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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 473531, 13 pages doi:10.1155/2012/473531 Research Article 1 2 3 Eldad J. Avital, Theodosios Korakianitis, and Touvia Miloh School of Engineering and Materials Science, Queen Mary University of London, London E1 4NS, UK Parks College of Engineering, Aviation and Technology, Saint Louis University, St. Louis, MO 63103, USA Faculty of Engineering, Tel-Aviv University, 69978 Tel Aviv, Israel Correspondence should be addressed to Eldad J. Avital, firstname.lastname@example.org Received 10 April 2012; Revised 9 August 2012; Accepted 10 August 2012 Academic Editor: Rama Bhat Copyright © 2012 Eldad J. Avital et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Sound wave scattering by a ﬂexible plate embedded on water surface is considered. Linear acoustics and plate elasticity are assumed. The aim is to assess the eﬀect of the plate’s ﬂexibility on sound scattering and the potential in using that ﬂexibility for this purpose. A combined sound-structure solution is used, which is based on a Fourier transform of the sound ﬁeld and a ﬁnite-diﬀerence numerical-solution of the plate’s dynamics. The solution is implemented for a circular plate subject to a perpendicular incoming monochromatic sound wave. A very good agreement is achieved with a ﬁnite-diﬀerence solution of the sound ﬁeld. It is shown that the ﬂexibility of the plate dampens its scattered sound wave regardless of the type of the plate’s edge support. A hole in the plate is shown to further scatter the sound wave to form maxima in the near sound ﬁeld. It is suggested that applying an external oscillatory pressure on the plate can reduce signiﬁcantly and even eliminate its scattered wave, thus making the plate close to acoustically invisible. A uniformly distributed external pressure is found capable of achieving that aim as long as the plate is free edged or is not highly acoustically noncompact. 1. Introduction Resonance frequency calculations were carried out using the Raleigh integral and Bessel function series expansions. This work looks at sound scattering by a ﬂexible plate embed- ¨ ¨ Mellow and Karkkaien  looked at the case of a circular ded on the surface of calm water. The latter is a simpliﬁed membrane embedded in an inﬁnite baﬄe. Good agreement model for ﬂexible structures embedded on free surface such was found between a Bessel series solution and a ﬁnite as manmade platforms which were investigated previously element model (FEM) as long as the latter was well resolved. for their interactions with gravity surface waves, for example, The acoustic pressure ﬁeld and the impedance of the [1–3]. Interest resides with the interaction between an membrane were calculated. The case of an inﬁnite elastic incoming sound wave and the structural dynamics of the plate was with dealt by Johansson et al. , seeking a plate embedded over the free surface and particularly how governing equation that involves only the acoustic pressure. it aﬀects the near and far sound ﬁelds. Examples for the They found two possible equations, one for the axisymmetric signiﬁcance of such interaction can be found in marine case and another for an antisymmetric case. Smith and engineering, for example, ships and oﬀshore structures, as Craster  developed numerical and asymptotic methods well as in aeronautical, mechanical, and nuclear engineering based on the Fourier expansions for sound scattered by . Further interest resides in the ability of the ﬂexibility elastic plates embedded in rigid baﬄes. Further solutions for of the thin structure to dampen sound scattering, whether circular and rectangular plates with various kinds of edge naturally, that is, with no external intervention or by adding support can be found in Junger and Feit . some external controlled load acting on the plate. The case of a plate embedded between two immiscible Sound generated by plate vibrations has been mainly ﬂuids of diﬀerent phases has been less investigated and, when investigated in relation to loud speaker applications. For it was, it has been mostly for incompressible ﬂuids. Kwak and example, Suzuki and Tichy  looked at sound generated by Han  used a Fourier-Bessel series expansion to investigate a circular plate supported on its edge by an inﬁnite baﬄe. the eﬀect of an incompressible ﬂuid depth on the vibration of 2 Advances in Acoustics and Vibration a circular plate resting on a free surface. Good agreement was investigation; the ﬁrst is in studying the sound scattering, found with experiments showing that the ﬂuid depth could and the second is in examining the potential ability of be neglected if it were larger than the plate’s diameter. Zilman diminishing or at least reducing the scattering using a simple and Miloh  derived a nondimensional stiﬀness parameter distribution of an external load acting on the plate. Interest to show the thresholds for a circular ﬂoating plate to behave resides with the eﬀect of the plate’s ﬂexibility, utilizing it to as a ﬂexible mat or a rigid body under the eﬀect of surface reduce the sound scattering as in the approach of Avital and waves. Khabakhpasheva and Korobkin  investigated the Miloh , while also looking at the eﬀects of the type of the hydroelastic behaviour of plates modeled as an Euler beam plate’s edge support and an impurity in the plate modeled whether ﬂoating or attached to the sea bottom by springs. as a hole. This is a mixed boundary problem, and a method Sturova  studied analytically the eﬀect of surface wave of solution for the general case when based on the Fourier pressure on ﬂexible rectangular plates, ﬁnding wave guides transform is discussed in the next section. Special intention for elongated plates. Bermudez et al.  considered the case is given for the circular plate that is still of an engineering of a rectangular plate placed as a lid over a cavity ﬁlled with interest , while presenting a simpliﬁed geometry that compressible ﬂuid. FEM was used to calculate the resonance includes3Deﬀects. Results are analyzed for the eﬀects of the frequencies. A cavity sealed by a ﬂexible rectangular plate incoming sound frequency and the type of the plate’s edge was also considered by Leppington and Broadbent . The support, followed by a summary. eﬀects of a plane wave coming from outside of the cavity and a point sound source located within the cavity were analyzed. 