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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 3966–3980

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Sound radiation from a vibrating plate with uncertainty

Wonjae Choi n, Jim Woodhouse, Robin S. LangleyDepartment of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

a r t i c l e i n f o

Article history:Received 3 September 2013Received in revised form11 March 2014Accepted 12 March 2014
Handling Editor: D. Juve
intensity fields from the sound radiation have not yet been widely studied using statistical

Available online 5 May 2014

x.doi.org/10.1016/j.jsv.2014.03.0180X/& 2014 Elsevier Ltd. All rights reserved.

esponding author. Present address: Departm4 207 594 5709.ail addresses: [email protected] (W. Cho

a b s t r a c t

Sound radiation into open space from a vibrating structure has been investigated sinceRayleigh. On the other hand the sound power transferring into neighboring reverberantsubsystems has also been rigorously studied using statistical energy analysis, particularlyfor the high frequency range. Falling between the two well-known problems, pressure and

methods. In this paper, the sound radiation from a vibrating thin plate having uncertaindynamic properties is investigated. Estimates are developed for the reverberant vibrationfield in the uncertain plate subjected to a point-excitation, and for the ensemble averageof pressure from the direct field and from the reverberant field, leading to an estimate ofthe average sound intensity. The power radiated from the plate and the radiationefficiency is also derived. Monte Carlo simulations are conducted with an ensemble ofplates with randomly-distributed point masses, and the simulation results compare wellwith the estimates.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Sound can be generated by several types of sources [1,2], but the most common is from a vibrating structure such as aviolin or loudspeaker. The flat plate is probably the most-investigated structure as a sound source. Radiation from a vibratingplate can be estimated using the Rayleigh integral [3], which treats the vibrating plate as a collection of point sources. Thismethod needs the plate to be embedded in an infinite baffle so that the ray paths to a remote point are all straight, andissues of diffraction do not arise. Radiation from some other simple shapes such as cylinders and spheres has also beeninvestigated [4–6]. However, if any of these structures have uncertain dynamic properties, it would be hard to describe theirradiation characteristics without taking a statistical approach.

Nominally identical structures, whether vehicles from one production line or violins made by the same maker, will havedifferent vibration and sound radiation properties to some extent, since the material properties and geometric details willnever be exactly identical in reality. In the high frequency range, wave fields are sensitive to such small differences. Thissensitivity can influence significant aspects of quality, so the effect of the uncertain properties plays an important role inquality control. The effect of such uncertainties can be systematically assessed by the Monte Carlo (MC) experiments orsimulations, for example by perturbing a plate with randomly-distributed masses [7,8], but such studies are generally very

ent of Mechanical Engineering, Imperial College, London SW7 2AZ, UK. Tel.: þ44 207 594 7227;

i), [email protected] (J. Woodhouse), [email protected] (R.S. Langley).

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W. Choi et al. / Journal of Sound and Vibration 333 (2014) 3966–3980 3967

time-consuming. If analytic approaches can be used to estimate the statistical properties of the wave field, this would haveobvious advantages.

The vibration of a structure with uncertain properties has been actively investigated, mainly by using the approach ofStatistical Energy Analysis (SEA) developed by Lyon [9] and others in the 1960s for example, [10–12]. In this method, asystem is divided into subsystems and the vibration behavior of the system is estimated in terms of statistical properties ofthe individual subsystems and the energy-flow relations between them. Thus the vibration statistics of a subsystem need tobe understood to properly estimate the behavior of the total system. Recently, Gaussian Orthogonal Ensemble (GOE)statistics have been introduced to model the statistics of individual subsystems [7,13–15], leading to better understandingand more accurate design guidelines for engineers.

In SEA, an acoustic cavity surrounded by vibrating structures can be one of the subsystems, and thus the statistics of afinite acoustic subsystem and its interaction with other subsystems have been investigated for example, [16]. However,sound radiation into infinite space has not been statistically investigated in anything like the same depth. Such statistics ofthe sound radiation would give a useful addition to the SEA framework. For example, sound generated from musicalinstruments or vehicle structures could be properly modeled in the high frequency range, a challenging problem for acousticengineers. This paper gives a contribution to filling that gap, by applying similar techniques to those used in SEA on finitesubsystems to derive results about the direct (i.e. non-reverberant) sound radiation from structures with uncertain dynamicproperties.

After a brief review of the necessary background theory, the analysis of vibration with uncertainties will be examined inthe light of the authors' previous work [17], and the first and second cumulants of pressure, ensemble intensity, totalradiated sound power and radiation efficiency from finite random plates will be estimated statistically. The estimates arevalidated by comparing with results of MC simulations.

2. Background

In an ensemble of structures with uncertain properties, the vibration response of any given ensemble member isdifferent from that of the unperturbed nominal structure. Statistical characteristics of the response across the ensemble ofstructures, for example the mean or variance, can give useful information to make engineering design decisions. Inparticular the sound radiation from each ensemble member will be different, and this is the target of this paper: to obtainstatistical expressions for sound radiation for an ensemble of systems sharing common characteristics. In this section,expressions for vibration statistics are summarized first, and then the equations governing sound radiation from vibratingstructures will be introduced. With this background, statistical estimates for the radiation will be obtained in Section 3.

2.1. Random plate vibration subjected to a point excitation

Consider a thin plate of thickness h, made of isotropic material with Young's modulus Y, Poisson's ratio s, and density ρ. Ifthe plate is driven by a harmonic force of frequency ω and unit amplitude at a point x0, the velocity u at a point x can beexpressed in terms of the set of modes [18] (omitting the assumed time dependence e� iωt)

uðx; x0;ωÞ ¼ � iω∑ϕnðxÞϕnðx0Þ

ω2nð1� iηÞ�ω2; (1)

where ϕn is the nth mode shape of the system, ωn is the corresponding natural frequency and η is the loss factor which isassumed identical for all modes. Note that velocity u rather than displacement is used here for convenience in calculatingpressure later in this paper. Each mode shape is normalized so that

Rρh ϕ2

n dSplate ¼ 1, where the integral is taken over the totalarea Splate of the plate. If most of the modal energy is concentrated near ω¼ωn, then Eq. (1) can be approximated [14] by

uðx; x0;ωÞ ¼ � iω∑ϕnðxÞϕnðx0Þ

2ωðωn�ω�ðiηω=2ÞÞ: (2)

The vibration behavior of a set of such plates with uncertain dynamic properties can be represented by the statistics ofthe ensemble. The present work investigates the ensemble statistics of the velocity u and pressure p under the assumptionsthat (1) mode shapes are random and (2) natural frequencies form a random process.

