Panel transmission measurements: the influence of the non plane wave nature of the incident field

Transcription

1 Acoustcs 8 Pars Panel transmsson measurements: the nfluence of the non plane wave nature of the ncdent feld V. F Humphrey a J. Smth b a Insttute of Sound Vbraton, Unv. of Southampton, Unversty Road, Hghfeld, SO7 BJ Southampton, UK b DSTL, Rm 4, Bldg 35, Porton Down, SP4 JQ Salsbury, UK vh@svr.soton.ac.uk 3673

2 Acoustcs 8 Pars For reasons of cost practcalty, the laboratory measurement of the acoustc transmsson reflecton propertes of materals for use n underwater applcatons are typcally performed on samples of lmted dmensons wth the source recever separated by relatvely short dstances resultng n a non-planar measurement feld. The nfluence of ths on the resultng measurements s nvestgated n ths paper. In partcular, for low frequency measurements the nfluence of the evanescent wave contrbutons can become sgnfcant. In ths paper two alternatve approaches are used to evaluate the transmsson propertes. The frst approach uses an asymptotc expanson of the felds n terms of wave front curvature: bounds are then placed on the error n ths expanson at low frequency usng thn plate theory. The second method decomposes the ncdent sphercal wave nto ts plane wave components ntegrates the resultng transmtted waves numercally to evaluate the transmtted feld. Results are compared contrasted for measurements n the frequency range to 6 khz for panels of smple elastc materals (steel Perspex (Poly(methyl methacrylate))). In addton the nature sgnfcance of the modes of the panel for evanescent waves are consdered. The consequences for laboratory measurements are also outlned. c Crown Copyrght 8, Dstl. Introducton The reflecton transmsson coeffcents of a plane wave ncdent on an nfnte panel provde a convenent means of charactersng the acoustc propertes of a materal. These are often measured usng a lmted sze panel; the fnte sze of the panel measurng system then mpose lmtatons on the measurement. Of the dfferent sources of error, ths paper s concerned wth those caused by the non-plane wave nature of the source. That sphercal wave components can cause devatons of the measured transmsson coeffcent from the plane wave result has been demonstrated before: n [, ] t was shown that the plane wave spectrum of the source can cause errors at hgh frequency, but the frequences consdered were hgher than those typcally of nterest n many applcatons. A numercal study by Pquette [3] ndcated the presence of a potentally large, low frequency effect. Ths was further studed by Shenderov [4, 5]; these calculatons show a much smaller effect than that seen n [3]. In ths paper the effect of an ncdent sphercal wave on a measurement of the transmsson coeffcent s reexamned. It s shown that the transmtted pressure can be wrtten as an asymptotc expanson of the wavenumber multpled by the separaton between the source the recever, the frst term of whch gves the plane wave transmsson coeffcent. When the plate s thn, explct analytc bounds are found on the error n ths expanson; these bounds suggest that, n order to ensure that a measurement gves a reasonable approxmaton to the plane wave transmsson coeffcent of a thn plate, the separaton between the source recever needs to be larger by a factor of the flud loadng parameter than that whch would be deduced by lookng at the asymptotc expanson alone. These bounds allow effects of the order of those seen n [3] for some materals: n order to examne the dscrepancy between [3] [4] the plane wave spectrum s ntegrated usng the full elastc transmsson coeffcent. Ths gves results n agreement wth [4]. Asymptotc analyss error bounds The startng pont s to express the reflecton transmsson coeffcents n terms of ntegrals over the plane wave spectrum of the source. Consder a plate of thckness, h, lyng at the orgn n the r φ plane of a cylndrcal co-ordnate system, (r, φ, z). It s surrounded on both sdes by an acoustc flud of sound speed c, s nsonfed usng a