An asymptotic formula for sound radiation from plates

Stephen C Creagh 1 , Neekar M Mohammed 2 , Martin Richter 3 and Gregor Tanner 4

School of Mathematical Sciences University of Nottingham Nottingham NG7 2RD UK

ABSTRACT We present a new approach to the modelling of sound radiated from vibrating structures that will provide a platform for simulation of larger problems by the method of Dynamical Energy Analysis (DEA), which characterises sound using densities in a ray-dynamical phase space. We consider in particular the radiated acoustic power from plates with di ff use bending vibrations, which we characterise using a two-point correlation function of normal velocities. Such field-field correlation functions are transformed to ray-dynamical phase space densities using a Wigner transformation, which allows insertion into ray-based methods such as DEA. Using correlation functions to characterise plate vibrations also allows the method to cater for stochastic, noisy driving of such systems. The results for the radiation e ffi ciency of a plate are presented in an asymptotic form, with leading contributions from the plate interior and its boundary. A notable feature of this analysis is that the bulk contribution vanishes below a critical frequency, and the asymptotic estimate of radiated power then leads with a boundary contribution. This is shown to agree well with a more traditional calculation based on modal analysis in the special case of a rectangular plate.

1. INTRODUCTION

We present an asymptotic treatment of sound radiation from vibrating plates that is designed to extend numerical simulations performed on complex, vibrating structures using the DEA (Dynamical Energy Analysis) method [1–4]. The DEA approach has been developed to simulate vibration in large, built-up structures in the mid-to-high-frequency regime, applicable where ray-based methods are an appropriate tool. Unlike traditional ray-tracing tools, it simulates energy flow using a density function on a ray-dynamical phase space - with coordinates corresponding to both position and direction - rather than by following rays individually. In this way it interpolates between full ray-tracing and Statistical Energy Analysis (SEA) [5,6], which assumes energy is distributed ergodically within subsystems.

1 stephen.creagh@nottingham.ac.uk

2 pmznmm@exmail.nottingham.ac.uk

3 martin.richter@nottingham.ac.uk

4 gregor.tanner@nottingham.ac.uk

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

Once the energy density on a vibrating structure has been simulated, the calculation of sound radiated from it is a separate, higher-dimensional problem. Although higher-dimensional, the case of radiation into a homogeneous medium is geometrically simpler and allows further analytical approximations to be incorporated into the overall simulation. This is the approach taken here, where we present a calculation of the associated radiated energy in a form that can eventually be adapted to larger DEA simulations. Detailed calculations are for the case of isotropic excitation of flat plates within the structure. This enables us to treat key radiation mechanisms in detail [7], but is also an important case for the treatment of radiated sound in general. For example, it allows us to treat radiation into the cabin of a tractor using DEA simulation such as presented in [8] and illustrated in Figure 1, or radiation from walls in buildings, panels in ships, etc.

2. THE DEA METHOD

The DEA method is based on a description of rays passing though or reflecting from faces or edges within a meshed structure. In the case of shell vibration, the two-dimensional structure is triangulated so that a ray passing an edge of the triangulation is characterised by a position coordinate s and a direction coordinate p equal to the direction cosine of the ray along the edge. A density of rays ρ ( s , p ) then determines the energy flux through the edge and is subject to an equation of the form

ρ = L ρ + ρ 0 , (1)

where ρ 0 denotes a driving term and the Frobenius-Perron operator L acts by transporting the density along rays from one edge to another, according to

[ L ρ ]( s , p ) = ρ ( φ − 1 ( s , p )) ,

and where φ is a map that transports a ray leaving an edge with direction p into it’s intersection with the next edge in its path. Once (1) has been solved to provide energy flux on mesh edges, one can calculate from it an energy density ϱ ( x , p ) for points x and ray directions p on the triangulation interior: it is this interior energy density that feeds into calculations of sound radiation described in the next sections. (Note that the choice of font for ϱ is deliberate and see [9] for a more detailed explanation of the notation). Note that in practice Eq.(1) is solved numerically by projecting on to a spatial basis that is constant on edges and a basis of Legendre functions in the momentum variable [8].

