The acoustic impedance characteristics of porous foams and fibrous materials

Transcription

1 5th AIAA/CEAS Aeroacoustics Conference (3th AIAA Aeroacoustics Conference) - 3 May 9, Miami, Florida AIAA The acoustic impedance characteristics of porous foams and fibrous materials Fumitaka Ichihashi, Christopher Porter and Asif Syed 3 University of Cincinnati, Cincinnati, Ohio, 45 This paper presents the results of an investigation into the acoustic impedance characteristics of porous materials such as porous foams and fibrous mats. Most of previous experimental investigations focused on the characteristics of fibrous materials only. However, Yu, Kwan and Yasukawa did include some foam materials in their investigation. Their experiments were conducted in an impedance tube, which allowed measurement up to 6 Hz. The empirical correlations of Delany and Bazley were derived from measurements conducted between 5 and 4 Hz. The data presented by Lee and Selamet 4 had an upper limit of just over 3 Hz. The apparatus developed by Syed and Ichihashi at the University of Cincinnati is capable of accurate measurements up to 9 Hz. Therefore it allows an investigation of the acoustic characteristics of porous materials over a much wider frequency range. Nomenclature a n = empirical constant obtained by curve fits through measured data; n =,, 3,.,., 8. c = the speed of sound in air c e = the speed of sound in the porous medium (bulk absorber) f = the acoustic frequency h = the depth of the cavity behind the fibrous or foam test sample in the wave tube i = k = the acoustic wave number; k = ω /c = ( π f /c) l = the thickness or the depth of a fibrous or foam sample tested in the wave tube p = the acoustic pressure oscillation p e = the acoustic pressure oscillation within the porous material (fibrous or foam) u = the acoustic velocity oscillation = the acoustic velocity oscillation within the porous material (fibrous or foam) u e A = the amplitude of the forward propagating wave in the porous medium (Figure ); = the intercept of the straight line fit through the steady flow resistance (cgs Rayls) test data. B = the amplitude of the backward propagating wave (Figure ) = the slope of the straight line fit through the steady flow resistance (cgs Rayls) test data. R = the steady flow resistance (cgs Rayls) of a porous test sample = the real part of the normalized acoustic impedance of a porous test sample R C = the real part of the normalized characteristic impedance, θ *, of a porous medium X = the imaginary part of the normalized acoustic impedance of a porous test sample Graduate student, Department of Aerospace Engineering, University of Cincinnati, P.O. Box 7, Cincinnati, OH 45-7, student member AIAA. Undergraduate co-op student, Department of Aerospace Engineering, University of Cincinnati, P.O. Box 7, Cincinnati, OH 45-7, student member AIAA. 3 Research Professor, Department of Aerospace Engineering, University of Cincinnati, P.O. Box 7, Cincinnati, OH 45-7, Senior Member AIAA. Copyright 9 by Asif A. Syed. Published by the, Inc., with permission.

