Deterining the unknown ipedance of the aterial is now recast as a iniization proble. Thus, should be chosen such that EW () is a global iniu. The global iniu is obtained using a two-step ethod. First, we use a coarse grid in the ipedance plane and tabulate the noralized wall error function to deterine the location in that grid of the iniu point ( 1 ; kd 1 ). We use a ne grid centered about ( 1 ; kd 1 ), where and kd are now uch saller. The location of the iniu point of the ne grid corresponds to the unknown ipedance. 7 Results A coputer code ipleenting the ipedance extraction ethod has been developed. The nite eleent atrix equation (14) is solved using a routine fro the highly developed software package \Lapack,"(ref. [11]) and iniization of the noralized wall error function is perfored internally by an in-house coputer code. The unknown ipedance,, is returned by the in-house code. Results were coputed using a Dec-Alpha work station and were not coputationally intensive (i.e., requiring only 0.5 seconds of CPU tie for each point in the ipedance grid). In this section, the integrity of the ipedance extraction ethod is tested on three aterials for which the ipedance,, is known. The rst two are aterials for which = 1 + 1i and = 3 + 2i, respectively. The last aterial is a rigid wall, for which the ipedance approaches 1 + 1i, which corresponds to an adittance of, 0 + 0i. Thus, for convenience, analysis of the rigid wall case is conducted in the adittance plane. Input data required to extract the ipedance of each liner was obtained by solving equation (18) with the known and calculating the source pressure and exit ipedance fro equation (19) and (20), respectively. Equation (21) was then evaluated at evenly spaced locations to provide the known wall pressure for the wall error function. In an attept to deterine the eects of error in the input data, a nuber of cases were run with the upper wall pressure distribution randoly perturbed according to ^p(x I ; H) = p(x I ; H)E r (29) where ^p(x I ; H) is the perturbed pressure and E r is the rando error. For the cases presented in this report, the range of rando error was set to 0:1 db, so that E r = 1 N r 10 0:1=20 (30) where N r is a rando nuber between 0 and 1. The 0:1 db rando error range was not arbitrarily chosen, but is typical of that experienced in the Langley Grazing Flow Ipedance Tube Facility. For each aterial, ipedance predictions are presented for a single ode nonprogressive wave eld (nodes = 1; A 1 = 1:0; B 1 = 0:5). The duct geoetry for which calculations were ade was chosen to be that of the Langley Flow Ipedance Tube Facility (i.e., H = 2:0 inches, L = 23:0 inches) test section. Results are presented for two source 8