One fundamental building block of these statistics is the ensemble average of the product of the mode shape at twolocations, the two-point correlation. It is well known that the correlation in a two-dimensional system can be expressedusing the Bessel function J0 [19]:

E ϕnðx1Þϕnðx2Þ� �¼ 1

MJ0ðk x2�x1 Þ

�� (3)

where E½⋯� denotes the ensemble average,M is the mass of the nominal plate, and k is the wavenumber of bending waves atthe frequency of mode ϕn satisfying

k2 ¼ ωffiffiffiffiffiffiffiffiffiffiffiρh=B

q; (4)

where B¼ Yh3=12ð1�s2Þ is the flexural rigidity.

W. Choi et al. / Journal of Sound and Vibration 333 (2014) 3966–39803968

The natural frequencies in Eq. (2) are considered as a series of random points along the frequency axis. The statistics of afunction on such points can be investigated using point process theory and described in terms of cumulants κi [20], whichare central moment functions; for example,

κ1½uðx1Þ� ¼ E½uðx1Þ�k2½uðx1Þ; uðx2Þ� ¼ E½ðuðx1Þ�E½uðx1Þ�Þðuðx2Þ�E½uðx2Þ�Þ�: (5)

Based on Eq. (3), the ensemble first and second cumulants of velocity u of a plate subjected to a point excitation at x0 canbe estimated through Eq. (2) as in [17] by

κ1 uðx0; x1Þ½ � ¼ νp2M

Πoðkr01Þ (6)

κ2 uðx0; x1Þ;unðx0; x2Þ� �¼ νp

2M2ωηð2þqðΛÞÞRe½Πoðkr01ÞΠiðkr02Þ� �þ J0ðkr12Þ� (7)

where rij ¼ jxj�xij, ν is the modal density, Λ¼ωην is the modal overlap factor, qðΛÞ is a factor derived from GOE statistics [7]which varies from 0 to �1 as Λ increases. Πo and Πi represent outgoing and incoming waves [21] composed of the Hankelfunctions:

ΠoðkrÞ ¼Hð1Þ0 ðkrÞ�Hð1Þ

0 ðkrÞ (8a)

ΠiðkrÞ ¼Hð2Þ0 ðkrÞ�Hð2Þ

0 ð� ikrÞ: (8b)

The modal density ν quantifies the number of modes within a frequency band; it is known to be asymptotically constantfor a plate [9]:

ν� LxLy4p

ffiffiffiffiffiffiρhB

r: (9)

The ensemble average in Eq. (6) is identical to vibration field of an infinite plate subjected to a point excitation in [21].This can be explained by the fact that the only coherent part of the vibration wave is propagating directly from the pointsource since scattered or reflected waves are assumed incoherent due to their random nature. The total vibration fieldexcept the direct field E½u� can be defined as a reverberant field, urevðx1Þ ¼ uðx1Þ�E½uðx1Þ�. Then Eq. (7) can be considered asthe ensemble average of the squared the reverberant field when x1 ¼ x2, E½jurevj2� ¼ κ2½u;un�. This assumption is reasonablewhen the boundaries are irregular or there are wave scatterers randomly located. The latter will be considered for MCsimulation and compared with the radiation estimates in Section 4.

2.2. Direct radiation from a plate

Once the vibration velocity of the structure is known, the sound radiation can be calculated. The plate is assumed to lie inthe x–y plane, embedded in an infinite zero-displacement baffle in the plane z¼0. Sound radiation is considered only in thehalf-space z40, and sound pressure will be calculated on a hemispherical surface as shown in Fig. 1. For far-field radiationthe dimension of the plate needs to be much smaller than the hemisphere, but the plate in the figure is exaggerated for

Fig. 1. Hemispherical field above a rectangular plate.


clarity. Energy transmission from the plate to the surrounding air can be readily obtained under the assumption that energyflow in the reverse direction can be ignored [5], as will be assumed throughout this paper.

For a monitoring point above the plate determined by angles (Θ; Φ) as shown in Fig. 1, the radiation is given by the Rayleighintegral [3]: with a surface velocity uðxÞ the approximate sound pressure p at a large distance R� r can be expressed as

pðΘ; Φ; ωÞ ¼ CZSplate

uðxÞeikr Ux dSplate (10)

where C ¼ ρ0ωeik0R=ð2pRÞ, ρ0 is air density, k0ð ¼ω=c0Þ is the acoustic wavenumber, c0 is the speed of sound in air, Splate is thevibrating area, and kr ¼ ikxþ jky, where

kx ¼ k0 sin Θ cos Φ and ky ¼ k0 sin Θ sin Φ: (11)

Note that the integral in Eq. (10) uses velocity u, rather than displacement u=ð� iωÞ. This integral equation can be readilyderived using a two-dimensional Fourier transform [21]. The sound intensity is now given in terms of the squared modulusof the pressure [4]:

I ¼ jpj2=ð2ρ0c0Þ: (12)

The total sound power radiated can be calculated by integrating the intensity over the hemisphere area, Shemi:

W ¼Z

jpj2 dShemi=ð2ρ0c0Þ: (13)

3. Theoretical estimation

In this section vibration statistics for a finite plate with uncertain dynamic properties subjected to a point excitation willbe considered first, and then ensemble statistics of sound pressure, intensity and radiated power will be estimated. Highfrequency excitation is assumed throughout the section so that kac1 and krac1 where a is a typical dimension of thestructure. Statistical equations for vibration and radiation from a rectangular plate will be derived using point process theoryand then they will be approximated using those of a circular plate under the assumption that detailed boundary shapebecomes unimportant for short wavelengths. Discrepancies resulting from the approximation will be discussed in Section 4,in comparison with MC simulation. Note that a rectangular plate of plane dimensions Lx � Ly will be assumed in this section,but the theoretical formulae derived can be generalized to plates of arbitrary shape.