2. Mathematical and Numerical Formulation Jeong and Kim  investigated the case of a circular plate separating two compressible ﬂuids inside a cylindrical Both linear acoustic and linear plate dynamics are consid- container. Both FEM and the Bessel series expansions were ered. The plate is assumed to be thin and of an isotropic used to yield good agreement. The compressibility eﬀect on and homogenous material. It is taken as embedded on the the plate’s motion was studied, and it was found that as the interface between water and air; both are assumed to be at plate approached the top or the bottom of the container the rest. The air’s acoustic impedance is very low as compared resonance frequencies decreased signiﬁcantly. to the water’s impedance, and thus the air-water interface Recently interest has grown in investigating the possi- is taken as a free surface, that is, a zero-pressure surface. A bility of acoustic invisibility of an object, meaning reducing monochromatic incident plane wave is assumed mimicking signiﬁcantly the scattered sound wave ﬁeld in order to dimin- the case of a sound source far from the plate. The water’s ish detection as well as improving structural integrity by depth is taken as inﬁnite as the source of the incident sound reducing the eﬀect of acoustic fatigue on nearby structures. wave is assumed to be in the far sound ﬁeld. Traditionally, invisibility or cloaking has been associated with Thereare variousapproaches to computesound waves electromagnetic waves, but interest has expanded towards reﬂected by rigid and elastic structures. They range from acoustics. The analogy between the acoustic equations to analytical methods based on eigenfunction expansions to Maxwell equations for a two-dimensional geometry can numerical ones based on ﬁnite diﬀerence, ﬁnite and bound- be used to derive a coordinate transformation needed to ary element, and source substitution methods . In design an acoustic cloaking coat of an anisotropic mass this study, we will concentrate on a solution based on a density. Zhang et al.  used a similar approach to design Fourier transform for the sound ﬁeld as coupled with a a 2D acoustic cloaking structure based on a network of ﬁnite diﬀerence solution of the plate’s dynamics. A similar small cavities connected by narrow channels. Water tank approach was already used successfully to investigate sound experiments showed a reduction of about 6 dB in the scattering by vertical cylinders  and compute the sound scattered ﬁeld for ultrasound waves impinging on a small generated by the interaction between water ﬂow and vertical scale cylinder of 1.35 cm radius. Active cloaking by point cylinders . A secondary solution based on a ﬁnite sources located outside of the cloaked region was also diﬀerence scheme in the time-space domain of the sound studied and demonstrated numerically for two-dimensional ﬁeld will be employed in this study for veriﬁcation purposes. acoustics . This required exact knowledge of the incident Further details on this solution are given in the appendix. wave. Another type of active cloaking was also studied by Avital and Miloh  who considered an approach where 2.1. A General Fourier-Transform Approach. Amonochro- the ﬂexibility of a cylindrical shell was utilized to search for matic incoming plane sound wave propagating towards the possible conﬁgurations of an external load acting on the shell free surface is assumed, and thus the acoustic pressure can be that would result in the demise of the scattered waves. Exact iωt taken as p(x, y, z, t) = P(x, y, z)e ,where i is the square root solutions were derived, while the possibility of a discrete load of −1and ω is the incoming sound wave’s frequency. Here x distribution was also discussed. and y are the rectangular horizontal directions and z is the A thin plate embedded on calm water surface is con- vertical direction pointing downwards, so z = 0 is the plane sidered in this study. Only compressibility eﬀects, that is, of the free surface (see Figure 1 for illustration). This leads to acoustic waves will be considered. The plate is taken as the following Helmholtz equation as the governing equation embedded on the free surface and thus the problem of gravity for the sound ﬁeld: waves can be linearly divorced from the acoustic problem ∂ P for calm water surface, for example, Avital and Miloh  2 2 (1) ∇ P + + k P = 0, h 0 and Avital et al. . There are two objectives for this ∂z Advances in Acoustics and Vibration 3 where Circular plate Free i(k x+k y) ik z x0 y0 zo surface P = e e , (7) incident Scattered sound i(k x+k y) iγ x+iγ y+sz x0 y0 x y wave P = e A γ , γ e dγ dγ , (8) scattered x y x y −∞ k and k are the possible horizontal waves numbers of the x0 y0 incident sound wave indicating an oblique propagation as relative to the free surface. They fulﬁll the relation: Incoming plane sound wave 2 2 2 2 k + k + k = k . (9) x0 y0 z0 0 The scattered wave’s vertical variation coeﬃcient s is Figure 1: Schematic description of the problem. 2 2 ⎨ 2 2 2 2 − γ + γ − k , γ + γ >k x y 0 x y 0 s = (10) 2 2 ⎩ 2 2 2 2 −i k − γ − γ , γ + γ <k , 0 x y x y 0 where k ≡ ω/c and c is the water’s speed of sound. 