REFLECTIVITY OF BARE AND COATED PLATES AND RESONANT SCATTERING FROM CYLINDRICAL SHELLS

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REFLECTIVITY OF BARE AND COATED PLATES AND RESONANT SCATTERING FROM CYLINDRICAL SHELLS Z. Yong Zhang Maritime Operations Division Defene Siene and Tehnology Organisation P.O. Box 1500, Edinburgh, SA 5111 Australia Email: Yong.

1 REFLECTIVITY OF BARE AND COATED PLATES AND RESONANT SCATTERING FROM CYLINDRICAL SHELLS Z. Yong Zhang Maritime Operations Division Defene Siene and Tehnology Organisation P.O. Box 1500, Edinburgh, SA 5111 Australia 1. INTRODUCTION Simple models for sonar ross-setions of underwater objets often assume that the objets are perfetly rigid, whih leads to effiient analytial formulas for objets with simple shapes 1, 2. A perfetly rigid body has a vanishing normal pressure derivative on its surfae, and sound reflets totally from the surfae with no phase shifts. Considering that the hulls of some underwater objets, e.g., submarines, are thin in omparison with the aousti wavelengths of typial surveillane sonar frequenies, one may ask the following question: How rigid are submarines aoustially and an we apply the simple formulas to objets with penetrable and non-perfetly refleting surfaes? In this paper we first alulate the refletion oeffiients of water-borne sound from bare and oated steel plates with either water or air bakings. The alulation is based on the appliation of linear elastiity theory to multi-layered visoelasti media. The results give us an assessment of the loal refletivity of submarine hulls with nominal thikness. We further propose that formulas developed for rigid bodies an be onveniently extended to aount for refletivity of penetrable surfaes. Loal refletivity gives insight into the harater of surfae refletions and this knowledge is then applied to interpret features of sattering from an air-filled ylindrial shell. We show that interferene of shell-borne irumferential elasti surfae waves with speular surfae leads to nulls at low frequenies and peaks at high frequenies in the alulated far field target strength. 2. REFLECTION FROM UNCOATED STEEL PLATES Various approahes (e.g., diret global matrix, propagator/transfer matrix, and invariant embedding) exist to alulate the refletion and transmission of waves in multi-layered elasti media. The results in this paper are alulated using a robust transfer matrix tehnique 3 and are further heked by an independent formulation 4.

We first onsider refletions from unoated steel plates, either water or air-baked. All the materials are assumed homogeneous with the following parameters.

The air has a sound speed of 340 m/s, and density of 1.2 kg/m 3. Energy absorptions in the above materials are ignored in our present alulations.

2 We first onsider refletions from unoated steel plates, either water or air-baked. All the materials are assumed homogeneous with the following parameters. The water has a sound speed of =1500 m/s and density of ρ =1000 kg/m 3. The steel has a bulk ompressional wave speed of 5880 m/s, shear wave speed of 3140 m/s, and density of ρ 7800 kg/m 3 s =. The air has a sound speed of 340 m/s, and density of 1.2 kg/m 3. Energy absorptions in the above materials are ignored in our present alulations. In this paper, a satterer is said to be aoustially soft if the total aousti pressure at the satterer s surfae vanishes, and to be perfetly rigid or aoustially hard if the normal derivative of the total aousti pressure at the satterer s surfae vanishes. Sound totally reflets from an aoustially soft surfae with a half wavelength phase shift and sound totally reflets from an aoustially hard surfae with no phase shifts. The phase shifts shown in this paper are for an exp( iω t) time onvention. If the exp(iω t ) time onvention is used, the phase shifts will have opposite signs. 2.1 Water-Baked Steel Plates Figure 1 shows the refletion and transmission oeffiients versus angle of inidene and frequeny for a water-baked, unbounded steel plate of 10 mm thikness. The frequeny overs from 100 Hz to 100 khz and is plotted on a logarithmi sale. We briefly note the following features. More detailed analysis, derivations, and results are in Zhang (2001) 5. Figure 1: Refletion and transmission oeffiients versus angle of inidene and frequeny for a waterbaked, unbounded steel plate of 10 mm thikness. (1) The ar-like feature above 10 khz is due to so alled oinidene, where strong oupling ours when the trae speed of the fluid-borne sound mathes that of the bending wave. It an be verified, using the equations given in the next setion, that the oinidenes shown here in the refletion oeffiient, whih is alulated using full elastiity theory, mathes preditions from Timoshenko-Mindlin thik plate theory.