monopole source of angular frequency ω, postoned at (,,z ). The ncdent pressure feld along the z-axs, for z<z, s thus gven by p (z) = a β eβ(z z) α dα = a R ekr () where k = ω/c, R = z z an overall tme dependence on exp( ωt) has been gnored. The branch cut for β = k α must be chosen such that β = α k as α. Usng these defntons the reflected pressure, p r, the transmtted pressure, p t, can be wrtten as p r (z) = a β R(α) eβ(z+z) α dα () p t (z) = a β T (α) eβ(z z) α dα. (3) R(α) T (α) are then found by satsfyng the stress dsplacement boundary condtons on the faces of the plate [3]. For propagatng components (α < k), they are the reflecton transmsson coeffcents. Attenton wll now be restrcted to the transmsson coeffcent, although smlar results can be obtaned for the reflecton coeffcent. Frst the ntegral n (3) s splt nto two parts an ntegral over α<k an ntegral for α>k: p t (z) = a k + a β T (α)eβr α dα k β T (α)eβr α dα. (4) In the frst ntegral the change of varable u = β/k s made. The result s a Fourer ntegral whose asymptotcs can be treated usng ntegraton by parts [6]. For the 3674

3 Acoustcs 8 Pars second ntegral (the ntegral over the evanescent part of the spectrum), changng varables to u = β/k gves an ntegral that can be treated usng Watson s Lemma [6]. Defnng ˆT (u) =T (k( u ) ) T (k cos θ) (where θ s the angle from the normal to the panel), these ntegrals become I = a k = a e kr R ˆT (u) e kr u du ( ˆT () ) kr ˆT () a ˆT () kr + a k ɛ I = a k ˆT (u) e kr u du = a ˆT () kr + a k ɛ, (5) (6) were use has been made of the fact that ˆT ()=. Puttng these two results together, the contrbuton from the lmt at u = from the propagatng ntegral, I, cancels the contrbuton from the lmt n the evanescent ntegral, I. Ths can be shown to be true at all orders of the asymptotc expanson. The result s (gnorng contrbutons of O(h/R)) p t = a R ekr { ˆT () kr ˆT () } + a k (ɛ + ɛ ). (7) Ths result s exact the error terms ɛ ɛ are gven by ɛ = ˆT (kr) (u) e kr u du, (8) ɛ = ( ˆT (u) ˆT ()u) e kr u du. (9) The term n curly brackets n Eq. 7 s the measured transmsson coeffcent, T m. Due to the theorems governng the asymptotc forms of ɛ ɛ mplct n Watson s Lemma ntegraton by parts [6], t can be seen that T m ˆT () ( ) kr ˆT ()+o, as kr. () kr Thus the leadng order sphercal correcton to the measured transmsson coeffcent depends on the rate of change of the plane wave transmsson wth angle, as has been observed prevously [, 4]. It s temptng to try to use () to determne the separaton needed to ensure that the correctons are smaller than a gven accuracy [4], however ths s ncorrect: the result () s a statement about the lmt kr. To estmate how accurately () holds bounds must be found on the error terms, ɛ ɛ. Snce these bounds necesstate an analyss of the pole structure of ˆT (u) n the complex u plane, the plate s now assumed to be thn. In ths case the poles are found to be related to the dsperson curves of a flud loaded thn plate. Crghton has shown how approxmate expressons for the bendng wave solutons could be found by expng n the flud loadng parameter ε, whch s gven by ε = ρ B mc m, () where ρ s the densty of the flud, m s the mass per unt area of the plate B s ts bendng stffness. Physcally ths s the rato of the mass loadng of the flud to the mass of the plate at the concdence frequency s always expected to be a small parameter for rgd plates [7]. Ths analyss was recently extended to nclude the symmetrc wave solutons [8]. It s useful to ntroduce ω, whch s the frequency scaled to the concdence frequency on the plate, the parameters α l α p (defned n [8]) δ = c /c p. The transmsson coeffcent s then gven by εu [ ˆT (u) = ( u δ) ωc(u)d(u) + α p ω ( u ) ( ω ( u ) ) where + α l ω ( u δ) ( ω ( u ) )] C(u) = [ u( ω ( u ) )+ ε ] ω D(u) = [ ( u δ)(u εα l ω) εα p ω( u ) ]. () (3) (4) Maxma n the transmsson coeffcent ts dervatves are assocated wth the factors C(u) D(u) n the denomnator gettng small. Snce ε s expected to be small (ε.3 for steel n water ε.3 for Poly(methyl methacrylate) (PMMA) n