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Sound radiation from plates with density and stiffness discontinuities
Journal of Sound and Vibration (1972) 21(2),
SOUND
RADIATION
FROM
STIFFNESS
193-203
PLATES
WITH
DENSITY
AND
DISCONTINUITIES M. S.
HOWE
Department of Mathematics, Imperial College of Science and Technology, London S. W.7, England AND
M. HECKL Miiller-B.B.N., Munich, West Germany
(Received 28 October 1971)
This paper is concerned with the theory of sound radiation from infinite, plane plates due to the interaction of bending waves with density and stiffness fluctuations in the material of the plate. Because of the great variety of configurations encountered in practice the theory is worked out in the case in which the variations in density and/or bending stiffness may be regarded as random functions of position on the plate. This should give valuable insight into the order of magnitude of acoustic losses to be encountered in engineering practice. Cases where the plate may be regarded as loaded with randomly sited point masses, and randomly sited parallel line masses, are considered in detail. Experiments
conducted with a large steel plate in air show a large measure of agreement with the theory. 1. INTRODUCTION
over a homogeneous thin plate of uniform thickness and bending stiffness is effectively decoupled from an ambient atmosphere provided that the wavelength is sufficiently long [l, Cremer and Heck1 19671. The wave field developed in the fluid decays exponentially with distance from the plate. This means that the wave propagates over the plate without the attenuation of total plate-wave energy, in other words, without the transfer of energy from the plate into the surrounding medium in the form of sound waves. In practice no plate has exactly uniform thickness or bending stiffness. For example, in practical structures plates are often held in position by means of rivets or coplanar beams, and these sometimes tend to behave as concentrated mass loadings. In this paper we attempt to represent such loadings as fluctuations in the mass density of the plate. These fluctuations may be regarded as waves of zero frequency so that wave-wave interactions with a long bending wave might be expected to generate a whole spectrum of bending waves which are acoustically coupled to the ambient fluid. Equivalently, the mass loadings may also be regarded as secondary sources due to the fictitious forces which must be applied at the loads to overcome the extra inertia of the plate. Mass loadings can also induce variations in the effective bending stiffness of the plate. However it is shown here that such variations in the bending stiffness behave as secondary sources of higher order and consequently generally radiate less energy than mass density fluctuations. It is often important to have a measure of the magnitude of the sound radiated into the 193 A bending wave propagating
194
M. S. HOWE AND M. HECKL
fluid by these mechanisms. This is conveniently expressed in terms of a loss,factor, E, of the incident bending wave defined by E=
acoustic energy radiated per cycle per unit area of the plate 27~x reversible energy per unit area of the plate ’
(1)
A great simplification in the analysis ensues if it can be assumed that the fluctuations in the mass density and/or bending stiffness of the plate are random. Actually no array of rivets or ribs can be perfectly regular, and it seems desirable, therefore, to construct a stochastic model to describe the effect of such loadings in rather general terms. This should give valuable insight into the order of magnitude of the acoustic losses to be expected in typical situations. In this paper a theory is proposed along such lines to account for the acoustic attenuation of long bending waves due to the presence of random mass and stiffness fluctuations (see sections 2 and 3). In section 4 the statistical specification of point and beam (rib) loadings is discussed, and the results are applied to point loaded and beam loaded plates. Finally, section 5 presents the results of an experimental investigation in which a 4 mm steel plate was used, loaded with a random array of point masses, and vibrating in air and also in water.
2. BENDING
WAVES
ON AN INHOMOGENEOUS
FLUID
LOADED
PLATE
Consider an infinite plane plate of density cr per unit area immersed in an unbounded compressible fluid of density p,-,.In the undisturbed state the plate occupies the plane z = 0 of a rectangular coordinate system. If h denotes the thickness of the plate, E and y respectively the Young’s modulus and Poisson’s ratio of the material of the plate, then the bending stiffness, B, is given by
(2)
B = Eh3/[12(1 - r*)].
