Investigatin f a Single-Pint Nnlinearity Indicatr in One-Dimensinal Prpagatin Lauren Falc, Kent Gee, Anthny Atchley, Victr Sparrw The Pennsylvania State University, Graduate Prgram in Acustics, University Park, PA, 168, USA, { lfalc, kentgee, atchley, vws1}@psu.edu The influence f nnlinear effects in the prpagatin f jet nise is typically characterized by examining changes in the pwer spectral density (PSD) f the nise as a functin f prpagatin distance. The rate f change f the PSD is an indicatr f the imprtance f nnlinearity. Mrfey and Hwell [AIAA J. 19, 986-99 (1981)] intrduced an analysis technique that has the ptential t extract this infrmatin frm a measurement at a single lcatin. They develp an ensemble-averaged Burgers equatin that relates the rate f change f the PSD with distance t the quantity Q p p, which is the imaginary part f the crssspectral density f the square f the pressure and the pressure. With the prper nrmalizatin, gemetrical spreading and attenuatin effects can be remved, and the nrmalized quantity represents nly spectral changes due t nnlinearity. Despite its ptential applicability t jet nise analysis, the physical significance and utility f Q p p have nt been thrughly studied. This wrk examines a nrmalizatin f Q p p and its dependence n distance fr the prpagatin f plane waves in a shck tube. The use f a simple, cntrlled envirnment allws fr a better understanding f the significance f Q p p. [Wrk supprted by the Natinal Science Fundatin, the Office f Naval Research, and the Strategic Envirnmental Research and Develpment Prgram.] 1 Intrductin In the study f the prpagatin f jet nise, the pwer spectral density (PSD) is typically the quantity used t assess impact n the surrunding cmmunity. A crrect assessment requires knwledge f whether the prpagatin is linear r nnlinear. If the prpagatin is nnlinear, the nnlinearity must be accurately accunted fr in any predictin mdel. It has been shwn [1] that, under sme cnditins, a linear prpagatin mdel des nt accurately predict the evlutin f the pwer spectral density f jet nise, especially fr higher frequencies. The imprtance f nnlinearity is usually determined by examining the evlutin f the PSD with prpagatin distance, a prcess that requires multiple measurements. Hwever, nnlinearity is nt the nly factr that must be cnsidered in the measurement and analysis f fullscale jet nise. Many ther effects, such as wind and temperature gradients, grund impedance, and the spatial extent and directivity f the surce influence the prpagatin f the nise. The cmplex nature f such an envirnment necessitates measurements (acustic and meterlgical) at many lcatins s that these effects can be quantified and selectively remved during analysis. In light f this cmplexity, it wuld be beneficial t be able t determine the presence r imprtance f nnlinearity with a measurement at a single lcatin. Mrfey and Hwell [] derive an expressin cntaining a quantity that has the ptential t serve as a singlepint nnlinearity indicatr. A nrmalizatin f this quantity, ften referred t as Q/S r the Mrfey- Hwell nnlinearity indicatr, has recently been used by several researchers [1,, 4] in the analysis f highamplitude nse. Hwever, its physical meaning is nt well understd and has nt yet been thrughly investigated. Thery The present analysis fllws that f Mrfey and Hwell []. Its basis is the Burgers equatin, p β x p δ p p =, (1) τ τ where p is acustic pressure, x is distance frm the surce, is the cefficient f nnlinearity, is ambient density, c is equilibrium sund speed, is retarded time, and is the diffusivity f sund [5]. The Burgers equatin is frequently used as a mdel equatin fr nnlinear prpagatin because it includes bth absrptive and nnlinear effects and because it has an exact slutin. The authrs next use a Furier transfrm t write the equatin in the frequency dmain and generalize it fr spherical spreading t btain 185
