Acoustic forcing of flexural waves and acoustic fields for a thin plate in a fluid
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1 Acoustic orcing o leural waves and acoustic ields or a thin plate in a luid Darryl MCMAHON Maritime Division, Deence Science and Technology Organisation, HMAS Stirling, WA Australia ABSTACT Consistency with conservation o energy or coupled acoustic ields and plate leural waves, discussed in another paper or this conerence, is used to derive the amplitude and phase o leural and acoustic waves or an ininite thin plate luid system ecited by an incident acoustic plane wave. The acoustic interaction o the plate luid system is deined by. specula relection rom the plate surace,.transmission through the plate material and 3. plate leural waves taking in to account luid loading. This reproduces the well-known peak in plate leural wave amplitudes above the coincidence requency where the trace wavenumber o the incident acoustic plane wave along the plate equals the plate vacuum leural wavenumber. This is essentially a resonance with the resonant requency that varies with the direction o the incident plane wave. The width o the resonance is governed by luid loading which maniests as radiation damping o the leural waves. t is ound that leural waves aect the acoustic relectivity o the plate through coherent intererence o the acoustic ield rom leural waves with the specula relected ield, but only i there is a nonzero phase shit in specula relection. Energy conservation considerations predict that a plate becomes acoustically sot close to the resonance condition. A simple ormula or the approimate resonance width is also derived. Keywords: Acoustic, Plate, Fluid -NCE Classiication o Subjects Number(s):., 3, 35..,. NTODUCTON The theory o vibration o a structure immersed in a luid is important or understanding acoustic phenomena such as scattering and radiation rom ships and submarines, and the eect o structures on sonar signals and sonar sel noise. Some useul concepts are ound by just considering a plane acoustic wave incident onto a lat plate immersed in a luid, then use well-known relations or acoustic relection and transmission at the plate luid surace. The latter relations need the densities and sound speeds or the luid and plate materials, plus plate thickness, Young s modulus and Poisson ratio. These parameters are suicient or modeling longitudinal (compression) waves in the luid and plate, and shear waves in the plate. Consideration o leural (i.e. bending) waves is oten omitted, motivating the theory o this paper enabling leural wave eects to be included into a convenient but artiicial model o plate acoustic relections. There are treatments o the problem o plane wave acoustic ecitation o ininite thin plate leural waves in tet books (,, 3), but they do not speciically discuss energy densities and energy density lues. Usually it can be taken or granted that energy is conserved in the solutions o wave equations, however or artiicial models such as acoustically hard plates conservation o energy must be imposed separately. This paper deines a more general artiicial plate model, where idealized acoustically hard and sot plates are special cases, and derives constraints on the model parameters rom conservation o energy. This paper applies previously derived energy conservation relations () to derive in Section. the amplitudes and phases o acoustically ecited thin plate leural waves while taking into account luid loading. Since all waves are travelling waves, luid loading in this case is just radiation damping o the leural waves. The eect o leural waves on the plate acoustic relectivity and the near acoustic ield close to a plate are then derived in Sections. and. respectively. Section.3 darryl.mcmahon@deence.gov.au nter-noise Page o