The (small) deflexion, q = v(x,y, t), of the surface from the undisturbed planar form then satisfies the bending wave equation a$
+ V*(BV*) q = Ap,
(3)
with V2 E 82/8x* + a*/+*. In this equation Ap denotes the net perturbation pressure of the fluid in contact with the plate: i.e., the difference between the fluid pressures on the negative and positive sides of the plate. Let us define the Fourier transformf(k, w) of a functionf (x, t) of x = (x,~) and the time t by means of the reciprocal relations
(4) f(x, t) = li f(k, w) ei(k.x-wt)dk do. -m
I
Then, by using the linearized, inviscid form of the Navier-Stokes equation it readily can be shown that the Fourier transform of Ap on the plate, z = 0, is related to that of r] by
2iw*PO77@, w) Ap@,w> = (&/a* - k *)I 12’
(5)
SOUND RADIATION
195
FROM PLATES
where a is the speed of sound in the fluid, and the branch of (w2/a2 - k2)1’2 having positive imaginary part for -ak < w < ak is taken. Consider now the effect on equation (3) of allowing the mass density u to contain small random fluctuations. To do this we denote the mean density by mo, and define a dimensionless random function f(x) by 0 = m0{1 + &)), (6) with the condition (c(x)) = 0. (Angle brackets and, where convenient, an overbar, denote averages taken over an ensemble of plates.) Then equation (3) may be written in the form
where L is a linear operator obtained with the aid of equation (5) and containing no random quantities. We now decompose the wave field rl into an ensemble average y and a random fluctuation about this average denoted by 7’: 7j = +j+ 17’.
(8)
The lowest order equations describing the evolution of these quantities have the forms (see, for example, [2, Howe 19711) L+j=-mo(l(x)?$),j 827
Lq’ = -m. S(x) p
Pa, b) . i
The wave problem associated with these equations is posed in the following way. An incident coherent wave +j propagates over the plate and generates a random field $, determined by equation (9b), by scattering off the random density fluctuations. These random waves are formed at the expense of the coherent wave whose amplitude therefore decreases. Actually equation (9b) is essentially a local Born approximation and is therefore unsuitable for determining 7’as such. However, because 9’appears in equation (9a) as a mean value in combination with the random fluctuations f(x), we see that equation (9b) is a good approximation for determining this mean value provided only that the Born approximation is valid over .distances of the order of the correlation scale of the fluctuations c(x). Now two cases must be distinguished. If the frequency of the coherent wave is sufficiently high, i.e. if the bending wave speed is higher than the speed of sound in the surrounding medium, the ambient fluid is forced to oscillate in an acoustic mode with consequent radiation of energy away from the plate. In this case the random fluctuations t(x) change the velocity field on the plate, but have hardly any effect on the sound radiated. If, however, the frequency is below a certain coincidence frequency the coherent wave would radiate no sound energy, and it is only the existence of the scattered waves that causes an energy transfer from the plate to the fluid. We shall concern ourselves only with this second class of low frequency, long bending waves. To analyse the properties of the mean wave field +jwe can first determine 7’ in terms of q from equation (9b) and substitute the result into the right-hand side of equation (9a) to obtain an equation for +j alone. This procedure may be executed formally by Fourier analysis. In terms of the definition (4) it is a simple matter to write the solution of equation ,(9b) in the form a0
SSSS [ [or> ‘2
-m
(X, T) e*(K.(x-X)-O(I-T))/L(K, J2) dXdTdK dQ
I
(10)
196
M. S. HOWE AND M. HECKL
In this result the radiation condition is automatically satisfied by taking the path of integration in the complex Q-plane parallel to the real axis but above all the singularities of the integrand: in other words above all the zeros of the dispersion function L(K,@, which are given by 2ipo Q* L(K, $2) E BK4 - m. ii” -
(Q21a2
_
K2)‘/2
=
”
(11)
By considering the mapping from the Q-plane to the Z-plane defined by Z = (BK4 - m. Q2) @*/a2 - K*)“* - zip0 a*,
(12)
it may be shown that when K is real, equation (11) has no roots in Im(SZ) > 0, a result which merely reflects the fact that the undisturbed planar form of the plate is one of stable equilibrium. Indeed, when D traverses a semicircular contour indented at the branch points Q = *aK and the real zeros, Q = iwo, say, [see Figure l(a)] in the anticlockwise sense, the
(a)
(b) Figure 1. Contours in the 9- and z-planes.