2 2 2 2 2 The operator ∇ P is ∂ P/∂x + ∂ P/∂y for rectangular where γ ≡ k + γ , γ ≡ k + γ . Expression (10) for the x x0 x y y0 y coordinates. The boundary condition for P on the free vertical variation coeﬃcient s of the scattered wave fulﬁlls surface is the radiation condition in the vertical direction z, showing that high wavenumbers of γ and γ are nonradiative, that is, P x, y, z = 0 = 0, x, y ∈ outside of the plate, (2) x y decay exponentially in the z direction, and low wavenumbers are radiative. and on the plate (see ), The problem comes down to ﬁnding the distribution of ∂P the Fourier modes A(γ , γ ) by making the acoustic pressure 2 x y = ρω W x, y , x, y ∈ on the plate, (3) ∂z P fulﬁll the boundary condition on the plate equation (3) z=0 and the boundary condition on the free surface equation (2). where ρ is the water’s density and the plate’s deﬂection w was There are several approaches of ﬁnding this Fourier mode iωt expressed as w(x, y, t) = W(x, y)e . distribution, and a general method will be communicated in Following Smith and Craster , the governing equation a followup study as discussed in the Section 4. The essence for the plate’s deﬂection is taken as of the present study is to assess whether the ﬂexibility of the plate has an eﬀect on the sound scattering and if there 4 2 D∇ W − ρ hω W = P + F. (4) is a potential to use that eﬀect to manipulate the scattering to a desired situation. Thus a solution that is rather of a The above equation holds for linear structural dynamics, and forward simple approach is used in this study. Firstly, a the reader is referred to Smith and Craster and Gratt common situation is that the sound emitter of the wave for further details on its applicability in structural acoustics propagating towards the plate-free surface and its associated 4 2 2 2 2 and dynamics. The operator ∇ W is (∂ /∂x + ∂ /∂y ) W sound receiver are located at the same place, thus that wave for rectangular coordinates. The plate’s bending stiﬀness D will be emitted as perpendicular to the plate in order to is maximize the reﬂection of the wave towards the sound emitter-receiver. This means that the horizontal incoming Eh D = . (5) wavenumbers k and k are zero. 2 x0 y0 12(1 − ν ) Secondly, the form of the scattered wave is of importance, E is Young’s modulus, h is the plate’s thickness, ν is the that is, whether it propagates in two dimensions as a cylindrical wave or in three dimensions capable of becoming Poisson’s ratio, and ρ is the plate’s density. The external load f acting on the plate was taken as f (x, y, t) = a spherical wave. Most of the free-surface thin structures are iωt ﬁnite in both horizontal directions; thus a three-dimensional F(x, y)e in (4). There are three possible types of plate edge propagation has to be allowed in the solution. The simplest conditions; clamped, hinged, and free. Clamped condition means no deﬂection and zero slope of the deﬂection. Hinged form of thin structure to achieve this kind of sound propagation is the circular plate, a form that is still of an condition means no deﬂection and no bending moment, and engineering interest . This means that the sound ﬁeld a free edge condition means no bending moment and no shear force. Since the mathematical formulation for these becomes axisymmetric and the Fourier transform in (8) can be replaced by a Bessel-Fourier transform. This is the conditions depends on the plate’s geometry, that formulation will be given in the implementation section for the circular approach presented in the following. plate. The solution of the acoustic pressure P for the Helmholtz 2.2. The Circular Plate Subject to a Perpendicular Incoming equation (1) can be expressed as Sound Wave. A ﬁnite number of the Bessel-Fourier modes will be computed by requiring the solution to comply with P = P + P , incident scattered (6) the pressure boundary condition on the plate and free 4 Advances in Acoustics and Vibration 3.5 1.5 2.5 1.5 0.5 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 Plate r (m) r (m) Plate (a) (b) Figure 2: Zoomed views on the contour levels of the pressure amplitude as were calculated by the spectral and time-marching scheme approaches in the presence a rigid ﬂoating plate of 1 m radius and an incoming sound wave of 3000 Hz. The computational domain is stretched up to r = 10 m and z = 10 m. There are nine contour plots ranging from 0.5 to 4.5, where the solid lines denote the spectral solution, and the dashed lines denote the time-marching solution. surface at a ﬁnite number of points. Thus the Bessel-Fourier propagating in the opposite direction of r in order to adjust transform can be written as a series as in (11) the pressure over the plate for the existence of an inﬁnite free surface. Solution (11) allows such waves, and again the computational domain radial length R has to be taken of ik z s z 0 m (11) P = e + A J γ r e , m 0 m suﬃcient length in order to account with suﬃcient accuracy m=0 for the eﬀect of the free surface on the pressure ﬁeld over the plate. where r is the radial direction. γ (R) can be seen as the radial are determined by requiring the The coeﬃcients A wavenumbers, and they are functions of the computational m acoustic pressure P to comply with the boundary conditions domain radial length R as explained after (12), where γ m=0 at z = 0, (2)and (3). Dividing the computational domain = 0. J (γ r) is the Bessel function of the ﬁrst kind. The 0 m side at z = 0to M + 1 segments then A for m = 0··· M can exponential coeﬃcient s of the z direction is to fulﬁll the m be determined by complying the boundary conditions at the same conditions as in (10) leading to mid points of those segments. At the free surface, this yields ⎨ 2 − γ − k , γ >k m 0 m 0 s = (12) −i k − γ , γ <k . m 0 0 m A J