2 shows that close to the resonance condition, where the trace phase speed o acoustic waves over the plate equals the phase speed o leural waves, energy conservation does not allow a plate to be acoustically hard.. FOCED VBATON OF A PLATE DEVED BY AN ENEGY CONSEVATON METHOD. Forced vibration o an ininite thin plate in a luid by an eternal acoustic plane wave n a previous paper () the ollowing energy conservation equation is derived: U(,t ) (,t ) U F (,t ) WF (,t ) WEF (,t ) WPF (,t ) WEF (,t ) () t t where U(, is the sum o kinetic and potential energy densities in the plate, U F (, is the added energy density o the plate luid system rom the added mass o the luid, (, is the energy density lu in the plate, (, is the energy density transer rate between the plate and the luid, W F and W EF (, is the energy density transer rate between the plate luid system and an eternal orce. Formulae or these unctions in terms o the plate wave amplitude are derived in reerence (). W PF (, is the energy density transer rate, either energy lost or gained rom the plate luid system, which is nonzero only when it is orced to conorm to a particular waveorm that is not a plate luid natural mode (). The HS o eqn. () is zero because we are considering the equilibrium situation o an energy density transer rate rom an eternal orce eactly balanced by the energy density transer rate o the plate luid system. Consider the eternally applied energy density transer rate (, needed to orce plate luid waves to satisy eqn. () (i.e. WEF (, WPF (, ) supplied by an eternal acoustic plane wave. ncident and specula relected acoustic waves bend the plate such that plate acceleration creates a pressure change that partially cancels these two eternal acoustic pressure sources. This pressure change is the luid loading eect contained in W PF (,. Consequently leural waves displace the acoustic relectivity o the plateluid system away rom non-bending plate-luid specula relectivity. What proportion o the energy o an incident acoustic wave that is converted into plate leural wave energy depends on plate stiness D, mass per unit area M and how much o the incident acoustic energy is absorbed and relected by internal plate compression and shear mechanisms. n general the incident acoustic energy gets distributed into specula relected energy, energy absorbed into plate material longitudinal and shear waves, plate leural waves that radiate acoustic energy back into the luid, and plate leural waves that propagate energy along the plate luid surace. An acoustic plane wave rom ar ield (i.e. large z) and incident onto a thin plate has a real wavenumber vector k, kz, k, kz 3. The specula relected wave has the same k, k, kz but k z. The acoustic wave generated by the leural wave, equivalent to luid loading, must also have real wavenumbers equal to that o the incident wave in the direction and opposite sign in the z direction. The comple incident, specula relected and leural wave generated acoustic wave pressures are given by respectively p (, p epik ikz z it i (a) ps (, p epik ikz z it i (b) pf (, p ep ik ikz z it i (c) where pressure amplitudes p, p and phases, distinguish the origin and directions o incident, specula relected and plate leural wave generated acoustic waves. For simplicity the luid is only on one side o the plate, and the other side is a vacuum. Five natural non-leaky and leaky leural wave modes were identiied in reerence (5) or a thin plate in a luid. The solution to any orced vibration problem implies deviations rom these natural modes (see reerences (6) and (7)). 3 Any comple number q is written as q q i q to distinguish the real part q rom the imaginary part q. nter-noise Page o W EF
3 For later brevity we deine a specula relection coeicient by p (3a) p and specula relection phase shit by (3b) We assume that the eternal pressure rom incident and specula relected waves contribute equally to causing plate bending (i.e. leural) waves, and the internal pressure o waves within the plate material do not directly contribute to plate bending waves. So a proportion o the incident acoustic wave pressure contributes to plate bending waves, and specula relection, also a proportion o the incident acoustic wave pressure, also contributes to plate bending waves albeit with a phase change. The proportion o the incident acoustic wave pressure drives compression and shear waves within the plate material leading to complete energy absorption within the material. Commonly used but somewhat artiicial cases are acoustic hard materials with (hence bending waves are the only energy loss mechanism) and acoustic sot materials with (hence no bending waves are ecited and all energy is absorbed by the plate material). The eternal acoustic pressure p D (, at z driving the plate