3. RAY DENSITIES AND DISORDERED WAVE FIELDS

The phase space density calculated in the DEA method provides a platform for describing the statistics of the chaotic random wave fields characteristic of stochastically driven and complex

Figure 1: The DEA method can be used to simulate transport of vibrational energy in large, complex structures at frequencies too high for wave-based methods such as FEM to be e ff ective. Figure based on [8].

vibrational structures. Let us focus on out-of-plane bending modes at a fixed circular frequency ω and characterise them by the normal velocity amplitude denoted w ( x ), where x parametrises points on the vibrating structure, assumed to be two dimensional in a thin-shell model. In the case of di ff use fields, the wave amplitude is described statistically, using the two-point correlation function

Γ ( x 1 , x 2 ) = ⟨ w ( x 1 ) w ∗ ( x 2 ) ⟩ . (2)

This can be related to phase space densities using the Wigner representation [10–12]. We simplify the discussion here by a assuming planar radiating structure - the general approach can be extended to curved shells using asymptotic analogues of the calculations to follow. We then alternatively represent Γ ( x 1 , x 2 ) as a Wigner function, defined by

W ( x , p ) = Z e − i k p · u Γ  x + u

2 , x − u

 d u .

2

An explicit link can be made between this representation of bending-wave statistics and the output of DEA simulation in the form of a ray density by observing that, after su ffi cient averaging over frequency or perturbations of system geometry, the Wigner function is asymptotically proportional (in a high-frequency limit) to the phase space density [13]:

⟨ W ( x , p ) ⟩∝ ϱ ( x , p ) . (3)

Reversing this connection, we see that phase space densities calculated as part of DEA simulations provide detailed local information about the statistics of the vibrational wave field.

4. ACOUSTIC RADIATION FROM DISORDERED BENDING MODES

We consider the case of acoustic radiation from a flat panel Ω within a larger, built-up structure, which has been simulated using the DEA method. For detailed calculations, we also simplify the

Figure 2: A comparison is shown of acoustic disturbances | p ( x , z = 4) | 2 / ( ρ 0 c 0 ) 2 with k B / k A = α = 0 . 7 and k B L x = k B L y = 40, near the boundary of plates with simply Dirichlet boundary conditions on the left and Neumann boundary conditions on the right. Plate vibrations are described by the correlation function in (5). Boundary conditions have a relatively weak e ff ect here.

calculation below by assuming that within that panel, the disordered field has isotropic and uniform statistics, corresponding to the case where the density is spatially homogeneous within it. Note, however, that the underlying approach is intended to generalise to cases where the density is not uniform, even within the panel. The total radiated power of the panel can be written

p ( x ) w ∗ ( x )d x ) ,

2 Im (Z

Π = 1

where p ( x ) denotes the acoustic pressure just next to the plate at x and ∗ denotes complex conjugation. By using a Rayleigh integral to express the pressure in terms of w ( x ) [7, 14] and averaging over the disordered wavefield to present the result in terms of the correlation function defined in (2), this can also be presented formally as a trace operation

Π = ωρ 0 Tr(ˆ g Γ ) , (4)

where ˆ g is an integral operator defined by kernel g ( x − x ′ ), where

g ( u ) = sin( k A | u | )

4 π | u |

and where ρ 0 and k A are respectively the surrounding fluid density and acoustic wave number. Using the relation suggested by (3), this provides a route to evaluating radiated power as a post- processing step to DEA simulation.

5. THE IMPORTANCE OF BOUNDARY CONDITIONS

Even in the case treated in this paper where the wave field on the radiating panel is statistically homogeneous and isotropic, significant di ff erences can be observed for the same geometry

z=4.0m

Figure 3: A comparison is shown of acoustic disturbances | p ( x , z = 1) | 2 / ( ρ 0 c 0 ) 2 with k B / k A = α = 2 . 5 and k B L x = k B L y = 40, near the boundary of plates with simply Dirichlet boundary conditions on the left and Neumann boundary conditions on the right. As for Figure 2, plate vibrations are described by the correlation function in (5). In contrast to Figure 2, results are completely di ff erent here for the two boundary conditions examined.

with di ff erent boundary conditions, below a critical frequency where the radiated wave from a corresponding infinite plate is evanescent. Figures 2 and 3 show the acoustic field intensity outside a square plate on which the phase-space density ϱ ( x , p ) derived from DEA simulation is constant. Qualitatively di ff erent results are obtained depending on the ratio

α = k B

k A between the wavenumber k B of bending in the plate and the acoustic wavenumber k A . When α < 1, radiated power is concentrated in a cone defined by angle θ = sin − 1 α to the normal. Boundary conditions at the plate’s edge have a small e ff ect in this case and an example is shown in Figure 2. When α > 1, however, which is the case below a critical frequency ω 0 , the radiated field is evanescent and dominated by edge corrections. These depend critically on boundary conditions and give markedly di ff erent results even for densities ϱ ( x , p ) obtained from DEA that are nominally identical. See for example the cases illustrated in Figure 3. Each of these examples is identical except that in the plot on the left we have imposed simply supported boundary conditions and in the plot on the right we have imposed idealised guided-wave boundary conditions in which the first and third normal derivatives of w ( x ) vanish. For short, we refer to these boundary conditions as "Dirichlet" and "Neumann" respectively in what follows. The latter boundary conditions are less realistic from a physical point of view but are included as a proof-of-principle that boundary conditions are important at leading order. We explain this di ff erence by describing the e ff ect of (in-plane) boundary conditions on the underlying correlation function. It is well established that the correlation function of a statistically homogeneous and isotropic random wave field is of the form [15]