2 X C Z = the imaginary part of the normalized characteristic impedance, θ *, of a porous medium = the normalized acoustic impedance of a porous test sample; Z = R + i X α = the imaginary part of κ β = the real part of κ ω = the circular frequency, ω = ( π f ) κ = the complex wave propagation constant in the porous medium at frequency, f κ * = the normalized wave propagation constant; κ * = ( κ /k) ρ = the density of air ρ e = the density of the porous medium (fibrous or foam) θ = the characteristic impedance of the porous medium (fibrous or foam). θ = ρ e c e = the normalized characteristic impedance; θ * = θ /(ρ c) = R C + ix C σ = the resistivity (cgs Rayls/cm) of porous material (fibrous or foam) ξ = the dimensionless frequency parameter ρ f /σ θ * ( ) ( ) I. Introduction Current designs of acoustic liners used in the nacelles of commercial aircraft engines are based on the Helmholtz resonator principle. They are of single or double layer design. The single layer design can provide good acoustic suppression over a relatively narrow range of frequencies, approximately one octave, centered on its tuning frequency. The double layer design can provide good suppression for a much wider frequency range. However, it has been known that bulk absorber materials, used in acoustic liners can provide superior acoustic attenuation over a much wider frequency range than the liners based on the resonator principle. The most commonly known bulk absorber materials are fibrous mats or porous foams. The principle reason why bulk absorbers are not used currently has to do with their tendency to soak up water during rain. This water soakage will immediately cause a substantial increase in the weight of the nacelle and a loss of acoustic suppression capability. Moreover, should they get soaked with liquid fuel; they can become a very serious fire hazard. Recently, there has been a renewed interest in bulk absorbers because of the possibility of using hydrophobic porous membranes at the surface of a bulk absorber liner. Therefore, a small experimental research project was undertaken at the University of Cincinnati (UC) to investigate the impedance characteristics of suitable porous materials. A set of material samples was acquired for this purpose. It consisted of commercially available fibrous mats and foams of different resistivity or porosity. Empirical correlations obtained by Delany and Bazley in the late 96s had established a methodology for designing bulk absorbers consisting of fibrous materials. Kirby and Cummings showed that these empirical constants depend on the materials being investigated. The objective of the research conducted at UC was to investigate the variation in the empirical correlation constants for different bulk absorber materials. The acoustic measurements were conducted in the recently developed wave tube apparatus, which is capable of impedance measurements up to 9 Hz. This paper describes the one-dimensional acoustic wave propagation theory on which the measurement methodology is based. It also describes the details of the test apparatus and the measurement procedures. The empirical correlations for the complex impedance characteristics of the various test samples are compared with those obtained by Delany and Bazley (D&B). II. Theoretical Notes In the analysis presented in this paper, we assume that the porous medium illustrated in Figure has uniform porosity, and therefore uniform characteristic impedance throughout. The diagram shows the porous material of length, l, in a wave tube. The acoustic pressure oscillation in the one dimensional (D) acoustic field at any location, x, in the porous medium is given by p e ( f, x) = A exp( iκ x) + B exp(iκ x) () f - is the acoustic frequency

3 κ - is the complex wave propagation constant in the porous medium A and B are the complex amplitudes of the forward and the backward propagating acoustic plane waves. The values of A and B are to be determined from the boundary conditions at the two ends of the porous medium. bulk or porous medium p u A B p u x l The corresponding acoustic particle velocity at location x is given by ρ e c e u e ( f, x) = {A exp( iκ x) B exp(iκ x)} () where u e ( f, x) is the acoustic particle velocity at frequency, f. ω - is the angular frequency ( π f ) i = κ - is the acoustic wave number (ω/c e ) c e - is the speed of sound in the porous medium (bulk absorber) - is the density of the porous medium (bulk absorber) ρ e Figure. A schematic diagram of one-dimensional wave propagation in a segment of a tube containing a porous material. The complex wave propagation constant, κ, can be expressed as follows κ = β i α (3) The principal characteristics to be determined by measurement are the wave propagation constant, κ, and the characteristic impedance, ρ e c e, of the porous medium (bulk absorber). Let us denote the characteristic impedance by the symbol θ. θ = ρ e c e (4) Let us consider that by means of a measurement we know the acoustic pressure, p, and the acoustic velocity, u, at the plane x =. We also know the corresponding values of the acoustic pressure, p, and the acoustic velocity, u, at the plane x = l. Moreover, the acoustic pressure and the acoustic particle velocity must be continuous at x = and at x = l. Therefore from equations () and () at x =, we have A + B = p (5) A B = θ u (6) From equations (5) and (6), we can express A and B in terms of p and u, as follows. A = p + θ u (7) 3

4 B = p θ u (8) Substituting the values of A and B in equations () and (), it can be shown that at x = l, the continuity of acoustic pressure and velocity is satisfied as follows. p = cos(κl) p i θ sin(κl) u (9) u = i sin(κl) p + cos(κl) u () θ Let us assume that we make a second measurement (at the same frequency) but with a different termination condition beyond the plane x = l. Let the second set of the measured values of the acoustic pressure and the velocity be: p *, u *, p * and u *. Then we can write a second set of equations in terms of the new measurements, as follows p * = cos(κl) p * i θ sin(κl) u * u * = i sin(κl) p * * + cos(κl) u θ Note that the astrix in equations () and () does not denote a complex conjugate. () () From equations (9), (), () and (), we can write two sets of equations in matrix notation, in terms of the measured acoustic pressures and acoustic velocities at locations and and the unknown quantities: sin(κ l) cos(κ l),θ sin(κ l), and. These equations can be solved, as shown below. θ cos(κ l) iθ sin(κ l) = p u * * p u cos(κ l) sin(κ l) i θ = u p * * u p u * u p * p (3) (4) From (3), we have cos(κl) = p u * p * u p u * p * u (5) and from (4), we have p θ sin(κl) = i p * * p p u p * * p u (6) cos(κ l) = p u * p * u p u * p * (7) u sin(κl) θ u = i u * * u u p u * * u p 4 (8)