3.1. Vibration response

The statistics of random plate vibration derived in [17] can be extended to investigate the degree of dominance of thereverberant field, for later comparison with the radiation. While the term Re½ΠoΠi� in Eq. (7) corresponds to the individualvelocity at points x1 and x2, subjected to a point source at x0; the second term J0ðkr12Þ is related only to the distancebetween the two measuring points. When the two points are far away from the source point but at an identical location, i.e.x0ax1 ¼ x2, the effect of the term 2þqðΛÞ becomes insignificant and thus the entire term in the square bracket on the right-hand side of Eq. (7) becomes approximately equal to unity. This means that the effect of the GOE factor qðΛÞ becomesnegligible in the far field region and thus the results assuming GOE statistics will agree with those from Poisson statistics.However, when the two points coincide with the excitation point, x0 ¼ x1 ¼ x2, this square bracket term takes the value3þqðΛÞ and thus GOE statistics well estimate the vibration behavior at the drive point as investigated in [14]. From Eqs. (6)and (7), the ratio of squared modulus of the reverberant field compared to the direct field Rvib ¼ E½jurevj2�=jE½u�j2, orequivalently κ2½u;un�=jE½u�j2, can be expressed at any arbitrary point as

Rvibðω; r01Þ ¼2pΛ

2þqðΛÞþ 1jΠoðkr01Þj2

� �: (14a)

Two extreme cases of interest can be considered: when kr01{1, the term 1=jΠoðkr01Þj2 in the above equation becomesunity, whereas when kr01c1, the term can be estimated by pkr01=2 using the high frequency approximation in Eq. (A3), andfurthermore it can be noticed that ð2þqÞ{pkr01=2. Thus Eq. (14a) can be approximated by

Rvibðω; r01Þ �2pΛ 3þqðΛÞ½ � kr01{1kr01=Λ kr01c1

(: (14b)

Eq. (14b) confirms again the importance of GOE statistics near the drive point. For points remote from the driving point,the equation is surprisingly simple at high frequencies, equal to the ratio of the Helmholtz number kr01 to the modal overlapfactor Λ. Equivalently, it is the number of wavelengths within the distance, relative to the number of modes within afrequency band.

In addition, the ensemble average of the total energy E of the plate vibration can be calculated using the spatialaverage of the squared velocity, ⟨E½juj2�⟩ where ⟨…⟩ means a spatial average. This average can be obtained by integrating


Eqs. (6) and (7) when x1 ¼ x2 with E½juj2� ¼ jE½u�j2þκ2½u;un�, leading to

⟨E½juj2�⟩¼ νp

2M2ωηAvib (15)

where

Avib ¼pΛ

2Splate

ZjΠoj2 dSplateþ

ð2þqðΛÞÞSplate

ZΠoj2 dSplateþ1��

:

�(16a)

The first term in Eq. (16a) is from the direct field in Eq. (6) and the second and the third from the reverberant field inEq. (7). The equation can be further simplified by using a circular-plate assumption. In a diffuse field, the ensemble averageof a two-point correlation is a cylindrically-symmetric Bessel function as in Eq. (3). In the high frequency range when2p=k{Lx � Ly, effects from the detailed boundary geometry of the plate should tend to become insignificant, and the platecould then be approximated by a circular plate with the same area, centered on the excitation point. At high frequencies, theintegral in Eq. (16a) can be expressed as Eq. (A4) (for details, see Appendix A),

Avib �2Λka

þ4þ2qp

2ka

þ1� �

� 2Λka

þ1� �

(16b)

where pa2 ¼ LxLy. The ensemble average of the total energy E½E� ¼ ⟨E½juj2�⟩ρhSplate=2 can be expressed as E½E� ¼ νpAvib=ð4MωηÞfrom Eq. (15), and furthermore the energy dissipated ωηE½E� can be found to equal E½W in�Avib, where the ensemble average ofthe power input through the point excitation E½W in� ¼ ReðFE½u�nÞ =2¼ νp=ð4MÞ for unit force F¼1 and the mean velocityfrom Eq. (6) at the drive point. Then, Avib in Eq. (16b) shows the contribution of the direct and the reverberant vibrationfields to the average dissipated energy, which depends again on the relation of ka to Λ. The factor becomes one, if 2Λ{ka,which means that the system is dominated by the reverberant field as Eq. (15) becomes ⟨E½juj2�⟩� κ2½u;un�jx0{x1 ¼ x2 , andtherefore it agrees well with the power balance analysis for the reverberant field in [14]. This case occurs whenever thesystem is lightly damped, and thus may be applied to a wide range of sound radiation problems. For other cases, the directfield term also needs to be considered for the total average energy, and the contributions of the two fields to the energydissipated can be estimated by Eq. (16b).

Such statistical vibration characteristics influence the statistics of sound radiation, to which we now turn.

3.2. Pressure response

The ensemble average of pressure response E½p�, or κ1½p�, can be expressed in terms of the average of plate velocity E½u�through Eq. (10), and thus the average pressure can be expressed by substituting Eqs. (6) into (10):

E½p� ¼ CU1ðk;Θ;ΦÞ; (17)

where the volume velocity U1 at angle ðΘ;ΦÞ is

U1ðk;Θ;ΦÞ ¼ pν2M

Z Ly=2

�Ly=2

Z Lx=2

�Lx=2Πoðkr01Þeikr Ux1 dx1: (18a)

The above equation can be solved when kac1 and krac1, and the circular plate assumption from the previous sectioncan be applied (for details see Appendix B). The high frequency assumption implies that jk�kr j{jkþkrj, and then Eq. (B4)can be further simplified to give

U1ðk; krÞ �ipνM

ð1�eiaΔkÞΔk

ffiffiffiffiffiffiffikkr

p ; (18b)

where Δk¼ k�kr and kr ¼ jkrj ¼ k0 sin Θ. The equation depends on the amplitude of the two wavenumbers k and kr , whichare plotted as functions of frequency in Fig. 2 for the example plate whose material and geometric properties are given inTable 1. The crossing point of k and kr at 901 defines the critical frequency ωcr ¼ c20

ffiffiffiffiffiffiffiffiffiffiffiρh=B

p, at which the radiated sound

energy is maximized. For other angles, the coincidence frequency ωco is defined when Δk¼ 0 by ωco ¼ωcr= sin2 Θ [6]. For the

frequency range larger than ωcr , the maximum magnitude of Eq. (18b) can be found at the coincidence frequency,U1ðk; krÞ � apν=ðMkÞ.