(2) The results indiate that the steel plate transits gradually from being aoustially transparent at low frequenies to aoustially hard at higher frequenies.

3 (2) The results indiate that the steel plate transits gradually from being aoustially transparent at low frequenies to aoustially hard at higher frequenies. We will give some simple expressions for quik engineering assessments of this transition proess later. (3) For normal inidene, the following simple expression an be used for quik engineering assessments of plate s transition from being aoustially transparent at low frequenies to aoustially hard at higher frequenies, that is, the refletion oeffiients an be approximated as R = 1/(1 + 2iβ ) (1) β= ρ /( 2π f ρ h) (2) Where ρ is the aousti impedane of the surrounding fluid, f is the aousti frequeny, ρ s is the density of the plate, h is the thikness of the plate, and β is the ratio of the harateristi impedane of the surrounding fluid to the mass reatane of the plate per unit area. Ignoring material absorption, the results shown in Figure 1 also apply to plates of other thiknesses by saling the frequenies so that the produt of frequeny-plate thikness remains the same. 2.2 Air-Baked Steel Plates Beause of the small aousti impedane of air, sound energy is essentially totally refleted for all the frequenies from 100 Hz to 100 khz, with a maximum loss of about db. The refletion loss is thus lose to that from a rigid surfae. However, we notie that the assoiated phase shifts hanges from π at low frequenies to zero at higher frequenies. In this sense, air-baked plates gradually transit from being aoustially soft to aoustially hard as the frequeny inreases. Similar behaviour has been shown for air-filled thin shells 6. s Figure 2: Refletion losses and phase shifts versus angle of inidene and frequeny for an air-baked, infinite steel plate of 40 mm thikness. These results are onsistent with those of water-baked plates, demonstrating again that the plates are transparent at low frequenies and rigid at high frequenies.

4 The phase shifts at lower frequenies are important when ontributions from different omponents are added oherently. They influene interferene patterns of the ehoes from long pulses and introdue time spreads in ehoes from short pulses. Besides oinidene with the lowest order bending wave, there are oinidenes with dilatational and other higher order plate modes, beause the plate now is four times thiker than that for Figs Coinidene It is well known that oinidene, or mathing of trae speeds along the interfae, leads to maximum oupling between the water-borne and plate-borne waves at ertain frequenies and inident angles 7. The oinidene angles shown in Figs.1-2 were based on alulations using linear elastiity theory. Here we provide losed-form expressions for them based on Timoshenko-Mindlin thik plate theory. The oinidene frequeny where the phase speed of the bending wave equals the sound speed in water ours at 8, / 2 f f {[1 ( / ) ][1 ( / ) } (3) = 0 p m ] where is the plate ompressional wave speed, is the modified shear wave speed, and f 0 p is the oinidene frequeny for thin plates, 12 2 f 0 = (4) 2π h p The modified shear wave speed is slightly less than the bulk shear wave speed and is best approximated by its high frequeny limit, the Rayleigh wave speed of a free elasti halfspae 10,11 R, whih in turn an be obtained by numerially solving the Rayleigh dispersion equation or by the following approximation 12, m m ( σ )(1 + σ) R 1 s. (5) where σ is Poisson s ratio. An alternative approximation for is given in Ross (1987) 13 m [ Eqs.(6.3), (6.11), and (6.66) of Ref. 13] 0.76(1 + 2σ / 5) 0.87(1 + σ / 5). (6) m s s For frequenies greater than the oinidene frequeny given in Eq.(3), the trae speeds of the fluid-borne wave along the interfae math those of the bending waves of the plate at some partiular angles of inidene. These are alled oinidene angles and their expressions are given by Timosheko-Mindlin plate theory as 8,10, 2 1/ f 0 sinθ = (7) m p m p f