image point Z describes a contour of the form illustrated in Figure l(b). Clearly the variation in the argument of Z over a complete circuit of the contour vanishes, and hence expression (1 I) can have no zeros inside the contour of Figure l(a), and hence in the upper complex Q-plane [3, Titchmarsh 1960, seep. 1161.It follows that it is sufficient to evaluate the integrals in equation (10) in the limit Im(Q) -+ +O. 3. THE ACOUSTIC
LOSS FACTOR
We now make the assumption that the random density fluctuations t(x) constitute a stationary random process, and define the correlation function R by
R(x - Y) = (Kx) 5(y)).
(13)
The spectrum function of the fluctuations is then given by means of the definition m
+m=& JR(x) e-lH.xdx. -0J
(14)
SOUND RADIATION FROM PLATES
197
Hence, if the formal solution (10) is substituted into the mean wave equation (9a), and the Fourier transform of the resulting equation is taken, we obtain L(k, co) = rni u4
m+(k-K)dK s Jw,~) -m
05)
’
where the integral on the right is to be evaluated in the limit Im(w) -+ $0 for real w. This is the dispersion relation for the mean wave field. Consider next a monochromatic wave propagating over the plate. More precisely, consider a trapped bending wave whose frequency wo, say, is less than the coincidence frequency, i.e., satisfies k > Iwolla, where k is the wavenumber. In the absence of random density fluctuations k and w. would be related by L(k, wo) = 0. (16) When account is taken of the presence of the random fluctuations the frequency w and the wavenumber k satisfy the mean wave dispersion relation (15), which is an O(p) non-secular correction to the homogeneous dispersion relation (16). The result may be interpreted as a small change in the frequency or in the wavenumber from the homogeneous case of relation (16). It is convenient here to regard it as producing a small change Au, say, in the frequency, the wavenumber being held fixed and real. We therefore set w = w. + Aw,
(17)
where the correction term Aw is O(p) and is to be calculated from equation (15). In fact correct to terms of O(F) we find -sgn(wo) rnz w$ 2
-=
(k
>
I~Ol/a>.
08)
/( --m
The lossfactor of the coherent field may be obtained from the imaginary part of Aw/lwo). The damping is due to two causes. The first comes from that part of the integral in expression
(18) taken around a small semicircular indentation avoiding the real zero of L(K,w,). This corresponds to the scattering of energy out of the coherent wave into trapped bending waves of the same frequency and wavelength, but of arbitrary direction of propagation. The second cause is due to a continuum of scattered waves contributing to that part of the integral over which L(K, wo) is complex. These waves satisfy K < Iwol/a, and are acoustically coupled to the fluid. They attenuate the wave field of the plate by radiating energy into the fluid in the form of sound waves. Now the acoustic loss factor Efor a monochromatic bending wave (k, o) defined by equation (1) may equivalently be expressed in the following analytic form : E = -2Im(Aw)/(wl,
in which the right-hand side is obtained by restricting the range of integration in equation (18) to those wavenumbers K satisfying K < lwl/a (the sufSx zero has been dropped): i.e.,
E =
13
sgn(w) miw2
m. +
* (19) )I s J%,w)
po(2k2 - w2/a2) ImX<‘w”a #C - K) dK j&/a2 - k2(3/2
198
M. S. HOWE AND M. HECKL
Note that the integrand in expression (19) is regular in the range of integration and that there is now no need to give w a small positive imaginary part. The explicit form of equation (19) is
We may briefly quote the corresponding results for the case in which the scattering is not due to the presence of random density fluctuations, but rather to random fluctuations in the bending stiflness, B. By analogy with equation (6) we set B = BoU + P(X))
(21)
and this leads to the following dispersion relation for the mean wave field : L(k, 0) = B,zk4
s
mK4&k - K)dK Jw,4 ’
(22)