The dispersion curve for the structural wavenumber k is proportional to the square root of frequency, while kr is linearlyincreasing, but the slope for an angle 901 is steeper than that for 101 where the radiation direction is closer to normal.Therefore, as the angle increases, kr becomes large and the approximation is valid for the high frequency range. On the otherhand, small kr at low angles violates the large wavenumber assumption used for Eq. (18b), and as a result the estimate in Eq.(18b) becomes less accurate, as will be discussed later in this section.

Note that the sound radiation from rectangular and circular plates can be explained in terms of that from an infinite platewith a rectangular and a circular window, respectively, extending the concept of an infinite plate used in Eq. (6). For

comparison, the sound pressure from a point-excited infinite plate is given by [21] pinf ¼ Cω=½2pBðk4�k4r Þ�. The equation can

be rewritten using a partial fraction decomposition, and for high frequency response jk2�k2r j{jk2þk2r j it can be reduced

Table 1Dimensions and properties for the simulations.

Plate Units

Dimensions Lx � Ly 1.2�1.35 mThickness h 0.005 mYoung's modulus Y 195 GPaPoisson's ratio s 0.28Density ρ 7850 kg/m3

AirDensity ρ0 1.29 kg/m3

Speed c0 340 m/s

Fig. 2. Wavenumber–frequency curve of structural wavenumber k (solid line) and kr at Θ¼ 103 (dashed line) and at Θ¼ 903 (dash–dot line).


with Eqs. (4) and (9) to

pinf � Cpν2M

4ΔkðkþkrÞ

: (19)

This equation can be compared with Eq. (18b) multiplied by the constant C. The infinite plate pressure in Eq. (19) has anarithmetic mean of the structural and the acoustic wavenumbers ðkþkrÞ=2 in the denominator since they are decoupled inthe integral equation in Eq. (10), whereas Eq. (18b) shows a geometric mean


pas a consequence of the windowing effect.

The circular window also changes the phase of the pressure so that the absolute value of Eq. (18b) becomes zero whenaΔk¼ 2pn where n is an integer.

Fig. 3 shows the ensemble average for the example plate at angles 11, 101 and 901. The amplitude of the pressure field ofan infinite plate at Θ¼ 0 and ω¼ 0 from Eqs. (4, 9 and 19) was used as a normalizing factor in this figure and throughout:

p0 ¼ρ0

4pρRh: (20)

In the figure, the rectangular plate result from Eqs. (17) and (18a) is the reference with which the other two should becompared. The circular plate estimate from Eqs. (17) and (18b) shows similar features to the rectangular one, even when theangle is as small as 101. The infinite plate from Eq. (19) also tends to follow the trend of the finite rectangular and circularplates, but the energy is concentrated at the coincidence frequency around 2500 Hz as shown in Fig. 3c, which correspondsto the crossing point in Fig. 2. Basically, the difference between the graphs shown in Fig. 3 results from the effects of the twodifferent windows. Thus, the peak from the infinite plate is smoothed out by windows for the other two. Fig. 3a shows theresponses at a very small angle. In such a case, the monopole component is dominant as shown in [6], meaning that theaverage velocity of the plate governs the pressure. The infinite plate radiation agrees well with the rectangular platebecause the rectangular window covers the most significant part of the average velocity. However, the pressures from thecircular plate are different since the high frequency assumption is violated. Zeros from the circular plate graph canbe seen when aΔk¼ 2pn, as predicted by Eq. (18b), except at the coincidence frequency. The locations of these zeros arereasonably similar to those with the rectangular plate at 901, but not at 11 due to the low measuring angle violating theassumption of high kr .

Fig. 3. jE½p�j=p0 at (a) Θ¼ 13 , (b) 103 and (b) 903 with the rectangular plate from Eq. (18a) (solid lines), the circular-plate assumption from Eq. (18b) (dashedlines), infinite plate from Eq. (19) (dash–dot lines).


3.3. Ensemble average of sound intensity

The ensemble-averaged intensity at a given measuring point can be represented through Eq. (12):

E I½ � ¼ E½jpj2�2ρ0c0

: (21)

The sound pressure from a vibrating plate consists of radiation from the reverberant vibration field as well as from thedirect field. Thus the pressure due to the reverberant field is the total response minus the contribution from the direct fieldprev ¼ p�E½p�, and the ensemble average of the squared modulus of the pressure equals the second cumulantκ2½p; pn� ¼ E½jprevj2�. The intensity can be estimated using the second cumulant κ2½p; pn� via

E½jpj2� ¼ jE½p�j2þE½jprevj2�: (22)

Since E½p� was obtained in the previous section, the second cumulant will be estimated in this section.

3.3.1. Reverberant responseThe reverberant response can be calculated by using Eqs. (7) and (10) (for detail see Appendix C) and expressed as

E½jprevj2� ¼2jCj2pΛ

U2ðk;Θ;ΦÞ ð2þqðΛÞÞΓþ1½ �; (23)

where

U2ðk; krÞ ¼ν2p2

4M2

ZS

ZS

J0ðkr12Þeikr U ðx2 �x1Þ dx1 dx2; (24a)

the ratio of the two integral Γ¼ jU1j2=U2 with U1 from Eq. (18a). The above equation U2 is dependent on the relativedistance between x1 and x2, but not on the excitation point x0. By contrast with the integrations in Sections 3.1 and 3.2, thetwo monitoring locations are independent and Eq. (24a) is a double integral over the plate area. As assumed when derivingEq. (18b), the circular plate approximation at sin Θc1=k0a in the high frequency range can also be used for Eq. (24a) (seeAppendix B) and, with the high frequency assumption ðk�krÞ2{ðkþkrÞ2, it becomes

U2ðk; krÞ �ν2p3

2M2

a½1� sincð2aΔkÞ�Δk2


p ; (24b)

where sincð2aΔkÞ ¼ sin ð2aΔkÞ=ð2aΔkÞ. The maximum of Eq. (24b) at the coincidence frequency becomesU2ðk; kÞ � ν2p3a3=ð3M2kÞ. The ratio Γ in Eq. (23) can be approximated by Eqs. (18b) and (24b) as

Γ� 4

paffiffiffiffiffiffiffikkr

p ð1� cos ðaΔkÞÞð1�sincð2aΔkÞÞ: (25)

Fig. 4 shows E½jprevj2� in Eq. (23), relative to jp0j2, at three different angles using both the rectangular plate from Eqs. (18a)and (B1) and the circular-plate assumption from Eqs. (18b) and (24b). The approximation remains surprisingly close even at101 as shown in Fig. 4b, although it deviates from the rectangular plate result at 11 as shown in Fig. 4a.