10 frequencies, f=500 Hertz and f=3,000 Hertz. A 231x21 evenly spaced grid is used (N = 231 and M = 21) in the nite eleent discretization for all calculations. This grid ensured that a iniu of ten eleents per wavelength was used in the nite eleent discretization at the highest frequency of interest for each of the wave elds considered. Nuerical experientation has shown that a 51x31 unifor grid (i.e., IM AX = 51 and JMAX = 31, with = kd = 0:1) is typically sucient for the coarse grid search procedure. Note that while this grid covers nearly all possible reactance values, the resistance only ranges fro 0 to 5 (i.e., ax = 5). If larger values of resistance are expected, a larger value of ax should be used. After the coarse grid procedure has been copleted, a ne grid search is conducted. Again, nuerical experientation has shown that a 21x21 uniforly spaced ne grid (i.e., IMAX = 21 and JMAX = 21, with = kd = 0:01) is sucient for convergence to the unknown ipedances. Convergence of the ipedance prediction ethod is best illustrated using contour plots in the (; ) plane. Figure 6 shows contour plots of EW () for the ne grid at a frequency of 500 Hertz. Ten evenly spaced points (i:e:; = 10) were used to construct the wall error function. The known ipedance is = 1 + 1i, and the resistance and reactance are plotted on the horizontal and vertical axes, respectively. The coarse grid contours collapse to a single point at = 1:00+0:97i. As can be seen in the gure 6, the global iniu point of EW () for the ne grid lies within the contour labeled 7. Thus the returned ipedance is the value at the grid point closest to the center of that contour, = 1:00 + 0:99i. It should be noted that separate tests were conducted to show that the error is even saller at = 1 + 1i, since the ne grid used as a standard in this study did not include this particular ipedance as a point in the grid. Figure 7 presents a coparison of the known resistance and reactance for the rst aterial ( = = 1) with the predicted values, for input data with and without rando error. The two plots in the upper half of the gure were coputed at a frequency of 500 Hertz, while the two in the lower half of the gure were coputed at 3,000 Hertz. The independent variable for the horizontal axis is the nuber of evenly spaced input wall pressure points,, used to deterine the wall error function. Results are shown for = 5; 10; 46 and 230. Note that the ipedance prediction ethod does an excellent job of deterining the noral incidence resistance and reactance, with and without rando error. Predicted resistance values are slightly less accurate at the higher frequency for sall values of. Collectively, these graphs show that the predicted ipedance is independent of. Figure 8 shows siilar results for the second aterial, whose known resistance and reactance was = 3, and = 2, respectively. Overall trends are consistent with that of gure 7. The largest error (i.e., approxiately 4%) occurs in the reactance prediction at 3,000 Hertz for = 5 when there is rando error. The third and nal aterial was a rigid wall, which was included in this study in an attept to cover the realistic range for the ajority of grazing incidence ipedance easureents. Since the ipedance for a rigid wall approaches = 1 + 1i, it was not feasible to perfor an ipedance plane grid search to try to deterine the appropriate noral incidence ipedance. However, the known adittance, = 1= = + i, for this case is zero ( = 0 + 0i). For this reason, an adittance plane grid search was perfored for this aterial. A coarse grid search was conducted with = = 0:1, over ranges of 0 2 and 1 1. A ne grid search was then conducted with = = 0:01, over ranges of 9

11 0 0:2 and 0:1 0:1. The results are shown in gure 9. Predicted adittances are in exact agreeent with the known value for each frequency, with and without rando error in the input data. Ipedance predictions were also obtained for each of the two soft aterials, but for the following sound elds 1. A single ode progressive wave eld (nodes = 1; A 1 = 1:0; B 1 = 0:0) 2. A ulti-odal wave eld without reections (nodes = 2; A 1 = A 2 = 1:0; B 1 = B 2 = 0:0) 3. A ulti-odal wave eld with signicant reections (nodes = 2; A 1 = A 2 = 1:0; B 1 = 0:5; B 2 = 0:0) Graphical results for these three elds are not presented for the sake of brevity. However, it was observed that ipedance predictions were in good agreeent to the known ipedances using each of these three sound elds. In fact, when the wall pressure was not subjected to rando error, predicted ipedances for each sound eld was identical to that obtained for the single ode nonprogressive wave eld (see gures 7, 8, and 9). Studies were also perfored for larger rando errors. When the rando error was increased to a level of 5 db (well above those typically experienced in noral applications), a weak dependence of the error in the prediction versus the nuber of wall pressure points was observed. This dependence was deterined to be a decreasing function of. Thus, as should be expected, an increasing nuber of wall pressure points should be used to increase the accuracy in the predictions for easureent systes with larger rando errors. 8 Conclusions A coarse/ne grid ipedance plane search technique has been developed for extracting the unknown ipedance of an acoustic aterial. A ain advantage of the ethod is that it is applicable to a nonprogressive wave environent, such as that usually encountered in coercial aircraft engines and ost laboratory settings. Although the ethod as presented here is restricted to two-diensional ducts without ean ow, it ay be extended to three diensions and to ean ows with shear. Data input for the predictions presented in this paper were obtained fro odal theory, but this data could be replaced with easureents taken in a grazing ow ipedance tube with the test specien installed. Results of this study show that the ethod is extreely eective in extracting the ipedance of a known aterial in coplicated nonprogressive wave elds. When there is signicant rando error in the input data, a large nuber of wall data points are required for an accurate ipedance prediction. The ethod is quite insensitive to rando error typical of that found in ost high quality easureent systes. The ethod has been found to be a siple and powerful tool for analytically based input data. There is now a need to test the ethod with easured data. 10