5 where f is the aousti frequeny. The oinidene angle dereases as the frequeny inreases beause the phase speed of the lowest order bending wave inreases with frequeny. It an be verified that the oinidene angles shown in Figs.1-2, whih are based on full linear elastiity theory, agree losely with those given by Eq.(7), whih is based on Timoshenko- Mindlin plate theory. The different approximations to the modified shear wave speed, Eq.(5) and Eq.(6), yielded negligible differenes. We note in passing that the oinidene angles given by Eq.(7) of Ref.8 ontain a misprint, whereas the expression given by Eq.(54) of Ref. 10 is in error. Our Eq.(7) is the orret form, whih is equivalent to Eq.(6.65) of Ref. 13 for general elasti material and simplifies to Eq.( 6.37) of Ref. 13 for steel. The onept of oinidene is also very useful in understanding sound interation with more omplex strutures. For air-filled spherial shells, oinidene is assoiated with enhanements of the baksattered target strength. The enhanements have been shown for steady-state 14,15 and transient sattering 16,17,18 and observed in laboratory 19. The enhanements due to oinidene with irumferential bending waves are muh weaker for baksattering from air-filled ylindrial shells 20. However, there are strong enhanements when the surfae guided waves strongly exited by oinidene travel along the axial diretion and reflet from impedane disontinuities 21,22. It an be shown that Eq.(3), from the Timoshenko-Mindlin plate theory, is also a good approximation to the oinidene ondition in thin-urved shells REFLECTION FROM COATED STEEL PLATES We next onsider the effets of thik absorptive oatings on steel plates. There are various kinds of anehoi oating materials with different aousti and absorption properties. These properties also hange with aousti frequeny, temperature, and ambient pressure. For simpliity, we use the phenomenologial formulations of Strifors and Gaunaurd (1990) 23 and treat oatings as homogeneous, highly dispersive and absorptive, visoelasti materials. Strifors and Gaunaurd (1990) 23 did not give the empirial elasti parameters for shear waves in their oatings. In their analysis, they onsidered only normal inidene and shear wave parameters are not needed. Rather than reating our own shear wave parameters, in this paper, we ignore the rigidity of the oatings and treat them as highly absorptive fluids. This simplifiation is expeted to be valid beause anehoi oating materials are generally ompliant and have low rigidity. Figure 3 shows the sound speed and attenuation alulated from the visoelasti parameters in Ref. 23. As an example, we only study oating B in this paper. We notie that for oating B the sound speed dispersion and sound attenuation are small for frequenies less than about 3 khz. Beyond 5 khz, sound attenuation inreases very rapidly with frequeny.

Sound Speed (m/s) 1000 800 600 400 200 A B C Attenuation (db/mm) 1.4 1.2 1 0.8 0.6 0.4 0.

6 Sound Speed (m/s) A B C Attenuation (db/mm) A B C Frequeny (Hz) Frequeny (Hz) Figure 3: Sound speed and attenuation for the three types of oatings, A, B, and C in Strifors and Gaunaurd (1990). 3.1 Coated, Water-Baked Steel Plates Figure 4: Refletion and transmission oeffiients versus angle of inidene and frequeny for a waterbaked steel plate of 10 mm thikness with a 50 mm thik oating. Figure 4 shows the refletion and transmission oeffiients versus angle of inidene and frequeny for a water-baked steel plate of 10 mm thikness with a 50 mm thik oating. We briefly note the following: (1) At low frequenies, both the oating and the steel are essentially aoustially transparent. The oatings absorb little sound energy. (2) The oating starts to be effetive at about 5 khz, where the fundamental resonanes of the oating layer our when the oating thikness is equal to half a wavelength of the longitudinal wave inside the oating layer. (3) At frequenies greater than 10 khz, sound energy is almost totally absorbed before it reahes the oating-steel interfae. The steel plates have little effet and the oating behaves like an infinite halfspae.