It is interesting to see the intensity ratio between the responses from the reverberant and direct vibration fieldsRrad ¼ E½jprevj2�=jE½p�j2, showing the relative importance of the reverberant field compared to the direct field. This can beobtained from Eqs. (17) and (23) as

Rradðω; krÞ ¼2pΛ

2þqðΛÞþ1Γ

� �: (26a)

Fig. 4. E½jprevj2�=jp0j2 at (a) Θ¼ 11 , (b) Θ¼ 101 and (c) Θ¼ 901 , for the integral equation with the rectangular plate from Eqs. (18a) and (B1) (solid line)and its circular plate estimate from (18b) and (24b) (dashed line).

Fig. 5. Rrad at (a) Θ¼ 11 , (b) Θ¼ 101 and (c) Θ¼ 901 , from the ratio in Eq. (26a) and Γ with the rectangular plate from Eqs. (B1) and (18a) (solid line)and from Eq. (26b) (dashed lines).


Eq. (26a) can be compared with the vibration field ratio Rvib in Eq. (14a). It can be seen from Eqs. (25) and (26a) that, asthe angle Θ decreases to zero directing normal to the plate, the term ð2þqÞ becomes more influential to Rrad, just as it is toRvib when r01-0, so that the term q remains important for the response at lower angles. However, the small angle case isrelevant to a very limited range at high frequencies. Therefore, in the high frequency region of interest, Rrad is moreinfluenced by 1=Γ than ð2þqÞ at large angles Θ, which is compatible with Eq. (14a) where Rvib is more affected by 1=jΠoj2when r01c1. The nominal value of the Rrad can be approximated as

Rradðω; krÞ �2pΛ

2þqðΛÞþpaffiffiffiffiffiffiffikkr

p4

" #: (26b)

Fig. 5 shows the ratio from Eq. (26a) at three angles, calculated using Γ with the rectangular plate from Eqs. (B1) and(18a) and with the circular plate from in Eq. (26b). The trends of the plots are well represented by the estimates at all theseangles. The ratio estimate at 4000 Hz increases from one at Θ¼ 13 to 2.26 at Θ¼ 903, showing reverberant field dominanceat large angles. Note that Fig. 5a also shows relatively good agreement at 11 even though the closed forms of E½jprevj2� andjE½p�j2 show discrepancies from the original integral equations as shown in Figs. 3a and 4a. This is because the errors fromthe two estimates are of similar order.

3.3.2. Ensemble intensityThe sound intensity is calculated from the average of the squared modulus of pressure by substituting Eqs. (22) into (21),

and can be rewritten using the ratio Γ¼ jU1j2=U2:

E I½ � ¼ jCj2ρ0c0pΛ

U2ðk; krÞAradðk; krÞ (27)

where Aradðk; krÞ ¼ ½pΛΓ=2þð2þqÞΓþ1�, and this factor can be approximated with Γ� 4=ðpaffiffiffiffiffiffiffikkr

pÞ from Eq. (23) as

Aradðk; krÞ �2Λ

affiffiffiffiffiffiffikkr

p þ4ð2þqðΛÞÞpa


p þ1

" #: (28a)

The amplitude Arad is in an identical form comparing with Avib in Eq. (16b). The average intensity E½I� in Eq. (27) can beapproximated by E½Irev�Arad, where E½Irev� ¼ E½jprevj2�=ð2ρ0c0Þ from Eq. (23) with Γ{1=ð2þqÞ at high frequencies, and Arad

indicates contribution of the pressure field from the direct and the reverberant field, to the total sound intensity. If the directfield term in Eq. (28a) is as small as 2Λ=ða

ffiffiffiffiffiffiffiffikkr

pÞ{1 , then the factor becomes one, meaning that the sound intensity results

only from the reverberant field, and otherwise direct field needs to be considered. Note that Eq. (28a) is derived assuming

Fig. 6. Normalized intensity E½I� = ðjp0j2=2ρ0c0Þ at (a) Θ¼ 11 , (b) Θ¼ 101 and (c) Θ¼ 901 , estimated with the rectangular plate from Eqs. (18a) and (B1)(solid lines), with the circular-plate assumption from Eq. (24b) and (28b) (dashed lines) and with jpinf j2=jp0j2 (dash–dot lines).


that the measuring point is far from normal direction to the plate and does not accurately estimate the factor Arad near thatdirection, although the area is small.