γ r +1 = 0, a<r <R, (13) m 0 m n n To ensure series convergence by the Strum-Liouville the- m=0 ory, the radial wavenumbers are determined by taking dJ (γ r)/dr = 0 at the radial edge of the computational 0 m domain r = R . Taking R →∞ will make the Bessel- and at the plate’s surface, Fourier series in (11) a Bessel-Fourier integral if the mode A is rewritten as A γ . However in reality R is taken as ﬁnite m m and it has to be much larger than the plate’s radius a to reduce eﬀects of computational domain edge conditions. In a full A s J γ r + ik = ρω W , r <a, (14) 3D problem, this will cause a signiﬁcant computational cost, m m 0 m n 0 n n m=0 but for our axisymmetric problem aimed at assessing the validity of using the plate’s ﬂexibility on the sound scattering, this computational cost is feasible. It should be also noted that the free surface is a perfect sound wave reﬂector and where a is the radius of the plate. In case of a central hole in it stretches to inﬁnity. Thus one should expect sound wave the plate then (13) holds on the free surface inside that hole. z (m) z (m) Advances in Acoustics and Vibration 5 |P| |P| 10 10 1.9 1.9 1.7 1.7 8 8 1.5 1.5 1.3 1.3 6 6 1.1 1.1 0.9 0.9 4 4 0.7 0.7 0.5 0.5 2 2 0.3 0.3 0.1 0.1 0 0 2 4 6 8 10 2 4 6 8 10 r (m) r (m) (a) (b) |P| |P| 10 10 1.9 1.9 1.7 1.7 8 8 1.5 1.5 1.3 1.3 6 6 1.1 1.1 0.9 0.9 4 4 0.7 0.7 0.5 0.5 2 2 0.3 0.3 0.1 0.1 0 0 2 4 6 8 10 2 4 6 8 10 r (m) r (m) (c) (d) |P| 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5 0.3 0.1 2 4 6 8 10 r (m) (e) Figure 3: Contour plots of the pressure amplitude as were calculated for (a) rigid plate and aluminium plates of (b) 5 cm thickness and hinged edge, (c) 5 cm thickness and clamped edge, (d) 5 cm thickness and free edge, and (e) 5 mm thickness and hinged edge. The incoming sound wave frequency is 3000 Hz, and the rest of the conditions are as in Figure 2. z (m) z (m) z (m) z (m) z (m) 6 Advances in Acoustics and Vibration |P| |P| 10 10 1.9 1.9 1.7 1.7 8 8 1.5 1.5 1.3 1.3 6 6 1.1 1.1 0.9 0.9 4 4 0.7 0.7 0.5 0.5 2 2 0.3 0.3 0.1 0.1 0 0 2 4 6 8 10 2 4 6 8 10 r (m) r (m) (a) (b) Figure 4: Contour plots of the pressure amplitude as were calculated for a hinged plate of 5 cm thickness and sound frequency of (a) 1500 Hz and (b) 6000 Hz. The rest of the conditions are as in Figure 2. |P| |P| 10 10 1.9 1.9 1.7 1.7 8 8 1.5 1.5 1.3 1.3 6 6 1.1 1.1 0.9 0.9 4 4 0.7 0.7 0.5 0.5 2 2 0.3 0.3 0.1 0.1 0 0 2 4 6 8 10 2 4 6 8 10 r (m) r (m) (a) (b) Figure 5: Contour plots of the pressure amplitude as were calculated for sound frequency of 3000 Hz and a hinged plate with a central hole of (a) 0.5 m radius and (b) of 0.25 m radius. The rest of the conditions are as in Figure 4. The plate deﬂection W at r = r is found by solving scheme as was used in this study. Thus the solution for W n n n the deﬂection equation (4). This solution is subject to three can be written as ⎡ ⎤ possible edge conditions : −1 ⎣ ⎦ W = L 1+ A J γ r + F . (16) n m 0 m s s ns dW m=0 Clamped edge: W = = 0 dr Thecentral ﬁnitediﬀerence schemes used to approximate (4) d W ν dW are ﬁve points stencils and are at least of second order. The Hinged edge: W = + = 0 dr r dr result is a pentadiagonal matrix for L which can be solved ns 2 3 2 rapidly using LU decomposition . d W ν dW d W 1 d W 1 dW Free edge: + = + − = 0. Equations (13), (14), and (16)resultinafull matrix 2 3 2 2 dr r dr dr r dr r dr equation for the coeﬃcients A , which can be solved (15) m using LU decomposition . This is basically a collocation These are implemented at the outer edge of the plate and method. However, numerical experimentation showed that at an inner edge if there is a hole in the plate. Symmetry the matrix may become ill conditioned, that is, close to be assumption is used for r = 0. The solution can be found singular and thus causing diﬃculties to the matrix solver. analytically using a Bessel series or by a ﬁnite diﬀerence Therefore a least square operation was added with respect z (m) z (m) z (m) z (m) Advances in Acoustics and Vibration 7 to A and with a linear weight function of r. This is similar Q stands for the acoustic power embedded inside the to the operation usually used to determine the coeﬃcients scattered wave as it diﬀers from the standing wave caused of a Fourier-Bessel series, and it led to a much better by the free surface. If Q = 0, the sound ﬁeld is as of conditioned matrix. The resulting equation with respect to only as in the presence of the free surface. As the incoming the wavenumber k ,where I = 0··· M,is pressure amplitude is taken as one, Q also represents a nondimensionalised scattering cross-section of the plate. Na Minimization of Q can be achieved by varying F and using ∂W r s J γ r − ρω n I 0 I n the Powell optimization algorithm, which is widely available ∂A n=0 . However for practicality reasons only the case of a ⎡ ⎤ uniform distribution of F is discussed in the next section. ⎣ ⎦ × A s J γ r + ik − ρω W (17) m m 0 m n 0 n m=0 ⎡ ⎤ M M 3. Results and Discussion ⎣ ⎦ + r J γ r A J γ r +1 = 0, n 0 I n m 0 m n 3.1. Validation. Comparisons between the spectral and the n=Na+1 m=0 time marching solutions are shown in Figure 2 for a rigid where circular plateofaradiusof1mand soundfrequency of 3000 Hz. The latter is in the lower end of low frequency ∂W −1 sonar and corresponds to a wave length of 0.5 m, where = L J γ r . (18) 0 I s ns ∂A the speed of sound was taken as of fresh water, that is, 1500 m/s. The incoming sound wave amplitude P was taken Na is the number of grid points on the plate, and it was as 1 Pa, taking into account that this is a linear problem. assumed that there is no hole in the plate. If there is a central The computational domain length was taken as 10 m in the hole in the plate, then an additional term similar to the vertical and