water), the peaks n ˆT (u) are near u = u = δ provdng ω < ( ω = O()). An estmate of ɛ s thus obtaned by separatng out these peaks then placng a crude bound on the remanng ntegral. The contrbuton from the peaks s then ncluded as the maxmum value of the peaks multpled by ther wdth. Thus, ɛ (kr) + { σ ( ) + ε ω α l + αp δ 4( δ)σ ε α pδ ω ω( 3σ) [ ε ω ( +4 ω ) ] +64ε } [ +O(ε ) ] (5) for some σ>ε ω [( δ)α l + α p ]. Thngs are slghtly more complcated for ɛ ; there are two poles on the path of ntegraton (n the case of no materal dampng), as can be seen from the plot of T over the evanescent part of the plane wave spectrum shown n Fg.. One s at u ( ω + ε ( ω) 3 ) + O(ε ), (6) ( ω = O()) whch s assocated wth couplng nto the subsonc mode n the plate for ω <. The other looks 3675

4 Acoustcs 8 Pars lke a Scholte-Stoneley wave s at u εα l ω + εα p ω ( 4δ ω ε [ α l + α ] p δ δ δ ) + O(ε 4 ). (7) () alone. When PMMA s replaced by steel however, the bounds on ɛ ɛ become large (Fg. 3): these bounds would allow the effects seen n [3] however they are upper bounds on the error only are expected to be conservatve. To nvestgate the errors further requres a numercal approach. It s a coupled flud-plate mode formed by couplng nto the plate at near grazng ncdence. The result may be evaluated usng contour ntegraton by deformng the path of ntegraton onto a new path at 45 to the real u axs. The error ɛ s then gven by fndng the maxmum value of ths (fnte) path addng the resdues of the poles crossed n deformng the path. Usng stard technques [6] the result s found to be ɛ ( ) ( δ)ε ω α l + αp δ (kr σn ) 3 (8) + πe krε ω(α l+α p/( δ)) where ln σ n ( ) (9) 4ε ω α l + αp δ kr has been chosen so that kr > σ n (otherwse the estmate for ɛ would depend on /(kr) ). T Plane Wave Asymptotc ε ε Error frequency (Hz) x 4 Fgure : Comparson of the plane wave transmsson coeffcent wth the frst order sphercal correcton () for a Perspex panel. Also shown are the magntudes of the error ntegrals, ɛ ɛ, the assocated error n T m. T T Plane Wave Asymptotc ε ε cos θ Fgure : Plot of the modulus of the transmsson coeffcent for evanescent wave contrbutons showng the presence of peaks due to Scholte-Stoneley subsonc modes. Surprsngly, rather than beng small when kr > as one mght expect from the asymptotc result (), the error terms (5) (8) have a functonal dependence on εkr. Ths seems to mply that, n order to be certan sphercal wave correctons can be gnored one needs to requre εkr >, rather than kr > as would be deduced from the asymptotc result (). Fg. shows a comparson of the thn plate transmsson coeffcent wth the modulus of T m from (), together wth the magntudes of the bounds on ɛ ɛ calculated usng the parameters for PMMA n Secton 3. Ths shows the close agreement of the asymptotc result to the plane wave result. The approxmatons used n dervng the bounds (5) (8) break down for ω = O(ε) hence the low frequency behavour s not physcal. Away from ths regon the bounds, though small, are larger than would be expected from frequency (Hz) x 4 Fgure 3: Plot of modul of the plane wave transmsson coeffcent frst order sphercal wave correcton, together wth magntudes of the error ntegrals for a steel panel. 3 Numercal calculatons An alternatve approach s to use numercal ntegraton to obtan values for the transmtted pressure p t from Eq.. (4). Ths enables the effectve transmsson coeffcent to be calculated, whch can be expressed as an Inserton Loss, IL, usng IL = log ( T m ). The numercal ntegraton of (4) s taken n two parts for convenence wth the ranges beng dvded as necessary for effcency. The transmsson coeffcent was mplemented usng the expressons n [9] wth a correcton for the water path replaced by the test panel. Fg. 4 shows example calculatons for a Perspex (PMMA) panel.7 m n thckness measured at a 3676