Γ ( x 1 , x 2 ) = AJ 0 ( k B | x 1 − x 2 | ) ,

where A is a normalising constant available from DEA simulation, and where J 0 denotes a zero-order Bessel function. This is a good model for wave statistics in a plate’s interior, but assumptions of isotropy and homogeneity break down near the plate’s edges. Here we can instead model the e ff ect of boundary conditions by including image corrections of the form [16]

Γ ( x 1 , x 2 ) = A ( J 0 ( k B | x 1 − x 2 | ) ± J 0 ( k B | x 1 −R x 2 | )) , (5)

where R denotes an operation of reflection through the edge. This is a local approximation that is appropriate when the plate’s dimensions are large compared to a wavelength, and the points x 1 and x 2 are within a few wavelengths of one another but many wavelengths from a corner (where additional corrections such as images of images are important). As implied by Figures 2 and 3, these image corrections have a modest e ff ect for α < 1 but dominate when α > 1. In the later case, the trace operation (4) undergoes complete phase cancellation and vanishes in the limit Ω → R 2 of an infinite plate, but cancellation is incomplete for finite plates and radiation occurs in a boundary layer of some wavelengths deep. In this boundary layer, image corrections are as important as the bulk contribution to Γ ( x 1 , x 2 ). By including bulk and boundary corrections in the trace operation (4), and assuming for simplicity the case of homogeneous and isotropic phase space density, we can calculate the initial terms in a series expansion for the radiated power. We express it as a nondimensionalised e ffi ciency

σ = Π

Π 0 ,

where Π 0 = ωρ 0

Z Γ ( x , x ) d x

2 k A

is an equivalent power radiated by a piston occupying the same area and with the same vibrational energy. Then

σ = g ( α ) + ℓ k A A f ( α ) + · · · , (6)

where A denotes the plate’s area and ℓ the length of its perimeter. The functions g ( α ) and and f ( α ) are independent of plate geometry (except for boundary conditions) and depend on frequency only though the ratio α = k B / k A . Above the critical frequency, when α < 1, then the first term here is nonzero and given simply by

g ( α ) = 1 √

1 − α 2 .

This is the classical radiation e ffi ciency of an infinite plate. In this case the image corrections in the boundary layer are responsible for a higher-order e ff ect and can typically be neglected. Below the critical frequency, however, it can be shown that g ( α ) = 0 and the boundary layer dominates. The function f ( α ) has been given in detail in [7, 17] (although note that slightly di ff erent scaling conventions have been used there). It is somewhat more complicated and is not described in detail here, except to emphasise that it depends in detail on the boundary conditions imposed on the plate’s edge. We illustrate these results by comparing in Figure 4 the radiation e ffi ciencies of a rectangular plate, plotted as a function of 1 /α , between the same two sets of boundary conditions as illustrated in Figure 2. These are benchmarked against an equivalent e ffi ciency obtained from a modal average. We see that above the critical frequency, for which for α < 1, the two boundary conditions and the modal average agree closely. Below the critical frequency, however, the two boundary conditions give markedly di ff erent results: in each case there is good agreement with the (now leading) boundary term in the asymptotic expansion, shown as dashed curves. Note the perhaps counter-intuitive outcome that the case of simply-supported boundary conditions, where the bending amplitude is forced to zero on the plate edges, gives the markedly higher radiated power. The detailed explanation for this comes from a careful analysis of the function f ( α ), but broadly speaking can be explained by noting that the boundary layer responsible for radiation is e ff ectively several wavelengths deep, and is not dominated by the value at the very end of the integration domain.

10 0

10 0

10 -5

10 -5

10 -2 10 -1 10 0 10 -10

10 -2 10 -1 10 0 10 -10

Figure 4: On the left, modal radiation e ffi ciencies σ nm of guided supported plates are shown for a selection of mode numbers ( n , m ), which are given in brackets in the legend. There is significant fluctuation from one mode to the next here, which is eliminated in the averaged e ffi ciencies. On the right, the radiation e ffi ciency is illustrated for a ba ffl ed rectangular plate with side lengths L x and L y subject to di ff use vibrational excitation and for Dirichlet and Neumann boundary conditions for k B L x = k B L y = 56. Results from a numerical simulation are compared with the approximations evaluated using (6) (green dashed curves for the Dirichlet case and blue dashed curves for the Neumann case).