radial directions with a grid resolution of about second term has to be added to (17) to account for the free 30 points per wave length of the incoming wave. A buﬀer surface in that hole. Equation (17) is a full square matrix with zone was added to the computational domain of the time M + 1 unknowns of A and it can be solved using a pivoted marching scheme for 10 m < r < 12.5 m having the same LU decomposition . grid resolution as inside the computational domain. The computational domain for the spectral solution extended 2.3. Forced Plate Deﬂection and Cloaking. One of the main up to 100 m, damping waves reaching its radial boundary aims of this study is to cause the sound wave reﬂection from by about 20 dB as compared to the plate’s edge. Reducing the plate to mimic a surface of zero impedance, that is, free the radial length of the computational domain from 100 m surface. This means acoustic cloaking and by expression (11) to 50 m showed very little eﬀect on the spectrum of the one gets various modes and in particular on the radiating modes. Thus the radial length of the computational domain was A =−1, A = 0, m = 1··· M. (19) 0 m determined as suﬃciently long to account for the free surface eﬀect on the plate’s scattered sound and damp any noticeable Thus the pressure gradient ∂P/∂z over the plate becomes 2ik , eﬀect of the radial wavenumbers in (11) being of a ﬁnite yielding a uniform deﬂection W over the plate by (3): set as determined by the radial edge condition used for series convergence. The spectral solution grid resolution was 2ik about 25 points per wave length of the incoming wave. W(r) = . (20) ρω Doubling that grid resolution showed very little eﬀect on the results. The pressure amplitude contours were computed and Such situation is possible only for a free edged plate, leading compared with the time-marching results. by (4) to a uniform external pressure F as required for Very good agreement is revealed between the two solu- cloaking the plate: tions as seen in the overall contour plots of the pressure amplitude seen in Figure 2(a) and in the vicinity of the −2ik ρ h 0 p F(r) = . (21) rigid plate in the zoomed view of Figure 2(b). The contour lines corresponding to the two solutions overlap each other as such that they are almost indistinct, showing the This solution of a uniform load is valid for any geometrical very good agreement between the two solutions. Thus the form of the plate and not just circular as long as the plate is spectral solution has been veriﬁed for a rigid plate as in free edged. comparison to the time-marching solution. The solution For clamped or hinged plates, a signiﬁcant reduction in of the plate’s deﬂection equation (4) was veriﬁed against the scattered sound may be possible if the following target known static solutions . Furthermore the analytical (cost) function Q can be minimized: solution for the optimised external pressure F required to M cloak the free edged plate (21) was also veriﬁed against Q = (1+ A ) + A . (22) the numerical solution of the spectral solution according to i=1 (17). 8 Advances in Acoustics and Vibration 3.2. Scattered Sound Field and the Plate’s Flexibility. The and free of Figure 3(d). However, taking the plate as rigid eﬀect of the plate’s ﬂexibility can be observed in Figure 3. increases noticeably the pressure load while reducing the The grid size, resolution, and sound frequency are similar to plate’s thickness to 5 mm reduces signiﬁcantly the pressure those as in Figure 2. The plotted contour levels of the pressure load. The latter yielded the reduced wave ﬁeld scattered by amplitude |P| are of the level expected for sound reﬂection the plate as was seen in Figure 3(e). The incoming wave from the free surface. This was done for purpose of further frequency has a stronger eﬀect on the plate’s pressure load clarity when considering the grey scale of the contour plots. than the plate’s edge condition as can be seen from Table 1. In fact the sound ﬁeld reﬂected by the rigid plate reached a Increasing the frequency from 1500 Hz (Figure 4(a))to level of |P| of about 4.5 times the incoming wave amplitude 3000 Hz (Figure 3(b)) and 6000 Hz (Figure 4(b))causesa around the centre line r = 0 as can be seen in Figure 2. monotonous increase in the pressure load. This is related to Replacing the rigid plate by a ﬂexible one made of aluminium the increase in the presence of the wave ﬁeld scattered by the AL2024 T3 and of 5 cm thickness reduced that pressure peak plate as is seen in the corresponding ﬁgures. to 3.5 but did not change much the pattern of the sound The case of an impurity in the plate modelled as a central ﬁeld as can be seen for the cases of clamped, hinged, and hole was also tackled and is illustrated in Figure 5 for hinged free-edges plates illustrated in Figures 3(b), 3(c),and 3(d), plates and a sound frequency of 3000 Hz. The computed respectively. The wave scattered by the plate concentrates in sound ﬁelds resemble those seen for the full plates of Figure 3, a narrow beam propagating perpendicular to the plate and that is, a beam of scattered wave above the plate, mildly widens mildly as it gets further away from the plate. The expanding as it propagates further from the plate. The zone plate ﬂexibility also reduced the level |P| just over the plate, of high pressure level inside the beam is shortened due to the while changing its edge condition from clamped to hinged hole in the plate as expected. Instead of two to three lobes and then free-edged extended the region of high pressure of high pressure over the plate, there is only one lobe for the over the plate from two lobes to three lobes as seen when plate with a hole of 0.5 m radius, extending to two lobes if comparing Figure 3(b) to Figure 3(d). Finally, reducing the the hole