Frum Acusticum 5 Budapest Atchley, Falc, Gee, Sparrw rp ~ 1 β + α ' = jω rq~, () r where r is radial distance frm the surce, is a generalized absrptin and dispersin cefficient, p ~ is the Furier transfrm f the pressure, and q ~ is the Furier transfrm f the square f the pressure. Multiplying Eq. () by the factr r times the cmplex cnjugate f the transfrmed pressure and taking the ensemble-average f the real part yields d dr αr β αr ( r e S p ) ω r e Q p p =, () where S p is the pwer spectral density and Q p p is the imaginary part f the crss spectral density (als knwn as the quadspectral density) f p and p. Equatin () is generalized fr spherical spreading; hwever, the present experiments invlve nly plane waves, s the fllwing frm f the equatin will be used: d dx αx β αx ( e S p ) ω e Q p p =. (4) The term in parentheses n the left-hand side f Equatin (4) is the absrptin-crrected PSD. Accrding t linear thery, the spatial derivative f this quantity shuld be zer. Thus, the right-hand side shuld accunt fr any changes in the PSD due t nnlinearity. As it is written, a psitive value fr the right-hand side f Equatin (4) indicates a gain f energy at that frequency and prpagatin distance; a negative value indicates a lss f energy at that frequency and prpagatin distance. Nrmalized Amplitude 1.8.6.4. B 1 B B B 4 Harmnic Amplitude B n vs. σ A physical interpretatin f this equatin can be drawn frm the nrmalized harmnic amplitudes f an initially sinusidal wave underging nnlinear prpagatin [6]. Figure 1 shws the fundamental alng with the secnd, third, and furth harmnics as a functin f nrmalized distance fr such a case. These are theretical curves derived by Blackstck t cnnect the Fubini and Fay slutins. They are fr plane waves and d nt include any explicit atmspheric r bundary layer lss mechanism. The independent variable is distance nrmalized by the shck frmatin distance (als a quantity derived using the lssless plane wave assumptin). Because these curves are pltted as a functin f distance, the slpe f a curve at any given pint is qualitatively represented by the spatial derivative n the left-hand side f Equatin (4). Thus, it appears that the fundamental and secnd harmnic will be f mst imprtance near the surce, and the higher harmnics will gain imprtance as prpagatin distance increases. Experimental Apparatus.1 Descriptin The data presented in this paper were btained in a plane wave tube cnstructed f PVC pipe with an inner diameter f 5.1 cm and a wall thickness f 4.1 mm. The ttal length f the tube is 1. m, with 9.68 m available fr acustic measurements. The last.5 m cntains a fibreglass anechic terminatin cnsisting f a 1 m linear taper and a 1.5 m sectin f cnstant 8 kg/m density which attenuates frequencies f interest (.9 khz) at 1 db/m [7]. Tw JBL 4H drivers are affixed t the ppsite end f the tube using a Tee jint. Fur B&K 6.5 mm type 498 micrphnes are placed at.1 m,.5 m, 6.4 m, and 9.55 m frm this jint, respectively. The first crss mde f the tube ccurs at apprximately.8 khz. All data shwn in this paper have a surce frequency f.9 khz. A schematic f the tube is shwn in Figure. 1 σ = x/xbar Figure 1: Nrmalized amplitudes as a functin f prpagatin distance fr the first fur harmnics f an initially sinusidal wave Figure : Schematic f plane wave tube 186
Frum Acusticum 5 Budapest Atchley, Falc, Gee, Sparrw. Validatin In rder t verify that sund in the tube cnsists f nly travelling plane waves, that reflectins frm the terminatin are negligible, and that absrptive and nnlinear effects are as wuld be expected, measured wavefrms and spectra were cmpared with numerical predictins. The predictin methd used was a versin f a cde written by Gee [8]. The cde is a hybrid time-frequency dmain numerical slutin t the Burgers equatin fr spherical waves that includes atmspheric absrptin and dispersin. The versin used here was created by remving spherical spreading and adding the effects f bundary layer absrptin and dispersin [9]. T generate the predictins, the measured wavefrm frm the first micrphne (.1 m frm the surce) was used as the input t the algrithm and prpagated numerically t the ther micrphne lcatins. Figure shws the PSD f a measurement at the third micrphne alng with a numerical predictin fr the same prpagatin distance. The surce is a sinusid at.9 khz; values f the PSD fr this frequency and its harmnics are shwn. There is generally gd agreement between the measurement and predictin, althugh the measured spectrum is smewhat less smth than the predictin. This is prbably a result f high-frequency scattering assciated with the micrphne hlders. This kind f spectral variability is evident abve 14 khz in many f the measurements frm the plane wave tube. PSD (db re µpa/rthz) 14 1 1 8 6 PSD at harmnics, 6.4 m, 16.9 db re µpa Measurement Gee Cde Predictin 5 1 15 5 5 4 Figure : Cmparisn f measured and predicted pwer spectral densities Figure 4 shws the measured and predicted wavefrms fr the same case. Agreement here is als generally gd, althugh the algrithm des verpredict the amplitude smewhat. This indicates the presence f additinal lsses frm the tube that are nt accunted fr in the algrithm, such as sund radiatin frm the micrphne prts r reflectins at the jints. Hwever, the agreement is sufficiently gd fr the preliminary investigatins reprted here. Pressure (Pa) 1-1 - Wavefrms fr 6.4 m, 16.9 db re µpa Measurement Gee Cde Predictin.5 1 1.5 Time (ms) Figure 4: Cmparisn f measured and predicted wavefrms 4 Results 4.1 Nnlinearity Indicatr The dependence f Q p p n surce amplitude and prpagatin distance was investigated by pltting the right-hand side f Equatin (4) fr varius experimental cnditins. Fr Figures 5 and 6, measurements frm the first micrphne (.1 m prpagatin distance) were used t calculate Q p p fr fur different surce amplitudes. (Pa /(Hz*m)) 4 - -4 d(e αx S p.1 m, 15.9 db, σ =..1 m, 14.9 db, σ =.14 5 1 15 5 5 4 Figure 5: Nrmalizatin f Q p p given by right-hand side f Equatin (4) fr tw different (mderate amplitude) surce cnditins 187
Frum Acusticum 5 Budapest Atchley, Falc, Gee, Sparrw (Pa /(Hz*m)) 1 5-5 -1 d(e αx S p.1 m, 145.8 db, σ =.5.1 m, 147.6 db, σ =.1 5 1 15 5 5 4 Figure 6: Nrmalizatin f Q p p given by right-hand side f Equatin (4) fr tw different (high amplitude) surce cnditins A similar trend is present in these fur data sets. In all cases, energy is being lst at the fundamental frequency and gained at the secnd harmnic frequency. This is cnsistent with the bservatins frm Figure 1. It shuld be nted that althugh the values fr the lwest amplitude case (15.9 db in Figure 5) appear t be zer, they actually exhibit the same trend seen in the ther plts but at an amplitude apprximately fur rders f magnitude less. Figures 7 and 8 shw the same quantity pltted fr all fur micrphne lcatins at the 147.6 db surce cnditin seen in Figure 6. The sund pressure levels given in the legends are lcal t each micrphne. While the value fr the fundamental frequency is always negative, the (nnzer) values fr the remaining harmnics are always psitive, indicating that they cntinue t gain energy nnlinearly at all pints alng the prpagatin path. (Pa /(Hz*m)) 5-5 -1 d(e αx S p.1 m, 147.6 db, σ =.1.5 m, 145.9 db, σ =.999 5 1 15 5 5 4 Figure 7: Nrmalizatin f Q p p given by right-hand side f Equatin (4) fr same surce cnditin at first () and secnd (+) micrphnes The nnlinear grwth f the higher harmnics begins t dminate as the prpagatin distance increases, especially at 9.55 m, fr which the steepened wavefrm is shwn in Figure 9. The spectral variability in Figure 8 is likely due t scattering and ther irregularities in the sund prpagatin in the tube, as mentined abve. (Pa /(Hz*m)) 15 1 5-5 d(e αx S p 6.4 m, 14.5 db, σ = 1.97 9.55 m, 19.7 db, σ =.94 5 1 15 5 5 4 Figure 8: Nrmalizatin f Q p p given by right-hand side f Equatin (4) fr same surce cnditin at third () and furth () micrphnes Pressure (Pa) - -4..4.6.8 1 1. 