6. CONCLUSIONS

The DEA method is a powerful tool for the simulation of vibrating structures in the mid-to-high- frequency regimes, but evaluation of acoustic radiation is higher dimensional and therefore often more challenging from a computational point of view. For this reason, any additional asymptotic or analytical progress is valuable as it enables us to leverage DEA simulation into a more complete analysis. We have argued that by providing a link to statistical characterisations of the wave amplitudes, the Wigner formalism provides an ideal tool to leverage DEA simulations into more complete calculations of sound radiation. Here we have concentrated on the case of radiation form flat panels in the vibrating structure, accounting for detailed boundary conditions of the wave problem to develop the first term in an asymptotic expansion for radiated power. The results have been benchmarked against corresponding modal averages and shown to give a good description particularly below a critical frequency where coupling to the acoustic problem is weak, and which is particularly dominated by boundary conditions. Although these boundary conditions do not play an explicit role in the DEA approach, we have shown that we can take advantage of the Wigner formalism to add them in a post-processing step. Although the detailed calculations presented here take advantage of simplifying assumptions that the radiating surface is flat and that the phase space density within it is uniform, we believe that the underlying approach will extend to curved structures and more general phase space densities.

ACKNOWLEDGEMENTS

The authors acknowledge the support of Romax Technologies Ltd and Yanmar Co. Ltd.

REFERENCES

[1] Gregor Tanner. Dynamical energy analysis—determining wave energy distributions in vibro- acoustical structures in the high-frequency regime. J. Sound Vib. , 320(4):1023 – 1038, 2009.

[2] David J Chappell, Gregor Tanner, Dominik Löchel, and Niels Søndergaard. Discrete flow mapping: transport of phase space densities on triangulated surfaces. Proc. Roy. Soc. A Math. Phys. Eng. Sci. , 469(2155):20130153, 2013. [3] D.J. Chappell, D. Löchel, N. Søndergaard, and G. Tanner. Dynamical energy analysis on mesh grids: A new tool for describing the vibro-acoustic response of complex mechanical structures. Wave Motion , 51(4):589 – 597, 2014. Innovations in Wave Modelling. [4] David Chappell, Jonathan J. Crofts, Martin Richter, and Gregor Tanner. A direction preserving discretization for computing phase-space densities. SIAM Journal on Scientific Computing , 43(4):B884–B906, 2021. [5] Richard H Lyon and Richard G DeJong. Theory and Application of Statistical Energy Analysis, Second Edition . Butterworth-Heinemann, 1995. [6] Richard H Lyon, Richard G DeJong, and Manfred Heckl. Theory and application of statistical energy analysis. J. Acoust. Soc. Am. , 98(6):3021–3021, 1995. [7] Neekar M. Mohammed. Vibroacoustics of complex structures - a wave chaos approach . PhD thesis, The University of Nottingham, 2021. [8] Timo Hartmann, Satoshi Morita, Gregor Tanner, and David J. Chappell. High-frequency structure- and air-borne sound transmission for a tractor model using dynamical energy analysis. Wave Motion , 87:132 – 150, 2019. Innovations in Wave Modelling II. [9] Stephen C Creagh, Martin Sieber, Gabriele Gradoni, and Gregor Tanner. Di ff raction of wigner functions. Journal of Physics A: Mathematical and Theoretical , 54(1):015701, nov 2020. [10] M. O. S. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner. Distribution functions in physics: fundamentals. Phys. Rep. , 106(3):121–167, 1984. [11] Thomas Dittrich, Carlos Viviescas, and Luis Sandoval. Semiclassical propagator of the wigner function. Phys. Rev. Lett. , 96:070403, Feb 2006. [12] Gabriele Gradoni, Stephen C Creagh, Gregor Tanner, Christopher Smartt, and David W P Thomas. A phase-space approach for propagating field–field correlation functions. New Journal of Physics , 17(9):093027, sep 2015. [13] S. C. Creagh, G. Gradoni, T. Hartmann, and G. Tanner. Propagating wave correlations in complex systems. J. Phys. A Math. Theor. , 50(4):045101, 2016. [14] Lord Rayleigh. Xxxvii. on the passage of waves through apertures in plane screens, and allied problems. London, Edinburgh Dublin Philos. Mag. J. Sci. , 43(263):259–272, 1897. [15] Michael V Berry. Regular and irregular semiclassical wavefunctions. J. Phys. A Math. Gen. , 10(12):2083, 1977. [16] Michael V Berry. Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A Math. Gen. , 35(13):3025, 2002. [17] Stephen C Creagh Neekar M Mohammed and Gregor Tanner. Acoustic radiation from random waves on plates.