is reduced to a radius of 0.25 m. Interestingly the plate’s thickness from 5 cm to 5 mm reduces considerably the plates with a central hole also exhibit radiation from their wave scattered by the plate as seen in Figure 3(e). Similar inner edge to form a peak in the scattered wave beam around behaviour was found for the plates with other types of edge r = 0 as can be seen particularly for the large hole case of support. This is of no surprise since the bending stiﬀness of 0.5 m radius as seen in Figure 5(a). the plate was reduced by a factor of 1000 as compared to the To further illustrate the eﬀect of the central hole, values of plate of 5 cm thickness, causing the thin plate to behave as the pressure load on the plate are given in Table 2 for several having low acoustic impedance. radii of the hole, hinged, or clamped conditions and a wave The eﬀect of the sound frequency on the scattered sound frequency of 3000 Hz. It is seen that for holes that are not ﬁeld is illustrated in Figure 4 for the aluminium plate of 5 cm too large, that is, 0.6 m or less, the eﬀect of the hole on the thickness and hinged edge. Computational grid resolution pressure load is mild. However, increasing the hole’s radius as relative to the sound wave length was kept the same as to 0.8 leads to a ring of a width of 0.2 m, which is compact in the previous calculations, and the plotted contour levels as relative to the incoming wave length of 0.5 m, resulting were kept the same as well. As in the sound ﬁeld with the in a signiﬁcant change in the pressure load. It signiﬁcantly frequency of 3000 Hz, both sound ﬁelds of 1500 Hz and decreases for the hinged condition, a trend that was also seen 6000 Hz frequencies show most of the wave scattered by the for the full plate case when it was made more compact by plate to concentrate in a beam above the plate. The widening reducing the wave frequency, see Table 1. On the other hand of that beam as it stretches away from the plate is not aﬀected the clamped condition stiﬀens the plate by its zero deﬂection- much by the frequency. The 1500 Hz sound ﬁeld shows a slope requirement, causing a higher pressure load. preservation of the wave fronts with a spatial lag diﬀerence between the zone above the plate and that further away from the plate in the radial direction. On the other hand, the 3.3. Cloaking and Wavenumber Spectra. The derivation in 6000 Hz sound ﬁeld shows a more complicated structure of Section 2 showed that uniform external pressure acting on sound waves with higher pressure levels near the centre of a free-edged plate can alter the sound wave scattered by the scattered wave beam with further longitudinal troughs the plate to mimic completely the wave reﬂected by the coming from the edge of the plate. This can be attributed free surface, that is, achieving complete cloaking within the to the fact that the plate is a noncompact object, that is, its limits of the theory. Therefore it is attractive to examine length scale is much longer than the sound wave length. In that approach for plates with other types of edge support. this manner, the reﬂected sound ﬁeld is similar to that found Although clearly it will not be able to achieve complete near noise barriers used to shield noise sources with a wave cloaking, it may still achieve signiﬁcant reduction in the length much shorter than the height of the noise barrier. scattered wave as compared to that reﬂected by the free To summarize the cases studied in Figures 3 and 4,values surface. Such approach was used to compute the sound of the pressure load acting on the plate and expressed as ﬁelds shown in Figure 6 for the hinged plate and subject the pressure amplitude averaged over the plate’s surface are to sound frequencies varying of 1500 Hz and 3000 Hz. The given in Table 1. It is seen that the pressure load is not target function Q of (22) was minimized by varying the much aﬀected by the edge condition of the plate, whether uniform pressure F until a minimum was achieved, where the hinged condition of Figure 3(b), clamped of Figure 3(c), that Q denoted the plate’s scattered acoustic power. The Advances in Acoustics and Vibration 9 |P| |P| 10 10 1.69 1.69 1.38 8 8 1.38 1.08 1.08 0.77 0.77 6 6 0.46 0.46 0.15 0.15 −0.15 −0.15 4 4 −0.46 −0.46 −0.77 −0.77 −1.08 −1.08 2 2 −1.38 −1.38 −1.69 −1.69 −2 −2 0 0 2 4 6 8 10 2 4 6 8 10 r (m) r (m) (a) (b) |P| |P| 10 10 2 2 1.69 1.69 8 1.38 8 1.38 1.08 1.08 0.77 0.77 6 6 0.46 0.46 0.15 0.15 −0.15 −0.15 4 4 −0.46 −0.46 −0.77 −0.77 −1.08 −1.08 2 2 −1.38 −1.38 −1.69 −1.69 −2 −2 0 0 2 4 6 8 10 2 4 6 8 10 r (m) r (m) (c) (d) Figure 6: The eﬀect of an optimised external uniform force as it is shown by the pressure amplitude diﬀerences between the ﬁelds reﬂected by the plate and free surface, and the free surface only. The sound frequencies and the hinged plate conditions are (a) 1500 Hz with no external force, (b) 1500 Hz with an external uniform force, (c) 3000 Hz with no external force and (d) 3000 Hz with an external force. The rest of the conditions are as in Figure 4. Table 1: The pressure amplitude averaged over the plate’s surface as corresponding to the cases in Figures 3 and 4. The pressure is expressed as relative to the incoming pressure amplitude of 1 (Pa). Case Figure 3(a) Figure 3(b) Figure 3(c) Figure 3(d) Figure 3(e) Figure 4(a) Figure 4(b) Pressure 2.1068 1.6907 1.6845 1.7778 0.3365 1.2457 1.9754 optimised value of F was found to be close to the analytical compared to the incoming sound wave length of 0.5 m. Thus value derived just for the free-edged plate in (21), but also a uniform pressure distribution is less likely to be eﬀective in having a small real part and not just an imaginary part as in inﬂuencing the scattered wave ﬁeld. Nevertheless a noticeable (21). reduction in the sound