Fig. 6 shows the intensity calculated by Eq. (27) with Eqs. (18a) and (B1) for the rectangular plate, (24b) and (28a) for thecircular-plate assumption, and (19) for jpinf j2. The circular plate estimates show good agreement with the original integralequations in Fig. 6b and c for 101 and 901, but Fig. 6a shows significant discrepancies, as can be expected from Fig. 4a, due tothe very small monitoring angle. At the coincidence frequency when Δk¼ 0, the ratio Γ can be calculated as Γjk ¼ kr ¼ 3=ðpkaÞusing the L'Hospital theorem, and Arad in Eq. (28a) can be expressed and approximated as

Aradðk; kÞ �3Λ2ka

þ3ð2þqÞpka

þ1� �

� 3Λ2ka

þ1� �

: (28b)

3.4. Radiated sound power

In addition to the local intensity, the total radiated power from the direct radiation can be obtained from Eq. (13),integrating the intensity from Eq. (27) over the hemisphere, which can be rewritten in terms of U2 from Eq. (B1) and Arad

from Eq. (28a):

W ¼ 2R2pjCj2ρ0c0

2pΛ

Z p=2

0U2ðk; krÞAradðk; krÞ sin Θ dΘ: (29)

This integral can be further simplified by the fact that the integrand has its peak at Δk¼ 0. Arad in the above equation canbe calculated at the coincidence frequency k¼ kr as in Eq. (28b) and then taken out of the integral. Furthermore, the integralin W can be expressed with U2 in Eq. (B6) in terms of kr ¼ k0 sin Θ, and then, using sin c function identities in [22, formulae6.554.2], Eq. (29) can be expressed with Eq. (28b) as

W � p2ρ0ων

8c0M2η

3Λ2ka

þ1� �

U3 (30)

where

U3 ¼Z 2a

0ð2a�rÞ2J0ðkrÞ sin cðk0rÞr dr: (31)

Radiation efficiency s¼W=ðρ0c0Splate⟨juj2⟩=2Þ represents the total power radiated from the plate, relative to that of anequivalent baffled piston source with the identical average velocity [4]. The spatial average of the squared modulus of thesurface velocity ⟨juj2⟩¼ R

E½juj2�dSplate can be calculated as in Eq. (15) so that the radiation efficiency can be estimated withAvib from Eq. (16b) as

s� pk204Splate

½3Λ=ð2kaÞþ1�½4Λ=ð2kaÞþ1�U3: (32)

4. Monte Carlo simulations

4.1. Plate model

For MC simulations, a thin rectangular Kirchhoff–Love plate is assumed with properties as listed in Table 1, with simply-supported boundaries and centered at the origin of x-y coordinates. The mode shapes of this bare plate are normalized by


Rρhϕ2

mn dSplate ¼ 1 and will be used for basis functions in what follows:

ϕmnðx; yÞ ¼2ffiffiffiffiffiM

p sinmpLx

x�Lx2

� � sin

npLy

y�Ly2

� � : (33)

Uncertain dynamic properties are modeled with point masses randomly distributed over the plate and, in terms of thebasis functions in Eq. (33), mode shapes ϕn of the plate with the randommasses are obtained by the Lagrange–Rayleigh–Ritzmethod (Meirovitch [18]). The number of point masses will be 10 or 40, to compare two different levels of uncertainty. Eachpoint mass has magnitude 0.3179 kg, 1/200 of the plate mass, and thus the plate with 10 masses has 5 percent more massthan the bare plate while the one with 40 masses has 20 percent more mass. By using the modal expression in Eq. (2), thevelocity of the random plate subjected to a point excitation at the origin was obtained. Two value of the loss factor have beenused: η¼0.1 or 1 percent.

Sound radiation from the plate vibration to the hemisphere was calculated with the Rayleigh integral using a 2D Fouriertransform as in Williams [21]. The simulation was repeated 1000 times for each case, with different random mass positions,to obtain the ensemble statistics. Note that although the point force at the center would excite only a quarter of the modesof a bare plate, most of the modes are excited for the plate with randomly-located point masses since the scatterers breakboth symmetries of the plate [17]. For the numerical simulations, a spatial grid spacing was chosen of approximately13.3 mm in the x and y directions, and thus the wavenumber along the plate surface can be as high as 78 m�1, if theminimum number of nodes per wavelength is to be six. The selected frequencies range from 50 to 4000 Hz in 50 Hz steps.Throughout the following section, estimates derived with the circular-plate assumption will be compared with thesimulation results.

4.2. Results and discussion

The ensemble average of the sound pressure from the vibrating plate was calculated via this MC simulation, andcompared with the estimates from Eqs. (17) and (18b). Fig. 7 shows the average pressure from two plates with 10 and 40masses, for the case when the loss factor is 1%. The estimate follows the trend of the MC simulations quite well, but in detailthe obvious peaks and dips do not match. These fluctuations arise because wave reflection from the plate boundary is to anextent coherent with the incident field, and the two coherent fields interact destructively or constructively. It would bepossible in principle to take this effect into account. Since the rectangular plate used in the example has boundaries at fixedlocations, the coherence of the reflected waves is directly related to the number and the dynamic properties of the scatterersalong the propagation path, and the degree of the coherence can be quantitatively estimated [17]. The wave reflections onthe plate can be expressed in terms of a series of image sources, including a statistical loss factor [17]. However, detailedpredictions incorporating this effect are not pursued here.

The sound radiation from the reverberant vibration field is considered next. Two different loss factors were tried: η¼0.1percent and 1 percent, with which the modal overlap factor Λ¼ωην becomes 0.1 and 1 at 925 Hz, respectively. The results areshown in Fig. 8, for the two different numbers of added masses. The MC simulation results show maximum magnitude at thecoincidence frequency because pressure is maximized at this frequency, and the estimates also have their largest magnitude atthis frequency. It is immediately obvious that the smaller loss factor gives a higher magnitude for the second cumulant, aswould be expected from the modal overlap factor in Eq. (23) meaning that the energy of the plate is inversely proportional toits loss factor. Note that the closed-form estimates in Fig. 8a and b are identical, calculated from Eqs. (30) and (31).

Fig. 7. Pressure response from a plate with η¼ 1% at (a) 101 and (b) 901: Monte Carlo simulation with 10 masses (dash–dot) and 40 masses (dashed), andits circular plate estimate from Eqs. (17) and (18b) (solid line).

Fig. 8. E½jprevj2�=jp0j2 with two different loss factor at 901: 1% and 0.1%, and (a) 10 and (b) 40 point masses. Monte Carlo simulation (solid line) and closedform with the circular-plate assumption from Eqs. (23), (18b) and (24b) (dashed line).

1970 1980 1990 2000 2010 2020 2030 20400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Frequency (Hz)

p(f n)

104080

Fig. 9. Probability density function of 1000 natural frequencies approximately at 2000 Hz. The larger the number of the scatterers, the broader thedistribution becomes.