12 References [1] Feder, E.; and Dean, III, L.W.: \Analytical and Experiental Studies for Predicting Noise Attenuation in Acoustically Treated Ducts for Turbofan Engines," NASA CR (Sept. 1969). [2] Phillips, B.: \Eects of High Value Wave Aplitude and Mean Flow on a Helholtz Resonator, " NASA TMX-1582 (May 1967). [3] Phillips, B.; and Morgan, C. J.: \Mechanical Absorption of Acoustic Oscillations in Siulated Rocket Cobustion Chabers, " NASA TN D-3792 (Jan. 1967). [4] Arstrong, D. L.; and Olsen, R. F.: \Ipedance Measureents of Acoustic Duct Liners With Grazing Flow," Boeing paper presented at the 87th Meeting of the Acoustical Society of Aerica (New York, NY), April [5] Watson, Willie R : \A Method for Deterining Acoustic-Liner Adittance in a Rectangular Duct With Grazing Flow Fro Experiental Data," NASA TP-2310, [6] Watson, Willie R : \A New Method for Deterining Acoustic-Liner Adittance in Duct With Sheared Flow in Two Cross-Sectional Directions," NASA TP-2518, [7] Parrott, Tony L., Watson, Willie R, and Jones Michael G. : \Experiental Validation of a Two-Diensional Shear-Flow Model for Deterining Acoustic Ipedance," NASA TP-2679, [8] Zienkiewicz, O. C.: \The Finite Eleent Method In Engineering Science," McGraw-Hill Book Copany, London, [9] Desai, Chandrakant S.; and Abel, John F.: \Introduction To The Finite Eleent Method," Van Nostrand Reinhold Copany, New York, N. Y [10] Kraft, R.E.: \Theory and Measureent of Acoustic Wave Propagation In Multi- Segented Rectangular Ducts," P.h.D. Thesis, University of Cincinnati, [11] Anderson, E. et al., \LAPACK user's guide", Society for Industrial and Applied Matheatics,

13 y Rigid Wall H x 1 x 2 x 3... x Source Plane Pressure, p (y) s Right Moving Wave Exit Plane Ipedance, ζ exit (y) Left Moving Wave 0 L Material of Unknown Ipedance, ζ(x) x Figure 1: Two diensional duct and coordinate syste with a nonprogressive wave eld y 6 H [1; M 1] [2; M 1] [N 1; M 1] [1; 2] [2; 2] [N 1; 2] [1; 1] [2; 1] [N 1; 1] 0 L x - Figure 2: Finite eleent discretization of two diensional duct 12

14 y Local Node Nubers y J b y J 1 2 a 0 x I x I + 1 x Figure 3: A typical nite eleent, [I; J], and local node nubering syste 13

15 Structure of the global stiness atrix with ajor blocks, A I ; B I and B T I? A 1 B 1 B T 1 A 2 B 2 B T 2 A 3 B B T N 2 A N 1 B N 1 6 MN Structure of each ajor block, each x is a coplex nuber? x x x x x x x x x x x x x x x x x x x x x x 6 M? B T N 1 A N? Figure 4: Structure of the global atrix and ajor blocks 14

16 kd 6 = + i cot(kd) - 6 kd? 6 J 0 I- ax - Figure 5: Ipedance grid in the coplex plane 15

17 χ Global Miniu D D D D C D B B C A 9 A C B D B C C C D D D C B A D D θ Figure 6: Contour plots of the wall error function, EW (), at 500 Hertz (ne grid) 16

18 Known value Predicted value without rando error 5 Predicted value with rando error 500 Hertz 500 Hertz ,000 Hertz 3,000 Hertz Figure 7: Resistance and reactance for the rst aterial ( = + i = 1 + 1i) 17

19 Known value Predicted value without rando error 5 Predicted value with rando error Hertz 500 Hertz ,000 Hertz 3,000 Hertz Figure 8: Resistance and reactance for the second aterial ( = + i = 3 + 2i) 18

20 Known value Predicted value without rando error 5 Predicted value with rando error 500 Hertz 500 Hertz ,000 Hertz ,000 Hertz Figure 9: Susceptance and conductance for the third aterial ( = 1 = + i = 0 + 0i) 19