ﬁeld can be observed. A signiﬁcant reduction in the plate’s acoustic signature The eﬀect of the sound frequency on the ability to reduce was achieved for the low sound frequency of 1500 Hz, as the plate’s acoustic signature, that is, achieving acoustic seen by comparing Figures 6(a) and 6(b), where the sound invisibility is shown in Figure 7 for hinged and clamped ﬁeld reﬂected by just the free surface was subtracted from plates. The ratio of the optimised scattering cross-section is the overall ﬁeld for clarity. A less profound reduction in deﬁned as Q(F )/Q(F = 0). This ratio was calculated optimised the acoustic signature is seen in Figures 6(c) and 6(d) for at various frequencies with intervals of 500 Hz, and a curve of the sound frequency of 3000 Hz. This can be understood on high-order polynomial was plotted to best ﬁt the computed the ground that the plate is no longer a compact object as ratios. As already presented in the analysis of Figure 6, this z (m) z (m) z (m) z (m) 10 Advances in Acoustics and Vibration 0.8 0.8 All modes 0.7 0.7 All modes 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 Radiative Radiative modes modes 0.2 0.2 0.1 0.1 0 0 1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000 Frequency (Hz) Frequency (Hz) (a) (b) Figure 7: The variation of the optimised scattering cross-section with the sound frequency for (a) hinged and (b) clamped plates, where the cross-section ratio is deﬁned as Q(F )/Q(F = 0). The symbols denote the actual computed values, and the lines are best ﬁtted to those optimised values using polynomials of 8th order. The rest of the conditions are as in Figure 4. Table 2: The eﬀect of the central hole’s radius on the pressure amplitude as averaged over the plate’s surface. The pressure is expressed as relative to the incoming pressure amplitude of 1 (Pa), and the rest of the conditions are as in Figure 5. Central hole’s radius (m) 0 0.2 0.4 0.6 0.8 Hinged plate, pressure 1.6907 1.6922 1.5616 1.7017 0.3038 Clamped plate, pressure 1.6845 1.6218 1.6810 1.5069 2.599 kind of active cloaking approach is more eﬀective in the more energy towards the nonradiating higher wavenumbers, compact range, that is, the lower frequency. Furthermore the thus explaining the reduction in the scattered wave ﬁeld reduction in the far ﬁeld scattering is much more profound amplitude of the ﬂexible plates seen in Figure 2 as compared than in the near ﬁeld for both plates in the compact case, to the rigid plate. Changing the edge condition from hinged which is good news. The hinged plate shows a higher ability to clampedorfree-edgedaﬀected the lobes pattern of the to achieve reduced scattering cross-section than the clamped amplitude distribution, pointing again to the eﬀect of the plate. This is because of the higher ﬂexibility of the hinged plate’s ﬂexibility on energy transfer between the Fourier- plate as compared to the clamped plate. The clamped plate Bessel modes. The modes’ amplitudes are much reduced losses the ability to reduce signiﬁcantly the scattered far for the cloaked cases, particularly for the low frequency of ﬁeld sound already at a frequency of 3000 Hz, that is, at an 1500 Hz, aﬀecting mostly the radiating modes as already incoming wave length of 0.5 m which is half of the plate’s seen in Figure 7. As the frequency increases, the success in diameter. On the other hand the hinged plate still manages reducing those modes is reduced although some reduction is to achieve a reduction of about 6 dB in the scattering cross- still achieved even for the frequency of 6000 Hz. section, the same level that was reported for the passive cloaking study of metamaterials by Zhang et al. . To further understand the eﬀect of cloaking, plate’s 4. Summary ﬂexibility, and sound frequency, the distributions of the spectral mode amplitudes composing the target function Q Sound wave scattering was considered as caused by a ﬂexible were plotted for the sound frequencies of 1500 Hz, 3000 Hz, plate embedded on the surface of calm water. The water and 6000 Hz. All mode distributions show a pattern of surface was taken as free surface and thus modeled as a lobes in Figure 8, which points to a wavenumber leakage zero-pressure surface. The aim of the study was to assess or energy transfer between the modes. This happens due the eﬀect on the plate’s ﬂexibility on the sound scattering to the presence of the plate which, if rigid, introduces a and particularly whether there was a potential in using coupling between the modes by forcing the condition of that ﬂexibility to aﬀect the scattering. An incoming plane zero-pressure gradient on the plate. Introducing ﬂexibility monochromatic sound wave was assumed. Linear acoustics increases the energy transfer between the modes and pushes and plate’s dynamics were used to obtain solutions for Ratio of optimised scattering cross-section Ratio of optimised scattering cross-section Advances in Acoustics and Vibration 11 Nonradiative Radiative Nonradiative modes Radiative modes modes modes 5 0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 γ (1/m) γ (1/m) Rigid plate Rigid plate Hinged plate Hinged plate Hinged plate, cloaked Hinged plate, cloaked (a) (b) Radiative Nonradiative modes modes 10 20 30 40 50 60 70 γ (1/m) Rigid plate Hinged plate Hinged plate, cloaked (c) Figure 8: Radial wavenumber spectra of the scattered sound wave for the hinged plate and frequencies of (a) 1500 Hz, (b) 3000 Hz, and (c) 6000 Hz. The mode amplitudes were normalized as relative to the plate’s radius. The rest of the conditions are as in Figure 4. the scattered sound ﬁeld and plate’s deﬂection. A solution nonradiating modes. The scattered wave concentrated in a approach based on a Fourier transform of the sound ﬁeld relatively narrow beam propagating perpendicular to the was