The MC simulations with 10 masses in Fig. 8a show more peaky response than the ones with 40 masses in Fig. 8b. For abare plate, energy is usually concentrated at the natural frequencies of the plate so that the power spectrum of the soundradiated from the plate also looks very peaky. However, when scatterers are added to the plate the natural frequencies arerandomly shifted, and as a result the ensemble average of the energy is spread into a smoother pattern. The extent of thisspreading is dependent on the level of uncertainty of the plate, in this case the location and density of the scatterers, andthus Fig. 8b shows smoother response compared with Fig. 8a. For example, the probability density function (PDF) of naturalfrequencies for a given mode number can be collected during each set of simulation runs, to give an indication of the degreeof the shift. Fig. 9 shows the PDFs of mode numbers 204, 213, and 225 (each at approximately 2000 Hz in the relevantensemble) across ensembles of systems with 10, 40 and 80 masses, respectively. As the number of scatterers increases, thewidth of the PDF increases: the standard deviations with 10, 40 and 80 masses are 37.9 Hz, 45.4 Hz and 56.5 Hz, respectively,and therefore the smoothing effect increases. Further investigation of the effect of the scatterers can be found in [17].

Fig. 10 shows the relative variance, i.e. the ratio of the second cumulant function to the squared modulus mean value. Theestimate without the fluctuation term from Eq. (26b) is shown in the figure, which is proportional to 1=ω1=4 fromffiffiffiffiffiffiffikkr

p�ω3=4 and Λ�ω for a plate. It follows reasonably well the trend of the MC simulation with 40 masses and seems less

close with 10 masses at high frequencies above 2000 Hz, since the effect of coherent reflections is stronger in the latter case.Fig. 11 shows the ensemble average of the power radiated from the random plates, relative to a reference, W0 ¼

jp0j22pR2=ð2ρ0c0Þ. The maximum power is radiated at the critical frequency (approximately 2500 Hz). The case with η¼1percent shows less power radiation than the one with η¼0.1 percent, as would be expected. Note that, in Fig. 11a and b, theestimates are identical while the MC simulations are different. Finally, radiation efficiencies for the plates with 10 and 40added masses are shown in Fig. 12. The estimate shows good agreement above 2 kHz, but in the low frequency region the

Fig. 10. Intensity ratio fields Rrad ¼ E½jprevj2�=jE½p�j2 at 901, using Monte Carlo simulation results with 10 masses (dash–dot line) and 40 masses (dashed line)and using circular plate estimate from Eq. (26b) (solid line).

Fig. 11. Radiated power W relative to W0 with (a) 10 and (b) 40 random point masses. MC simulation (solid lines) and the circular-plate estimation fromEqs. (30) and (31) (dash–dot lines) for loss factor η¼1% and 0.1%. Note that the estimates in (a) and (b) are identical.

Fig. 12. Radiation efficiency from Monte Carlo simulation for plate with 10 (dash–dot) and 40 (dashed) masses and η¼ 0:1 % and the estimate fromEq. (32) (solid).


estimate show discrepancies with the MC simulation, which can also be seen in Fig. 11. This is because coherent reflectionshave more influence in that frequency region [24], but the estimates do not consider these reflections at all. In general,however, the estimates from Eq. (27) show encouraging agreement, particularly at higher frequencies.


5. Conclusion

The ensemble statistics of sound radiation into unbounded space from vibrating plates with uncertain dynamicproperties, subjected to harmonic point excitation, have been investigated. The ratio of reverberant and direct vibration fieldat any arbitrary point on the plate was equated, which can be expressed in terms of ka and Λ, and furthermore theircontribution on the dissipated energy was derived. The ensemble average of sound pressure from the plate was thenestimated at a measuring point in the far field, on a surrounding hemisphere above the plate. Next, the pressure from thesquared reverberant vibration field (or the second cumulant function) was derived for the sound pressure. Since theinfluence of the boundary details of the plate reduces at high frequencies, a circular plate assumption could be introduced tosimplify the integral expressions, and using this assumption the equations could be expressed in closed form. The soundintensity was estimated in terms of a


pand Λ: Poisson statistics for natural frequencies were shown to model the

ensemble average of the reverberant pressure field well at high frequencies and large angles, but GOE statistics make asignificant difference at small angles close to normal direction. Similar behavior had previously been shown for the vibrationstatistics at a point on the plate, depending on whether it was at a large or small distance from the drive point. In addition,power radiated from the plate was expressed in terms of ka and Λ. Radiation efficiency was also derived in a simple form.Monte Carlo simulations were performed to test the result, and they show generally good agreement over most of thefrequency range, even though the closed forms were derived under a high-frequency assumption.

Acknowledgments

This work was supported by Korea Science and Engineering Foundation Grant (No. D00005) and Cambridge OverseasTrust (COT). The authors thank two anonymous reviewers for constructive comments.

Appendix A. Circular plate assumption—vibration

The integral term in Eq. (16a) is rewritten as B,

B¼Z Ly=2

�ðLy=2Þ

Z Lx=2

�ðLx=2Þ

��Hð1Þ0 ðk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þy2

qÞ�Hð1Þ

0 ðikffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þy2

qÞ��2

dx dy: (A1)

As mentioned in Section 3.1, assuming that the boundaries have insignificant effect on the vibration behavior, a circularplate assumption can be adopted provided it has the identical area, and the integral in Eq. (A1) can then be rewritten as

B� 2pZ a

0

��Hð1Þ0 ðkrÞ�Hð1Þ

0 ðikrÞ��2r sr; (A2)

where pa2 ¼ LxLy. In addition, the near-field wave component Hð1Þ0 ðikrÞ becomes negligible when krc1, and using the high

frequency approximation [23, formulae 9.2.3],

Hð1Þ0 ðkrÞ �

ffiffiffiffiffiffiffiffi2pkr

rexp ikr�p

4

�: (A3)

This approximation obviously does not apply when r approaches zero, but in the high frequency range this exceptionalregion has little effect on the value of the integral in Eq. (A1). Thus, B in Eq. (A1) can be obtained as

B� 2pZ a

0

2pkr

r dr¼ 4ak: (A4)

Appendix B. Circular plate assumption — radiation

For E½p�, The Cartesian coordinates in Eq. (18a) can be replaced by cylindrical coordinates, if the boundary haveinsignificant effect on the response, and then it can be simplified by setting x0 ¼ ð0;0Þ and using a Bessel function identity[22, formula 8.411.1], independent of Φ:

U1ðk; krÞ �νp2M

Z a

0

Z p

�pΠoðkrÞeikrrr dΘ dr � 2p

Z a

0ΠoðkrÞJ0ðkrrÞr dr; (B1)

where pa2 ¼ LxLy and kr ¼ jkrj ¼ k0 sin Θ.When Θ¼ 0, the Bessel function J0ðkrrÞ ¼ 1 and then the integral can be expressed by using [22, formulae 5.52.1],

U1ðk;0Þ �p2aνkM

Hð1Þ1 ðkaÞ�Hð1Þ

1 ðikaÞh i

: (B2)

For the case of Θa0, the near-field wave component in the outgoing wave Πo, the second term in Eq. (8a), becomesnegligible when krc1, and then it can be approximated by Hð1Þ

0 ðkrÞ. The Bessel functions in the integrand can also be

Fig. A1. Schematic diagram for (a) Lx�jxj and (b) its equivalent, ð2a�rÞ ffiffiffip

p=2.


approximated by Eq. (A3) and by a trigonometric function as in [23, formulae 9.2.1]

J0ðkrrÞ �ffiffiffiffiffiffiffiffiffi2

pkrr

scos krr�

p4

�(B3)

for krc1 and krrc1. The second condition is dependent on the monitoring angle as well as the frequency and can besatisfied if krac1, meaning sin Θc1=k0a. Then Eq. (B1) can be estimated in closed form,

U1ðk;ΘÞ �4ffiffiffiffiffiffiffikkr

p Z a

0eiðkr�ðp=4ÞÞ cos krr�p

4

�dr¼ 2ffiffiffiffiffiffiffi

kkrp 1�eiaðkþkr Þ

kþkrþ i

1�eiaðk�kr Þ

k�kr

� �: (B4)

For E½jprevj2�; Eq. (24a) can be reduced to an area integral by using the variable x¼ x2 �x1 as in [19, p.56]:

U2ðk;Θ;ΦÞ ¼ ν2p2

4M2

Z Ly

�Ly

Z Lx

� LxðLx�jxjÞðLy�jyjÞJ0ðkjxjÞeikr Ux dx dy: (B5)

Using the circular-plate assumption used in Eq. (B1), the Cartesian coordinates are replaced by polar coordinates andthe term ðLx�jxjÞðLy�jyjÞ can be estimated by the equivalent approximation ð2a�rÞ2p=4 when pa2 ¼ LxLy as in Fig. A1 so thatEq. (B5) becomes

U2ðk; krÞ ¼ν2p4

8M2

Z 2a

04a2J0ðkrÞJ0ðkrrÞr drþ

Z 2a

0ð�4arþr2ÞJ0ðkrÞJ0ðkrrÞr dr

" #: (B6)

When Θ¼ 0, J0ðkrrÞ ¼ 1, and Eq. (B6) can be solved by integration by part, expressed with the Bessel and Struve functionHi [22, formulae 5.52.1] and [21, formulae 11.1.7],

U2ðk;0Þ ¼aν2p4

2M2k32akJ0�2J1ðkaÞþpkafH0ðkaÞJ1ðkaÞ�H1ðkaÞJ0ðkaÞg� �

: (B7)

At high frequencies, this equation can be expressed in the form of trigonometric functions, using [23, formulae 12.1.30,9.2.2] and then further simplified by using [23, formulae 9.2.1],

U2ðk;0Þ ¼aν2p4

2M2k31� J1ðkaÞ� �

: (B8)

When Θa0, the first integral involving the term 4a2 in Eq. (B6) is solved byRJ0ðkrÞJ0ðkrrÞr dr¼ ½krJ1ðkrÞJ0ðkrrÞ�

krrJ0ðkrÞJ1ðkrrÞ�=ðk2�k2r Þ in [22, formulae 5.54.1], which can be further estimated by trigonometric functions when kr andkrrc1 as in Eq. (B3). The Bessel functions in the second integral in Eq. (B6) are again approximated by cosine functions andthen readily solved by integration by parts. As a result, Eq. (B6) can be simplified as

U2ðk; krÞ �ν2p3

4M2ffiffiffiffiffiffiffikkr

p 2að1� sin cððk�krÞ2aÞÞðk�krÞ2

�1� cos ððkþkrÞ2aÞðkþkrÞ3

" #: (B9)

Appendix C. Derivation of jU1j2 for Eq. (23)

The second cumulant κ2½p; pn� can be estimated by using Eqs. (7) and (10),

κ2 p; pn� �¼ νpjCj2

2M2ωηð2þqðΛÞÞ

Zx2

Zx1

Re½Πoðkr01ÞΠiðkr02Þ�eikr U ðx2 �x1Þ dx1 dx2þU2

� ; (C1)

where U2 is in Eq. (24a). The integrand in the first integral term of the equation can be expressed as

Re½Πð1Þo Πð2Þ

i � ¼ Re½Πð1Þo �Re½Πð2Þ

i �� Im½Πð1Þo �Im½Πð2Þ

i �: (C2)


where Πð1Þo ¼Πoðkr01Þ and Πð2Þ

i ¼Πiðkr02Þ, and since Πo and Πi are a conjugate pair the identity can be rewritten as


i � ¼ Re½Πð1Þo �Re½Πð2Þ

o �þ Im½Πð1Þo �Im½Πð2Þ

o �: (C3)

Thus, the following integral can be solved and compared with Eq. (B1) using this equality:

ν2p2

4M2

Zx2

Zx1


i �eikr U ðx2 �x1Þ dx1 dx2 ¼ν2p2

4M2

Zx1

Re½Πð1Þo �eikr Ux1 dx1

��2

þZ

x1Im½Πð1Þ

o �eikr Ux1 dx1

��2

" #

¼ ν2p2

4M2

Zx1

Πð1Þo eikr Ux1 dx1

��2

¼ U1j2�� (C4)

Eq. (C1) becomes

κ2 p;pn� �¼ νpjCj2

2M2ωηðð2þqðΛÞÞ U1j2þU2Þ:

�� (C5)

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