5 Acoustcs 8 Pars Inserton Loss / db 4 3 Sphercal wave (part) Sphercal wave (full) Inserton Loss / db 4 3 Sphercal wave (full) Experment Fgure 4: Calculated Inserton Loss for a.7 mm Perspex plate for a plane wave (dashed lne) sphercal wave for a source-recever dstance of. m (sold lne). The contrbuton from the real part of the spectrum for the sphercal wave case s shown as a dotted lne. dstance of. m from the source. Note that the range of the test panel from the source s not sgnfcant. In ths case the Inserton Loss for a plane wave ncreases steadly from khz wth a smooth maxmum at about 54.5 khz. The result of the frst ntegral n (4) s shown as a dotted lne for frequences above khz. Ths shows a dstnct departure from the plane wave result between 5 6 khz; wth a maxmum departure of about.6 db. The results are, however, obscured by a sgnfcant oscllatory component that grows n ampltude as the frequency s reduced. Ths s the reason for only presentng the results above khz. Ths oscllaton s assocated wth the non-zero contrbuton from the upper end of the ntegral over the real angle alpha. Ths dsappears when both parts of the ntegraton n (4) are performed, as the contrbuton from the lower lmt of the magnary ntegral almost completely cancels out that from the end of the real ntegral. The resultng Inserton Loss follows the plane wave result very closely for frequences up to khz, then les slghtly below the plane wave result untl the sgnfcant devaton around 55 khz. Ths result ndcates that at low frequences the dfferences between the effectve transmsson coeffcent for sphercal waves the plane wave transmsson coeffcent s very small. In ths case less than. db up to khz. The exstence of the specfc devaton at 55 khz has already been noted n []. In that case the transmsson coeffcent was predcted usng an approxmate soluton for the effectve transmsson coeffcent that was approprate to a parametrc array used as the source n []. In that case the ntegral was only performed over real angles was, therefore, lmted to frequences above about khz. The expermental data obtaned for a source-hydrophone separaton of.9 m n [] s compared wth the results of the current calculaton for a sphercal wave ncdence a separaton of.9 m n Fg. 5. Although not strctly equvalent (the expermental data were obtaned for a parametrc beam rather than Fgure 5: Comparson of calculated Inserton Loss for a.7 mm Perspex plate for a sphercal wave observaton ranges of. m (sold lne).9 m (dot-dashed lne) wth the plane wave result (dashed lne). a pont source) the plots show a number of features n common. Frstly, over the range from to 5 khz both the experment the model ndcate an Inserton Loss that s less than that for a plane wave. Secondly, both experment theory show a characterstc devaton from the plane wave result at about 55 khz. The sphercal wave approach underestmates the peak loss; however, f the apparent acoustc centre of the array s assumed to be at ts md pont, shortenng the apparent range, then the predcted loss ncreases. Inserton Loss / db Sphercal wave m Sphercal wave.9m Fgure 6: Comparson of calculated Inserton Loss for a.7 mm Perspex plate for a plane wave (dashed lne) for a sphercal wave (sold lne) wth expermental results (ponts) obtaned usng a parametrc array as a source []. The source-recever dstance s.9 m. The sphercal wave approach also predcts that the effectve loss should smoothly decrease wth reducng frequency, be very smlar to the plane wave result for the lowest frequences. The effect of the measurement range s shown n Fg. 6 whch presents numercal results for the effectve sphercal wave Inserton Loss for separatons of.9 m. m. Ths ndcates how the peak loss at 55 khz ncreases as the range reduces. It also shows that the underestmate of loss between 5 khz ncreases as the separaton reduces. Fnally the results show that at the lowest frequences there s 3677