introduced. The solution was simpliﬁed for the case of plate. A high-pressure zone was found near the centre of the a circular plate subject to an incoming perpendicular wave, beam, and such zone decreased but did not vanish when a representing a simpliﬁed case of three-dimensional sound large hole was introduced at the centre of the plate. This was propagation. The pressure Fourier modes were coupled to attributed to sound radiation from the inner edge zone of the the plate’s deﬂection using central ﬁnite-diﬀerence schemes. plate. The combined spectral-ﬁnite diﬀerence solution was The possibility of reducing the plate’s acoustic signature, validated against a time-marching simulation of the scattered that is, cloaking was also investigated. This means making wave and known solutions of the plate’s deﬂection. It was the plate acoustically invisible by changing the character of found that the plate’s ﬂexibility reduced the plate’s scattered its scattered sound wave to resemble as much as possible wave by transferring some of its energy from radiating to that of the sound wave reﬂected by the free surface. For Spectral mode amplitude |A| Spectral mode amplitude |A| Spectral mode amplitude |A| 12 Advances in Acoustics and Vibration this purpose, the approach of applying an external pressure radial distances the incoming plane wave propagating in the distributed uniformly over the plate and oscillating at the z direction dominates the sound ﬁeld, a simple boundary 2 2 sound frequency was suggested. It was shown theoretically condition of d p/dr = 0 was used at the computational that such approach could achieve complete cloaking for domain’s radial edge r = R.Abuﬀer zone was added in a free-edged plate subject to a perpendicular propagating front of the computational domain’s radial edge in order sound wave and within the limits of the linear theory. to minimize any artiﬁcial wave reﬂections [17, 19]. Scheme (A.1) is an explicit time marching. Thus the CFL limit on Computations showed that the approach of an external Δt was observed in the time marching which started from a pressure uniformly distributed over the plate could be also zero-pressure condition at t = 0 and continued until a steady eﬀective to achieve good degree of cloaking as long as periodic state was achieved. the plate is not highly noncompact. The radiating modes were particularly reduced as was shown for a hinged plate. It should be noted that the ratios between the plate’s Symbols length scale (radius) and the incoming sound wave length investigated in this study were of levels similar to those in A:Coeﬃcient of the expansion series for the the study of Zhang et al. , where a cloaked cylinder scattered sound wave of 13.5 mm radius was subject to incoming sound wave a: Circular plate’s radius length of 23.5 mm or higher. Thus the proposed method c: Water’s speed of sound of applying a uniform pressure can present an alternative D: Plate’s bending stiﬀness approach to metamaterials, provided that further work is E:Young’smodulus put on investigating eﬀective implementations. A general F: External pressure acting on the plate in the method of computing sound scattering for noncircular plates frequency-space domain and an oblique incoming wave can be based on ﬁnding the f : External pressure acting on the plate in the time-space domain inﬂuence (Green) functions of discretized segments of the plate. Summing those inﬂuences into the pressure condition h: Plate’s thickness over the plate will yield a full matrix equation for the pressure J : The Bessel function of the ﬁrst kind and zero values acting on the plate’s segments. The contribution order of another inﬂuence function accounting for the eﬀect of k : Wavenumber of the incoming wave the plate’s deﬂection will have to be added. This approach L : Matrix representation of the ﬁnite diﬀerence mn is of the boundary element methodology, and it will be discretization of the plate’s deﬂection equation communicated in a separate study. M: Numberofresolvedspectralmodes P: Acoustic pressure in the frequency-space domain p: Acoustic pressure in the time-space domain Appendix Q: Target (cost) function to be minimized in order to cloak the plate A time marching scheme was used as a validation tool for R: Radial length of the computational the solution of the acoustic pressure obtained by the Fourier- domain Bessel series solution for a rigid plate. The linear wave r: Radial direction equation in the time-space domain can be discretized in time s:Exponentialcoeﬃcient of the scattered wave in using a central ﬁnite-diﬀerence scheme of second order as the vertical direction follows: t:Time l+1 l l−1 p − 2p + p nm nm nm 2 2 l (A.1) W: Plate’s deﬂection in the frequency-space − c ∇ p = 0, nm Δt domain where l is the time stage and Δt is the time step of w: Plate’s deﬂection in the time-space domain the time marching. A fourth-order central ﬁnite diﬀerence x and y: Rectangular horizontal directions scheme was used to discretize ∇ p in space. The boundary z: Vertical direction conditions at z = 0, that is, (2)and (3) as written in the time- α:Buﬀer zone damping coeﬃcient space domain were discretized using second-order central γ: Wavenumber in the radial direction of the ﬁnite-diﬀerence schemes and thus the 4th-order scheme used expansion series for the scattered wave to discretize ∇ p was replaced by a 2nd-order near the free Δt: Time-marching step surface and the plate. ρ: Water’s density A 1D inlet boundary condition was used for upstream in ρ : Plate’s density the z direction: ν: Poisson’s ratio ω:Frequency. ∂p ∂p ∂p i(k z+ωt) + c = 2 , p = e . incident ∂t ∂z ∂t incident (A.2) References Boundary condition (A.2) was discretized using a 2nd-  G. Zilman and T. 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