5 From (5) and (7) we write p u * p * u cos(κl) =.5 p u * p * u + p u * p * u p u * p * u (9) Manipulation of equations of (6 and (8) gives p p * * p p u θ = p * * p u = p p * * p p u u * * u u u u * * u u p u * * u p () Thus, from equations (9) and (), the values of the complex propagation constant, κ, and the characteristic impedance, θ, can be determined. III. The Acoustic Measurement Method Acoustic Wave Tube Apparatus with an array of 6 acoustic pressure transducers Test sample of bulk absorber material A Movable Piston with single acoustic pressure transducer l A h 3.8 cm. Section A -A Figure. A schematic diagram of the Wave Tube apparatus. It shows the setup to measure the complex propagation characteristics in a porous material of uniform porosity. The measurement technique employed by the authors is similar to that employed by Lee and Selamet [4]. The acoustic excitation in the wave tube is by means of a JBL compression driver. The acoustic field in the wave tube is measured by a set of six acoustic pressure transducers in three planes. Note that up to 45 Hz. (approx.), the acoustic field in the wave tube must consist of plane waves only. The first transverse mode is cut-on above 45 Hz. The second transverse mode is cut-on above 9 Hz. Thus up to 9 Hz. only the plane wave and the first transverse mode is sensed by the six transducers shown in Figure. The six-transducer-array allows plane wave measurements 5 up to a frequency of 9 Hz. The 7th transducer is flush mounted at the center of the acoustically hard movable piston. Because of its central location, this transducer also measures only the plane wave component 5

6 (up to 9 Hz.) of the acoustic field beyond the measurement plane #. Thus, from the acoustic pressures, measured by these six transducers the acoustic pressure, the acoustic particle velocity, and the acoustic impedance at the measurement plane # are computed [5]. The acoustic pressure and the acoustic particle velocity at the measurement plane # are computed from the acoustic pressure measured by the 7 th transducer. Thus, up to 9 Hz., all measurements are limited to the plane wave mode only. First, acoustic data are measured with the piston located at a cavity depth of h = h. Then, the test is repeated by moving the piston to a cavity depth of h = h. Thus the two sets of acoustic data, required in equations (3) and (4), are obtained. From these the complex acoustic propagation constant, κ, and the characteristic impedance, θ, are computed. IV. Flow Resistance Testing Foam or Fibrous Test Sample.54 cm. thick Test Sample Holder. cm. diameter Air Flow Through Test Sample Plenum Chamber of the Test Stand Figure 3. Schematic diagram of a bulk absorber test sample installed in the sample holder of the Raylometer test stand. A very thin wire-mesh screen was used on both sides of the test sample to contain it within the sample holder during the test. ρ f The resistivity, σ, of the porous material is incorporated in the dimensionless frequency parameter,, σ defined by Delany and Bazley []. Therefore, tests were conducted on the flow resistance test stand, a.k.a. the Raylometer, at UC. The method of holding the porous test sample in place during a flow resistance test is illustrated in Figure 3. During a test, the sample must be held in place. This is done by means of a wire-mesh screen on each side of the test sample. The diameter of the test sample is cm. The procedure for mounting a circular test sample of a fibrous or foam material is described below. There are three metal plates that detach from the apparatus. These are used to mount the bulk (foam or fibrous) material. These three pieces are similar. Each consists of a square plate with a circular hole ( cm. diameter) cut into it. The first plate is attached to the stand. Then a fine metallic mesh is placed on the first plate to cover the circular hole (to prevent materials from falling into the plenum). The next plate (. inch thick) is placed on top of the first plate and the bulk absorber is placed inside the cavity. A piece of mesh material is placed to cover the hole and ensure that the bulk material is not blown out of the cavity. The third plate is placed on top of second plate. There are thin rubber gaskets in between each of the aluminum plates and the clamps seal the configuration to ensure zero leakage. Some of the samples were.7 cm. (.5 inch) thick. Two layers of these samples were used to fill the.54 cm. (. inch) deep cavity of the sample holder. After completing testing on one side, the samples were flipped upside down to test Side. Thus each test sample/configuration was tested multiple times and data were averaged to provide a good statistical measurement of the sample. The metallic wire-mesh screens used in the tests were very 6