leural wave is i, p (,, ps (,, e p i e ik it i pd( (,, () p ep The parameter is used to keep track o the contribution o the specula relected acoustic wave to driving leural waves, and their eect on the net relectivity o the plate and net pressure at the plate surace. t is implicit that. p D (, can be rewritten in terms o an amplitude p and phase deviation rom by i pd (, pe epik it i (5a) where p p cos sin p cos (5b) i i e e, (5c) cos Another, later useul, orm o eqns. (5a, b, c) introduces a variable related to phase by p cos cos, (6a) p p sign (sin ) sin sin, (6b) p a denotes the positive square root o any number a so the sign is given eplicitly. The phase is nonzero only i n, n,,... where or an acoustic hard surace and or an acoustic sot surace. The plate leural wave displacement (, is given by (, epik it i (7) (, is deined as positive or a plate displacement in the z direction. Equations (37a) and (37b) o reerence () relate p and to and by p M a D k (8a) kz Energy dissipation within the plate material needs to eist i since a vacuum is present on the ar side o the thin plate. nter-noise Page 3 o
4 (8b) k k (8c) z k Here k /c is the acoustic wavenumber in the luid with phase speed c, k M / D is the plate vacuum leural wavenumber, M is the plate mass per unit area and D its bending stiness, and is the luid density. The plate leural response to a plane wave acoustic pressure can only be to radiate, so the added mass is only the imaginary part im a, and is the ratio M a / M. t remains to determine, p, and rom p and by calculating the work rate W EF (, o the driving pressure wave p D (, on the plate and equating it to (,. Below we use rom reerence () a phase,t or energy density and lu variations where, t ( k W EF (, is given by W PF (9) * * p p WEF (, (, pd (, D D () The minus sign in eqn. () arises rom the acoustic orce o the driving pressure being in the opposite direction z to plate positive velocity direction z. From eqns. () and () ( ) ( ) W (, W (, W (, (a) ( ) WEF, p cos sin, t ( ) WEF, p cos sin, t EF EF EF cos, t sin ( ( ) W EF cos, t sin ( where (, are the incident and specula relected wave parts o the eternal acoustic work rate or plate bending. The minus sign in eqn. () is absorbed by in the phase terms o eqns. (b, c). (b) From eqn. (5b) o reerence (), noting that by k z, D WPF (, k k sin, t k cos, t (),t leading to The HS o eqns. (a, b, c) and () must be equal or all phases k k p cos k p sin (c) D cos (3a) D sin (3b) Using eqn. (3 b), can be eliminated rom eqns. (3a, b) giving k k p cos k p cos( ) sin D sin sin sin( ) cos we ind k k cos k cos D sin (3c) D sin cos (3d) When eqns. (3c, d) are solved or and p sin cos sin cos k k sin cos sin D cos p From eqns. (3e, ) and (5b) the plate leural wave amplitude is k (3e) (3) nter-noise Page o
5 D and the phase can be determined rom tan k k k cos k sin k k cos k k sin k Equation (a) shows that reaches a maimum at k k which is a resonance that does not diverge owing to damping by radiation rom the plate into the luid. This resonance is only possible at requencies above the coincidence requency deined at the condition k k p. For brevity deine dimensionless and always positive parameter, which is equivalent to a phase, by k sin, k k k k k k (a) (b) (5a) k k sign ( k k ) cos, (5b) From eqns. (8a) and (a) p p sinp (6) From eqns. (6a, b), (a) and (5a, b) we ind that eqns. (3e, ) become sin sign( k k ) sign(sin ) sin (7a) cos sign( k k ) sign(sin ) cos (7b) Eqns. (7a, b) show that the phase dierence between plate leural waves and the incident acoustic wave satisies n, n,,,...,, (8a) Combining eqns. (8a) with (8b) we obtain the phase dierence between acoustic radiation by plate leural waves and the incident acoustic wave 3 n, n,,,...,, (8b) The phase o the driving pressure is so using eqn. (8b) the phase dierence between the leural wave acoustic ield and the driving ield is 3 n, n,,,...,, (8c) At the resonance p p rom eqn. (6) so we ind the leural wave generated acoustic ield amplitude at the plate surace equals the acoustic wave amplitude driving the plate leural waves. Also at the resonance by eqn. (5a), / and by eqn. (8c). Hence the leural wave driving ield and acoustic ield rom the plate have opposite phase and so cancel eactly. There remains at the surace only the component o the acoustic ield that drives transmission and absorption by the plate material. Our results derived rom energy transer rate considerations should agree with results or the acoustic ecitation o leural waves or luid loaded plates derived by other methods. The leural wave mobility is a convenient comparison unction and this is ound to be i (, i e (9a) p (,, D k k i k Substituting or k, k and, eqn. (9a) can be rearranged to i (, cos e c ik h / cos p (,, sin / s c (9b) nter-noise Page 5 o