7 3.2 Coated, Air-Baked Steel Plates Figure 5 shows the refletion losses and phase shifts for a oated, air-baked steel plate. (1) At low frequenies, both the oating and the steel plate are essentially aoustially transparent. The refletion is similar to that from a water-air interfae, with zero refletion loss and a phase shift of π. Most of the energy is being refleted bak by the pressure release air surfae. (2) For frequenies above the first longitudinal resonane of the oating layer, the refletion loss and phase shifts are similar to that in Fig. 4. Obviously at these high frequenies, little energy is being refleted from the oating-steel interfae. The sound annot reah the steel. The thikness of the steel and whether it is water or air-baked beome irrelevant. (3) Refletion loss dereases with inidene angle beause grazing inidene has lesser penetration and hene lesser energy loss. Figure 5: Refletion losses and phase shifts versus angle of inidene and frequeny for an air-baked, steel plate of 40 mm thikness with a 50 mm thik oating. Finally, we note that the above results for oated plates annot be saled exatly to oatings of other thiknesses beause the aousti absorption inside the oating is not linearly proportional to frequeny (the Q-value or loss tangent hanges with frequeny), as shown in Fig SCATTERING FROM AIR-FILLED CYLINDERS 4.1 Loal Refletions and Elasti Wave Reradiation Sattering from elasti bodies onsist of superposition of refletions from surfaes of the objets and re-radiation of the elasti waves. Therefore knowledge of loal surfae refletivity aids our understanding and interpretation of sattering features. To illustrate this, we onsider the example of an air-filled ylinder.

8 Figure 6: Baksattered target strength at broadside for an air-filled ylindrial steel shell of length 60 m, radius 3 m, and thikness 30 mm. Figure 6 shows the baksattered target strength at broadside from a finite, air-filled, steel ylindrial shell. Sattering from infinitely long ylinders an be desribed analytially 24, but there is no exat analytial solution to sattering from finite ylinders. The result shown here is obtained by applying the Huygens priniple and assuming that a finite ylinder satters sound as if it were part of an infinitely long ylinder of the same radius. Under this assumption, the field on the surfae of the finite ylinder an be approximated by that of an infinitely long ylinder and the resulting Helmholtz-Kirhoff integral an be evaluated in losed-form. The fundamental ring frequeny of the shell is 280 Hz. We note the regularly spaed interferene minimums at low frequenies and interferene peaks at high frequenies. This feature an be explained as follows. At low frequenies, the ylinder wall is aoustially soft and refletion from the ylinder wall has a phase reversal ompared with the irumferential modes of the shell. Therefore their interferene leads to anellations. At high frequenies, the ylinder wall transits to aoustially hard, refletions from the ylinder wall are in phase with the irumferential modes, leading to reinforements. 5. SUMMARY (1) Unoated, water-baked steel plates transit from being aoustially transparent at low frequenies to aoustially hard at high frequenies. Simple expressions are given for quik engineering assessments of the transition proess. (2) Unoated, air-baked plates transit from being aoustially soft at low frequenies to aoustially hard at high frequenies. (3) Studies of the effet of absorptive oatings using nominal parameters show that the oating starts to be effetive when the frequeny exeeds the fundamental resonant frequeny of the oating layer. (4) For an air-filled ylindrial shell, interferene of the irumferential elasti wave with the speular refletions leads to anellations at low frequenies and reinforements at high frequenies.