where t/t(K)is the spectrum function of the stiffness fluctuations defined in a manner analogous to that for #II) in equation (14). If we now isolate the scattered waves responsible for the acoustic attenuation we obtain a loss factor eB, say, given by
(23) This result is considered further in section 5. 4. ACOUSTIC LOSS FACTORS FOR POINT AND BEAM LOADED
PLATES
We now apply the above theory to plates loaded with a random array of point masses and of parallel line masses (beams or ribs). It will be assumed that the loads do not alter the stiffness of the plate. In order to apply the formula (20) giving the acoustic loss factor we must fkst determine the appropriate spectrum functions, #K), for these cases. Consider then a uniform plate of density M per unit area. In the first instance let point loads each of mass m be randomly distributed at points x, on the plate, so that the effective plate density u has the form a=M+L:m&x-x,). (24) I If there are on average n loads per unit area of the plate, then clearly mo=(a>=M+nm.
(25)
Accordingly the random function f(x) of equation (6) becomes mo&x)=o-
(u)=m[$B(x-xj)-n).
(26)
The correlation function, R(x - y), is now given by R(x - Y) = =-
,2 (m”H.
K(s(x-x~)~(Y-x3>-nz).
(27)
SOUND RADIATION FROM PLATES
199
When the loads are randomly distributed on the plate the probability that the jth mass occupies an element of area dx, is proportional to dx,. The probability that the ith mass occupies dxl, given that thejth mass occupies dx,, is then proportional tof(x‘ - x,)dx,dx,, wheref (z) is a joint probability distribution function which is an even function of its argument and tends to zero as ]z] + 0, and to unity when z is large. In terms of this function it is known that 1<6(x - Xj) S(y - Xi)) = n 6(x - y) -I-nZf(x - y) (28) 1.1 (see, for example [4, Landau and Lifshitz 19591): i.e., that R(x - Y) = n(Mr~o)~ @(x - Y) - ng(x - Y)}.
(29)
In this final expression g(x - y) = 1 -f(x - y) is the “packing function”, which is essentially equal to unity over the region “occupied” by a point mass situated at x = y and vanishes elsewhere. It is related to the mean square fluctuation number density of the point masses, ((AZV)2), say, by the formula m
g(z)dz=
1
_
(30)
n
-CC
([4], see Section 115), and this result may be used as a starting point for determining the form of the packing function in practical situations. Using the form (29) we are now in a position to determine the acoustic loss factor for a trapped bending wave (k,w). Equation (20) gives this in terms of the spectrum function C+(K)of the density fluctuations. From equation (29) we have in fact
(31) where g(K) is the Fourier transform of the packing function. We deduce, therefore, the following integral expression for the acoustic loss factor: (w2/a2- K2)“2 [ 1/b2 - ngQ -
K)]dK (32)
This result shows that the largest loss factors are expected in cases where the relative positions of the point loads are uncorreluted, for, in general, g(K) > 0 (but suthciently small, of course, to ensure that d(K) is always positive), so that the value of the integral in equation (32) is normally less than that obtaining when g(K) = 0. In practice such correlations will always exist. It is of interest, however, to note the form of the result in this singular case of g(K) s 0. Equation (32) then becomes
Now for long waves satisfying k2 < moa2/B, we have, in the integrand of this result, K2 < &/a2 N Bk4/mo a2 < k2
200
M. S. HOWE AND M. HECKL
and hence lwl/a
s
nm2PO
’ 21 n(mo + 2po/k) o
(w2/a2 - K2)1’2 KdK
{mi(w2/u2 - K2) + 4~3 ’
i.e.,
nm2p0 1 - J$$ ’ N nami(m + 2po/k)
tan-i (2))
(34)
which should be compared with the result of Cremer and Heck1 ([l], see Chapter VI). They are concerned with the acoustic power radiated into the fluid by a point force applied to an infinite plate. Our result shows that in the present approximation (g z 0), the interaction between the mean wave and the randomly sited point loads produces an acoustic field equivalent to one that would be generated by a random distribution of point forces each of intensity proportional to the inertia of the mass load. Indeed each point mass m experiences an acceleration w2v so that the equivalent force is just F= mw2q. Cremer and Heck1 show that the acoustic energy generated per cycle by such a force is equal to AE= “1”_‘i’ l - $itan-i awmO (