7 thin and of high porosity. The measured flow resistance for these mesh materials was less than cgs Rayl at 5 cm/s. Therefore we can disregard the effects of the mesh screens in any measured flow resistance data for the bulk absorber samples. Typical Flow Resistance data for a.54 cm thick fibrous sample are shown in the plots of Figure 4. A straight line fit through the test data is obtained to represent the flow resistance characteristics of the test sample Measured Straight line fit Flow Velocity, U (cm/sec) Figure 4. Flow resistance data for Polyurethane foam sample (Item #).. The resistivity, σ, of the porous material is defined as the flow resistance per unit thickness of the material. Therefore, Resistivity can be computed from the measured flow resistance data as follows: where σ = R(U) l = A + BU l = A + l B U () l l is the thickness or depth (cm.) of the test sample used in flow resistance measurements A is the intercept of the straight line fit through test data shown in Figure 4 B is the slope of the straight line fit through test data shown in Figure 4 From the measured flow resistance data, the linear and the nonlinear components of the resistivity (cgs Rayl/cm) of the test samples were determined. These data together with brief descriptions of the test samples are contained in table. Notice that apart from the Polyurethane material (Item #), all the fibrous and the foam samples are very linear. That is, their resistivity characteristics are relatively insensitive to the magnitude of the flow velocity. Item # is very nonlinear, as shown in the plot of Figure 4. 7

8 Test Sample ID # Item #8 Description of material, manufacturer, or supplier Melamine Foam; white colored soft foam. GK Industries, Inc. Item #9 Melamine Foam; white colored soft foam. AMI Item # Melamine foam much like items #8 & #9. However, It is of a yellow-orange color instead of white. BASF. Item # Item #6 Item # Item # Polyurethane Foam; soft foam of black/dark grey color. GK Industries, Inc. Fibrous mat - can be torn apart easily; most likely due to different sheets of fiber stacked together. Color- Yellow fiber with a white impervious sheet as a back wall. Acoustic Solutions Inc. Mineral fiber of brown color; rigid fibers that can cause splinters. Must handle carefully. Lancaster GTB Systems Ltd, UK. Silsoft glass. Fiber of white color, not easily compressed. BFG Industries, Inc. Resistivity Intercept Resistivity Slope Table. Summary of the fibrous and foam materials tested at UC. The resistivity data, linear (intercept) and the nonlinear (slope), are in cgs Rayl units. V. Testing in the Wave Tube Apparatus Before discussing the acoustic data measured by the authors, the forms of the empirical correlations obtained by Delany and Bazley (D&B) are described. The normalized complex wave propagation constant, κ *, can be expressed as follows and the normalized complex characteristic impedance, κ * = κ k = β α i () k k θ, can be expressed as ρ c θ * = θ = R c + ix c (3) ρ c Based on their excellent measurements, Delany and Bazley derived their empirical correlations in the following forms. α k = a β ξ a ; k =+ a 3 ξ a 4 ; R c =+ a 5 ξ a 6 ; X c = a 7 ξ a 8 where ξ = ρ f, is the dimensionless frequency parameter; and k = π f σ c (4) is the acoustic wave number. The empirical constants derived by Delany and Bazley from their measurements are presented in Table. a a a 3 a 4 a 5 a 6 a 7 a Table. The empirical constants for the correlation derived by Delany and Bazley. 8

9 Several repeat tests were conducted on each sample listed in Table. For each test, two measurements were made. The first measurement was made with the cavity depth, h = h. The second measurement was h : h h h Number of combination # (cm) (cm) repeat tests made with the cavity depth, h = h. From the data acquired from this pair of measurements, the complex values of κ and θ were computed. We used five combinations of h and h. For each combination of h and h, the test was repeated a Table 3. The values of the cavity depth, h, which were number of times. The values used for h and h and used for different repeat tests to measure the complex the corresponding number of repeated tests for each combination are listed in Table 3. Finally, each values of κ and θ for a porous material sample. Total of sample was tested with the hard wall piston located 4 repeat measurements were conducted for each test at h =. These tests provided the normal impedance sample. data for the material samples for 3.5 which the above measurements were made. This procedure allowed a minimum of twenty-four repeat measurements of the complex acoustic parameters, κ and θ, for each test sample at each measurement frequency value. From these data the mean values of κ and θ were obtained at each frequency for further analysis. A. Repeatability of Test data Measured values of the real and the imaginary parts of the normalized wave propagation constant, κ *, from six different tests are plotted against the acoustic frequency in figures 5a and 5b. These plots show the quality of the repeatability of testing with different values of the cavity depth, h, in Figure. The total number of tests carried out for this test sample, Item #6, was 8. The minimum, the maximum, the mean values, and the standard deviation of the data in Figures 5a and 5b are shown in the plots of Figures 6a and 6b. These data show that the repeatability of the test data is excellent between Hz. and 85 Hz. Real (κ*) Test # Test # Test #3 Test #4 Test #5 Test # Figure 5a. Measured values of the real part of the normalized propagation constant, κ*, plotted against frequency Item #6 (fibrous). {Imag(κ*) -} Test # Test # Test #3 Test #4 Test #5 Test #6 Figure 5b. Measured values of {Imag(κ*) } plotted against frequency from 6 different measurements Item #6 (fibrous). 9