6 Acoustcs 8 Pars a small negatve loss,.e., a transmsson coeffcent of slghtly greater than one for sphercal wave. Ths effect s larger at shorter ranges s attrbuted to the apparent change n poston of the source caused by the panel. The numercal evaluaton of the ntegrals n (4) s more dffcult for steel because of the lower materal loss hence reduced wdth of the Inserton Loss peaks contrbutng to the magnary part of the ntegral. However the results presented n Fg. 7 show that the resultng Inserton Loss for the sphercal wave case s very close to that for a plane wave, n agreement wth [4]. Inserton Loss / db 5 5 Sphercal wave Fgure 7: Comparson of calculated Inserton Loss for a.7 mm steel plate for a plane wave (dashed lne) wth that for a sphercal wave (sold lne) for a source-recever dstance of. m. 4 Dscusson Conclusons The asymptotc expanson for the sphercal wave transmsson coeffcent () depends only on the plane wave transmsson coeffcent ts dervatves at normal ncdence would gve a pressure at the recever that was lower, thus a hgher Inserton Loss, than would be obtaned usng a plane wave. The bounds on the error ntegrals however depend on the whole plane wave spectrum turn out to be domnated by couplng nto the longtudnal wave for the ntegral over the real part of the spectrum (ɛ ) the Scholte-Stoneley mode for the ntegral over the magnary part of the spectrum (ɛ ). The second of these would not be predcted by the usual thn plate analyss [7] though t s seen n the numercal evaluaton of the full elastc equatons [5] arses n [8] through the matchng of the acoustc pressure to the normal stress at the surface of the plate, allowng the formaton of a Scholte-Stoneley type wave below the concdence frequency n addton to the subsonc wave. Couplng nto these modes can ncrease the pressure at the recever above that for a normally ncdent plane wave. These bounds are upper bounds on the error only, however numercal ntegraton of the plane wave spectrum for PMMA gves an Inserton Loss lower than the plane wave result. Ths n part s due to shft n the apparent poston of the source whch s of O(h/R) has been gnored n Eq. (7). The numercal ntegraton also shows a characterstc devaton from the plane wave result at about 55 khz that agrees well wth experment. Ths s above the concdence frequency the error analyss would need to be extended to nvestgate ths effect. The error analyss also ndcates that the devatons from the plane wave results should be larger for steel the ntegrals more dffcult to evaluate numercally due to the proxmty of the poles to the path of the ntegral over the evanescent part of the plane wave spectrum when the materal dampng s small. Ths s lkely to be the cause of the dscrepancy between the results of [3] [4]. The current results show that for realstc separatons the devatons from the plane wave results are small. Only at very small source-recever separatons do the devatons become realstcally measurable at low frequences, then they are unlkely to be of sgnfcance. Acknowledgments JS was funded by the Defence Technology Innovaton Centre (DTIC). References [] V. F. Humphrey H. O. Berktay, The transmsson coeffcent of a panel measured wth a parametrc source, J. Sound Vbr., 85-8 (985). [] V. F. Humphrey, The nfluence of the plane wave spectrum of a source on measurements of the transmsson coeffcent of a panel, J. Sound Vbr. 8, 6-7 (986). [3] J. C. Pquette, Sphercal-wave scatterng by a fnte-thckness sold plate of nfnte lateral extent, wth some mplcatons for panel measurements, J. Acoust. Soc. Am. 83, 84 (988). [4] E. L. Shenderov, Transmsson of a sphercal wave through an elastc layer, Sov. Phys. Acoust. 37, 47 (99). [5] E. L. Shenderov, A sound feld near an elastc layer excted by a sphercal source, Acoustcal Physcs 4, 44 (994). [6] F. W. J. Olver, Asymptotcs Specal Functons, Academc Press. Reprnted by A K Peters, 997. [7] D. G. Crghton, The 998 Raylegh medal lecture: flud loadng the nteracton between sound vbraton, J. Sound Vbr., -7 (989). [8] J. D. Smth, Symmetrc wave correctons to the lne drven, flud loaded, thn elastc plate, J. Sound Vbr. 35, (7). [9] L. M. Brekhovskkh, Waves n layered meda, London: Academc Press (Second Edton), 98. [] V. F. Humphrey, The measurement of acoustc propertes of lmted sze panels by use of a parametrc source, J. Sound Vbr. 98, 67-8 (985). 3678