9 REFERENCES 1. R. Urik, Priniples of Underwater Sound (MGraw-Hill, New York, 1983), 3rd Ed. 2. D. M. Drumheller, M.G.Hazen, L.E. Gilroy, The bistati aousti simple integrated struture target strength model, Naval Researh Laboratory, Washington, DC, Report NRL/FR- MM/ ,019 (May 2002). 3. Daniel Lévesque and Lu Pihé, A robust transfer matrix formulation for the ultrasoni response of multilayered absorbing media, J. Aoust. So. Am. 92, (1992). 4. S.A.L. Glegg, The effetive depth approximation for sound propagation in shallow water over a sediment layer and a hard rok basement, J. Aoust. So. Am. 94, (1993). 5. Z.Y. Zhang, The aousti rigidity of submarine hulls and sattering from simple shapes with penetrable surfaes, 30th Meeting of the TTCP Maritime Systems Group, Tehnial Panel 9 (Sonar Tehnology), Otober 2001, Halifax, Nova Sotia, Canada. 6. M. F. Werby and G. Gaunaurd, Transition from soft to rigid behavior in sattering from submerged thin elasti shells, Aoustis Letters 9, 89-93(1986). 7. L. Cremer, M. Hekl, and E. E. Ungar, Struture-borne sound, Springer-Verlag, Berlin, 2nd ed., P. R. Stepanishen, Influene of rotary inertia and shear deformation on aousti transmission through a plate, J. Aoust. So. Am. 58, (1975). 9. D. Feit, Pressure radiated by a point-exited elasti plate, J. Aoust. So. Am. 40, (1966). 10. Anthony J. Rudgers, Aousti refletion and radiation by thik fluid-loaded omposite plates, J. Aoust. So. Am. 66, (1979). 11. Pieter S. Dubbelday and Anthony J. Rudgers, An analysis of effetive shear modulus for flexural and extensional waves and its appliation to refletion of sound by a plate, J. Aoust. So. Am. 70, (1981). 12. I. A. Viktorov, Rayleigh and Lamb Waves, (Plenum, New York, 1967). 13. D. Ross, Mehanis of Underwater Noise (Peninsula Publishing; Los Altos, Calif., 1987). 14. G. C. Gaunaurd and M. F. Werby, Sound sattering by resonantly exited, fluid-loaded, elasti spherial shells, J. Aoust. So. Am. 90, (1991). 15. P. L. Marston and N. H. Sun, Resonane and interferene sattering near the oinidene frequeny of a thin spherial shell: An approximate ray synthesis, J. Aoust. So. Am. 92, (1992). 16. L. G. Zhang, N. H. Sun, and P. L. Marston, Midfrequeny enhanement of the baksattering of tone bursts by thin spherial shells, J. Aoust.So. Am. 91, (1992). 17. Gaunaurd, G.C. and Strifors, H.C., Frequeny- and time-domain analysis of the transient resonane sattering resulting from the interation of a sound pulse with submerged elasti shells, IEEE Transations on Ultrasonis, Ferroeletris and Frequeny Control, 40(4), (July 1993). 18. L. R. Dragonette, D. M. Drumheller, C. F. Gaumond, D. H. Hughes, B. T. O Connor, N.-C. Yen, and T. J. Yoder, The appliation of two-dimensional signal transformations to the analysis and synthesis of strutural exitations observed in aoustial sattering, Pro. IEEE 84, (1996). 19. G. Kaduhak and P. L. Marston, Observation of the midfrequeny enhanement of tone bursts baksattered by a thin spherial shell in water near the oinidene frequeny, J. Aoust. So. Am. 93, (1993).

10 20. P. L. Marston and N. H. Sun, Baksattering near the oinidene frequeny of a thin ylindrial shell: Surfae wave properties from elasti theory and an approximate ray synthesis, J. Aoust. So. Am. 97, (1995). 21. Gregory Kaduhak, Chad M. Wassmuth, and Charles M. Loeffler, Elasti wave ontributions in high-resolution aousti images of fluid-filled, finite ylindrial shells in water, J. Aoust. So. Am. 100, 64-71(1996). 22. Sot F. Morse and Philip L. Marston, Meridional ray ontributions to sattering by tilted ylindrial shells above the oinidene frequeny: ray theory and omputations, J. Aoust. So. Am. 106, (1999). 23. Hans C. Strifors and Guillermo C. Gaunaurd, Seletive refletivity of visoelastially oated plates in water, J. Aoust. So. Am. 88, (1990). 24. Fernand Léon, Florene Leroq, Dominique Déultot, and Gérard Maze, Sattering of an obliquely inident aousti wave by an infinite hollow ylindrial shell, J. Aoust. So. Am. 91, (1992).

Duct Acoustics. Chap.4 Duct Acoustics. Plane wave

Duct Acoustics. Chap.4 Duct Acoustics. Plane wave Chap.4 Dut Aoustis Dut Aoustis Plane wave A sound propagation in pipes with different ross-setional area f the wavelength of sound is large in omparison with the diameter of the pipe the sound propagates