6 where is the angle o the incident wavenumber vector to the plate normal, s is the plate density and h the thickness, and c c M / D c where c is the coincidence requency. For c reerence () eqn. (6.9) agrees with eqn. (9b) or, (acoustically hard plate) and (specula relection contribution). eerence () eqn. (6.7) would lead to eqn. (9b) or any ecept the denominator o eqn. (6.7) has an error 3 instead o / c. / c. Eect o leural waves on acoustic absorption and relection by an ininite thin plate in a luid We seek ormulae or leural wave contributions to acoustic energy absorption and relection. Denoting the energy density transer rate o the incident eternal acoustic wave onto the plate as (,, and denoting W (, as the relected energy density transer rate, then W (, W (, W (, is the energy density transer rate injected acoustically into the plate luid system. (, includes both an acoustic wave part W F (, generated by plate leural waves and a specula relected acoustic wave part W S (,. Similarly W E (, consists o a leural wave part (, and a plate material acoustic absorption part W EP (,. The acoustic absorption coeicient E o the plate luid system is deined rom random W / W W W / W and the acoustic relection phase averaged energy density transer rates by E E coeicient is E W EF W / W. We deine leural wave and plate material acoustic absorption coeicients EF W EF / W and EP W EP / W respectively. From eqns. (a, b, c) and (a) using phase relation eqn. (8a) kz cos WEF (, p sin(, cos(, t () c k sin The acoustic energy density rate incident onto the plate is phase sensitive and depends on,t and giving kz W (, p sin sin(, cos cos(, t (a) c k From eqns.(7a, b), is related to and eqn. (a) becomes kz W (, p sin sin(, cos cos(, t (b) c k Comparing the HS o eqns. () and (b) we see that W EF (, and (, are not in phase with each other which makes the ratio WEF (, / W (, phase sensitive. Consequently random phase averaged energy rates are used to deine EF. From eqns. () and (a, b) kz kz WEF p p (a) c k c k kz W p (b) c k The coeicient EF EF or the average acoustic input energy density transer rate into leural waves is W p EF p EF cos (3) W p p is the sum o two parts EF EF EF (a) where EF is an incident acoustic contribution and EF is a specula relected wave contribution. These are easily distinguished by the terms that depend on and give EF (b) E W W W nter-noise Page 6 o
7 EF cos (c) The irst term on the HS o eqn. (c) is rom the specula relected generated leural wave, and the second term is the eect o coherent intererence o the incident wave generated and specula relected generated leural waves. To derive it is irst necessary to calculate the coherent sum o specula relected and plate leural wave generated wave pressures. From eqns. (b, c) the pressure or the relected wave is p(, ps(, pf (, (5a) Using eqns. (5a) and (b, c) a relected wave amplitude p and phase are determined rom p(, p epik ikz z it i p epik ikz z it i (5b) p epik ik z it i z p p p cos( ) p p (5c) Since by eqn. (8b) p p p sin( ) p p p p sin( ) p p (5d) From phase relations eqns. (3b) and (8a) we obtain sin( ) sin( ) which can be used in eqn. (5d) together with eqns. (6), () and (3a) to give p p p p p sin p sin / (5e) This shows that the net relected wave is determined by specula relection, the acoustic radiation rom leural waves in the plate, and coherent intererence o the specula relection with the radiation o the leural wave due to incident and specula relected acoustic sources. Using eqns. (6) and (5b) p p p p cos p (5) Then eqn. (5e) becomes p p cos sin p p sin p p p p p sin (5g) The parameter allows easier identiication o 6 dierent contributions to the relected acoustic wave. The irst 3 HS terms o eqn. (5g) are specula relection, acoustic radiation rom the leural wave generated by the incident wave and