10 4 3.5 Real (κ*) Mean Min. Max. Standard Deviation Figure 6a. Measured values of the real part of the normalized propagation constant, κ*, plotted against frequency Item #6. These plots are based on data from 4 different measurements {Imag(κ*) -} Mean Min. Max Standard Deviation Figure 6b. Item #6 (fibrous). Measured values of {Imag.(κ*) } plotted against frequency. These plots are based on data from 4 different measurements.

11 B. Correlation of the test data The process of obtaining the values of the correlation constants, a through a 8 is explained with the help of Figures 7a and 7b which show the real and the imaginary parts of the measured (averaged) values of κ * and θ *, plotted against the dimensionless frequency parameter, ξ. For each one of these plots we seek a curve fit of the form: y = a x a. This was done by using the Generalized Reduced Gradient (GRG) algorithm provided in Microsoft Excel Solver. This algorithm was developed by Leon Lasdon, 6 of the University of Texas at Austin, and Allan Waren, of Cleveland State University. The curve fits through the data were obtained by using the constants obtained in this manner. The constants, a through a8, for items #8, #9, #, #6, #, and # are presented in Table 4. Although there is some variation in the values of these constants for these Normalized wave propagation constant (κ *) (β* -) κ * = κ = β * i α * k measured real part measured imag. part Curve fit through real part Curve fit through imag. part Non-dimensional frequency parameter, ξ Figure 7a. Item #6, a fibrous material. The measured values of the real and the imaginary parts of the normalized wave propagation constant, κ*. different materials, this variation is considered relatively small compared to that observed for the item #, the Polyurethane foam. For this reason, the data for Item # are presented separately in Table 5. α* ( ) ; ξ = ρ f σ Normalized Characteristic Impedance: R c & X c 3 θ * = θ = R c + ix c ρ c measured real part, Rc Measured imag. part, Xc - Curve fit through Rc data at UC Curve fit through Xc data at UC -3 Non-dimensional frequency parameter, ξ Figure 7b. Item #6, a fibrous material. The measured values of the real and the imaginary parts of the normalized characteristic impedance, θ*.

12 Delany and Bazley Item #8 Item #9 Item # Item #6 Item # Item # a a a a a a a a Table 4. The correlation constants derived from the measured data for items #8, #9, #, #6, #, and #. The power curve-fit process obtained these constants. Delany and Bazley Item# a a a a a a a C. Computed Versus Measured Impedance Data a Table 5. The correlation constants derived from the measured data for items #. The computed and the measured values of the normal impedance are compared with the measured data in Figures 8 through 4. As expected, the values computed with the new correlation constants derived by UC, compare better with the measured data than the computed values by using the Delany and Bazley (D&B) constants. For the 6 samples of Table 3, the impedance data computed by the D&B correlations are considered to be in good agreement with the measured impedance data. The computed impedance data by using the D&B correlations for item # clearly do not compare well with the measured data, as shown in Figure R ; computed by correlation from D&B X ; computed by constants from D&B R ; measured X ; measured R ; new constants by UC X ; new constants by UC Figure 8. Measured and computed data for the Item # 8. The Melamine foam test sample is 4.95 cm. (.96 inch) deep. Broadband excitation level of 39.3 db OASPL at the surface of the test sample.