1.4 Time (ms) Figure 9: Measured wavefrm crrespnding t 19.7 db at 9.55 m cnditin in Figure 8 4. Predictins Equatin (4) was cast int the frward-difference frm α x β S p( x + x) = e S p( x) ω x Q ( x) p p (5) and used t prpagate the PSD. S p and Q p p were measured at a distance x, and Equatin (5) was used t predict S p at a distance (x + x). This predicted PSD was then cmpared with the measured PSD at (x + x). Figure 1 shws ne such predictin and cmparisn. In rder t make this cmparisn with a measurement pint, x must be equal t the distance between micrphnes n the plane wave tube, which is currently.15 m. In Figure 1, x = 6.4 m and (x + x) = 9.55 m, and the lcal sund pressure level at 9.55 m is 17 db. The measurement and the predictin agree well fr the first fur harmnics, and the verall sund pressure levels agree t within. db. The fact that the measured levels are generally higher than the 188
Frum Acusticum 5 Budapest Atchley, Falc, Gee, Sparrw predicted levels fr the higher harmnics can be attributed t the cntinued grwth f Q p p at these frequencies between 6.4 m and 9.55 m. Overall, the agreement is gd fr this preliminary result, especially in light f the relatively large value f x. PSD (db re µpa/rthz) 1 1 11 1 9 Pwer Spectral Density Measurement, 9.55 m, σ = 1.7 Predictin frm 6.4 m, σ = 1.16 8 1 4 Figure 1: Predictin (+) frm 6.4 m t 9.55 m generated using Equatin (5), cmpared with the measured spectrum () at 9.55 m 5 Cnclusins An experimental apparatus has been develped t investigate the physical meaning and ptential applicatins f the quantity Q p p. This quantity has been identified as being related t the nnlinear distrtin rate f a prpagating finite amplitude wave. The nrmalizatin f Q p p given by the right-hand side f Equatin (4) has been shwn t behave qualitatively as wuld be expected given the evlutin f the harmnic amplitudes f an initially sinusidal wave. Its dependence n surce amplitude and n prpagatin distance have been demnstrated. Preliminary predictins using a finite-difference frm f Equatin (4) have been presented and have shwn gd agreement with measurements. References [1] Gee, K.L. et al., Preliminary analysis f nnlinearity in military jet aircraft nise prpagatin, AIAA J. 4(6), 198-141 (5). [] Mrfey, C.L. and G.P. Hwell, Nnlinear prpagatin f aircraft nise in the atmsphere, AIAA J. 19(8), 986-99 (1981). [] McInerny, S.A., and Ölçmen, S.M., Highintensity rcket nise: nnlinear prpagatin, atmspheric absrptin, and characterizatin, J. Acust. Sc. Am. 117, 578-591 (5). [4] Petitjean, B.P. et al., Acustic pressure wavefrms measured in high speed jet nise experiencing nnlinear prpagatin, 4rd AIAA Aerspace Sciences Meeting and Exhibit, 5, AIAA 5-9. [5] Lighthill, M.J., Viscsity effects in sund waves f finite amplitude, frm Surveys in Mechanics, G.K. Batchelr and R.M. Davies, eds., Cambridge University Press: Cambridge, 5-51 (1956). [6] Blackstck, D.T., Cnnectin between the Fay and Fubini slutins fr plane sund waves f finite amplitude, J. Acust. Sc. Am. 9, 119-16 (1966). [7] Tarnw, V., Measured anistrpic air flw resistivity and sund attenuatin f glass wl, J. Acust. Sc. Am. 111(6), 75-79 (). [8] Gee, K.L. et al., Nnlinear mdeling f F/A-18E nise prpagatin, 11 th AIAA/CEAS Aeracustics Cnference, 5, AIAA 5-89. [9] Blackstck, D.T., Fundamentals f Physical Acustics, Jhn Wiley & Sns, Inc.: New Yrk, -5 (). 6 Future Wrk Wrk n the plane wave tube will cntinue with tw gals: identifying and reducing excess lsses and abnrmalities in prpagatin; and adding the ability t vary the spacing f the micrphnes t allw fr better accuracy in the determinatin f Q p p and in predictins. Experiments will be perfrmed in the tube with bi-frequency and nise surces, and the behavir f Q p p fr these cases will be investigated. These investigatins may help t determine what value f Q p p indicates the threshld f imprtance fr nnlinearity and the dependence f Q p p n distance. 189