coherent intererence o specula relection with leural wave acoustic radiation generated by the incident wave. The last 3 HS terms are rom acoustic radiation o the leural wave generated by the specula relected wave, coherent intererence o leural wave acoustic radiation generated by the specula relection with the specula relection, and coherent intererence o specula relection ecited leural wave acoustic radiation with the incident wave ecited leural wave acoustic radiation. The acoustic energy density transer rate relected rom the plate is W (, that can be derived in terms o p, and plate phase similar to eqn. (a) and the result is kz W (, p sin sin(, cos cos(, (6a) c k The random phase averaged relected energy density transer rate is kz W p (6b) c k Then using eqns. (b), (6b), (6) and (5g) the relectivity coeicient is W p cos ( ) sin sign k k W p (7a) cos sinsin sin Setting, eqn. (7a) reduces to ( ) sin sign k k sinsin (7b) nter-noise Page 7 o
8 Comparing eqn. (7b) with eqn. (7a) we see that leads to cancellation o most leural wave terms leaving only the leural wave contribution rom the intererence o the incident wave induced leural wave acoustic ield with the specula relected ield. This leural wave eect on the relectivity only eists i sin and (i.e. not eactly at the resonance condition). nterestingly the idealized acoustically hard and sot plate models with sin have no leural wave eect on the relectivity..3 Energy conservation constraints on acoustic ecitation o leural waves o an ininite thin plate in a luid By energy conservation, the average rate o work o the incident acoustic wave on plate leural waves cannot eceed the average energy lu contained in the incident acoustic wave. This requires then,and rom eqn. (3) gives EF Setting cos (8) (i.e. no specula relection contribution to leural wave ecitation) eqn. (8) is always satisied since and. Setting, when there is no problem satisying eqn. (8) or any and any cos. However or near the resonance the middle unction in eqn. (8) would be greater than or suiciently large and cos. This leads to an interesting conclusion or understanding acoustic relection rom a plate when the leural wave resonance is taken into account. Specula relection can create a higher pressure amplitude at the plate surace than in the incident wave, so it might seem the rate o work by this higher pressure on plate bending waves could eceed the power available rom the incident wave unless the parameters and cos are constrained to satisy eqn. (8). This implies that specula relection must be signiicantly modiied at the leural wave resonance condition even i the plate is acoustically hard away rom the resonance 5. Setting then eqn. (8) requires cos (9) Equation (9) eliminates the possibility o an acoustic hard plate deined by,cos cos were still possible at the resonance, then is constrained by (3) that represents a airly sot acoustic relection by a signiicant energy loss mechanism within the plate material. there is no signiicant energy loss mechanism within the plate, then and eqn. (9) becomes cos (3) Eqn. (3) is realistic or a thin plate with a vacuum on one side since it approimates a pressure release (i.e. acoustic so surace. For the ideal acoustic sot plate deined by cos so that EF since the incident and specula relected acoustic ields cancel and do no net work on plate leural waves. Consider another constraint which requires the relected acoustic energy lu to not eceed the incident acoustic energy lu. This is easily satisied or weak specula relections giving. However or no signiicant energy losses in the plate, and eqn. (7b) requires. 5 An acoustically hard surace, makes the leural wave driving pressure twice the incident wave pressure. eal suraces that approimate this model over some requency range are used or sonar signal enhancement at a sensor close to a signal conditioning plate. t might seem the average work rate on the sensor can be our times the average incident wave energy rate, but a typical sensor is almost acoustically transparent such that energy conservation is assured. However i the plate is itsel a sensor to detect signal induced vibrations, energy conservation limits and making an acoustically hard surace impossible near the resonance condition. nter-noise Page 8 o