13 R; computed by correlations from D&B - X; computed by correlations from D&B R; measured X; measured R; computed using new constants by UC X; computed using new constants by UC Figure 9. Measured and computed data for the Item # 9. The Melamine foam test sample is 4.95 cm. (.96 inch) deep. Broadband excitation level of 39. db OASPL at the surface of the test sample R; computed by correlations from D&B X; computed by correlations from D&B R; measured X; measured R; computed by constants from UC X; computed by constants from UC -3 Frequency Figure. Measured and computed data for the Item #. The Melamine foam test sample is 5.84 cm. (.3 inch) deep. Broadband excitation level of 39. db OASPL at the surface of the test sample. 3

14 R; computed by correlations from D&B X; computed by correlations from D&B R; meaasured X; measured R; computed by constants from UC X; computed by constants from UC -4 Figure. Measured and computed data for the Item # 6. The fibrous mat test sample is.9 cm. (.9 inch) deep. Broadband excitation level of 44. db OASPL at the surface of the test sample R; computed by correlations from D&B X; computed by correlations from D&B R; measured X; measured R; computed by constants from UC X; computed by constants from UC -4 Frequency Figure. Measured and computed data for the Item #. The fibrous mat test sample is.9 cm. (.9 inch) deep. Broadband excitation level of 44.5 db OASPL at the surface of the test sample. 4

15 R; computed by correlations from D&B X; computed by correlations from D&B R; measured X; measured R; computed by constants from UC X; computed by constants from UC Figure 3. Measured and computed data for the Item #. The fibrous mat test sample is.78 cm. (.7 inch) deep. Broadband excitation level of 44. db OASPL at the surface of the test sample R ; computed with D&B correlations X; computed with D&B correlations R ; measured data X ; measured data R ; computed with new constants from UC tests -6 X ; Computed with new constants from tests at UC Figure 4. Measured and computed data for the Item #. The Polyurethane foam test sample is 5.4 cm. (.5 inch) deep. Broadband excitation level of 39. db OASPL at the surface of the test sample. 5

16 VI. Discussion and Conclusions The correlations derived by Delany and Bazley are of the form: y = a x a. Thus for each correlation between the real or imaginary parts of the complex parameters, κ * and θ *, against the parameter ξ = ρ f, σ we need to determine two correlation constants. This is a simple curve fit but it is by no means the most accurate representation of the measured data. It is thought that a 3rd or 4th order polynomial may provide a much more accurate correlation. In the analysis presented in this paper, the nonlinearity of the materials was ignored completely. This was done by using the linear component of the resistivity in developing the correlations presented in Tables 3 and 4. In the case of the Polyurethane foam (Item #), nonlinearity is clearly very significant. Therefore, in any future work, it is suggested that the nonlinearity should be accounted for when required. It has been shown that the wave tube apparatus and measurement technique developed at University of Cincinnati is capable of measuring the complex wave propagation (κ * ) and the characteristic impedance (θ * ) parameters for porous materials, such as foams and fibrous mats. The accuracy and the repeatability of test data are very good. It has also been shown that the correlations between the values of the normalized wave propagation constant, κ *, the normalized characteristic impedance, θ *, and the dimensionless frequency parameter, ξ, are different for different materials. These differences can be quite significant, as shown by the data for item #. Therefore, it is important that before designing an acoustic liner based on fibrous materials or porous foams, tests should be carried out to determine the correlations between κ * and ξ and between θ * and ξ for the porous materials to be used. Acknowledgements The authors wish to acknowledge Mr. Curtis Fox for his invaluable help and advice during the initial setup of the wave tube test apparatus. Mr. Fox is a senior research associate in the Department of Aerospace Engineering at the University of Cincinnati. References Delany M. E., Bazley E. N., Acoustical properties of fibrous materials, Applied Acoustics (3), 97. Kirby R., Cummings A., Prediction of the bulk acoustic properties of fibrous materials at low frequencies, Applies Acoustics (56), Yu J., Kwan H. W., Yasukawa R. D., Use of HTP ceramic foam for aeroacoustic applications, AIAA CP. 4 Lee I., A. Selamet A., N. T. Huff N. T., Acoustic impedance of perforations in contact with fibrous material, J. Acoust. Soc. Am 9 (5) May 6. 5 Syed, A. A. and Ichihashi F., The Modeling and Experimental Validation of the Acoustic Impedance of Multi- Degrees-of-Freedom Liners, paper number AIAA-8-97, presented at the Aeroacoustics Conference in Vancouver, Canada, May 8. 6 L. S. Lasdon, A. D. Warren, A. Jain, and M. Ratner, Design and testing of a generalized reduced gradient code for nonlinear programming, ACM Trans. Math. Software 4 (978), pp