9 sin sin (3) Using eqn. (5b), eqn. (3) shows that ( k )sin( ) (33) At the resonance k k, k, sin, so eqns. (3) and (33) do not constrain sin sin( ) then sin( ) must have opposite signs just above and below the resonance, sign (sin( )) above the resonance requency, and sign (sin( )) below the resonance requency. Also or eqn. (3) rules out, so there must be a discontinuity in at the resonance where. This again shows that specula relection and leural waves must be linked to get consistency with energy conservation near the resonance condition. A more detailed analysis o specula relection rom thin plates in a luid, with a vacuum on one side, using more detailed acoustic and elasticity theory is needed to check the conclusion that a plate must be acoustically sot near the resonance condition. the width o the resonance rom radiation damping is quite narrow, the sotness eect may not have been previously noticed eperimentally. n many practical cases the requency and direction o incident acoustic waves are well away rom the resonance condition enabling almost any level o acoustic hardness / sotness o the plate.. Acoustic ield close to the wavenumber resonance condition The total acoustic ield at the plate surace z is given by (see eqns. (a, b, c)) p (, p (,, p (,, p (,, p (,, p (, p (,, T S F i i (3) p pe sin e epik it i At the resonance condition /, eqn. (3) shows that the acoustic ield p D (, rom the incident and specula relected waves driving the leural waves is eactly cancelled by the acoustic ield produced by the leural waves, leaving only the acoustic ield component p absorbed as compression and shear waves in the plate material. Consider the requency and wave directions or the resonance condition. Equating the incident wave trace wavenumber on the plate surace to the plate-vacuum leural wavenumber k k we ind the resonance condition or angle is sin / c D F.. Hence the resonance condition is only physically possible or, although the requency width o the unction rom radiation damping (see (5a)) c causes some eect at lower requencies. For k k, can be approimated by k / k k k / (35a) showing the approimate resonance wavenumber width is k k / (35b) Using the previously derived ormula or (see Section.) this width at the resonance is k tan (35c) h s 3. SUMMAY This paper has analysed theoretically acoustic wave ecitation o thin plate leural waves taking into account luid loading causing radiation damping. An acoustic plane wave ecites a leural wave in an ininite thin plate with a requency and direction dependent amplitude and phase given by eqns. (a, b). The acoustic ield generated by the leural wave has a phase that tends to cancel the incident and specula relected acoustic ields driving the leural wave. The maimum cancellation eect occurs at a resonance condition where the trace acoustic phase speed along the plate equals the phase speed o plate-vacuum leural waves. Near the resonance condition, energy conservation requires that the plate specula relection be acoustically sot ruling out the possibility o an ideal hard plate in this case. This eample shows that solutions o the wave nter-noise Page 9 o
10 equation can be unphysical or artiicial acoustic material assumptions. More detailed acoustic analysis using realistic plate material properties should be able to relate the parameters and and demonstrate their consistency with energy conservation eqns. (8) and (33). EFEENCES. Fahy F, Gardonio P. Sound and Structural Vibration: adiation, Transmission and esponse. nd edition. Elsevier (a Knovel ebook); 7. Chapter 6.. Cremer L, Heckl M. and Ungar E. E. Structure Borne Sound. nd edition. Springer-Verlag; 988. Chapter Skelton E.A. and James J.H. Theoretical Acoustics o Underwater Structures. mperial College Press; 997. Chapter 8.. McMahon D. Acoustic and leural wave energy conservation or a thin plate in a luid, Proceedings o nter-noise, Melbourne Australia, 6-9 November. 5. Crighton D. G. The ree and orced waves on an ininite thin luid-loaded elastic plate. J. Sound Vib. 979; 63: p Feit D, Lui Y. N. The nearield response o a line driven luid loaded plate. J. Acoust. Soc. Am. 985; 78: p Chapman C. J, Sorokin S. V. The orced vibration o an elastic plate under signiicant luid loading. J. Sound Vib., 5; 8: p nter-noise Page o
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