(za)),
provided that the frequency is well below the coincidence frequency. Hence, noting that the mechanical energy of the plate is equal to +(mo + 2po/k)02j712 and using the definition (1) of the loss factor E we obtain a result identical with equation (34) when there are n masses per unit area. The case of parallel beam or rib loading may be considered in a similar fashion. Suppose that, as a particular example, the beams are parallel to the y-axis and are distributed almost periodically at a distance 1 apart. Then the appropriate form for the correlation function, R(x - X), is given by
R(x-X)=f(; 0)2(s(*-X)-fH(l-2~x-X~)) where His the Heaviside unit function. This ensures that ((AN)2) vanishes in the case of an almost periodic structure. Note that n = l/l is the number of beams per unit length in the x-direction, and m is the mass per unit length of beam. Hence, l-
(36)
where k = (k,, k,), K = (K,, K,.). Substituting this result into (20) gives the acoustic loss factor. It is clear that the presence of the S-function implies that there are no acoustic losses when 1k,, 1 > 1w 1/a, In the important case of long waves (k2 < moa2/B) considered above, the result simplifies to the following expression for the acoustic loss factor: -PO
m2
’ = 2rni I(mo + 2po/k)
1 _ sin($klcos 8) fkl cos 6’
](l +[1 +$($-k2sin2B)r1’2),
(37)
where 0 is the angle between the x-direction and the wavenumber vector k: i.e., k = k(cos8, sin 0). 5. EXPERIMENTAL
INVESTIGATION
OF A POINT
LOADED
PLATE
Before describing the experimental arrangements it is of interest to compare in orders of magnitude the relative importance to the acoustic radiation problem of the random per-
SOUND RADIATION
FROM PLATES
201
turbations being in the bending stiffness B or, as discussed in detail in the previous sections, in the mass density. Referring to equations (20) and (23) makes it clear that the ratio of the acoustic loss factors E and +, due, respectively, to mass and stiffness fluctuations, is given in order of magnitude Ed B,2k4K4 -Np<-N E - mgio4
B;k4 mia4-
(38)
where w, is the coincidence frequency. The final term on the right of expression (38) is therefore small provided that the frequency is well below the coincidence frequency. In the experiments described below we shall assume that the effect of loading the plate with an array of “point” masses does not have an appreciable effect on the bending stiffness of the plate, which effect is therefore neglected in calculating the radiated sound power. In practice bending waves on plates are attenuated not only by the radiation of sound, but also by internal losses, by viscous losses at the plate-fluid interface, and by energy transfer into supports. Since some of these mechanisms give rise to loss factors of the same order of magnitude as the E of equation (32), it would have been impossible to measure the radiation damping by observing the (time or space) decay of bending waves. For this reason a different method was used which is based on the definition of the loss factor given by equation (1). Thus the radiation loss factor E can be expressed in terms of the reversible energy m,, w2q of the plate and the radiated sound power P, say (all quantities being per unit area of the plate). Since the energy lost per cycle as sound is 27rP/w we obtain
?? =P/co3mo~7- -P/t_Omov72,
(39)
where 3 is the mean square plate velocity. The experimental arrangements are illustrated in Figure 2. The mean square plate velocity was measured by means of a pick-up at five different points on the plate. The sound power, P, was measured in two different ways. In air it was possible to place the experimental rig in a fairly reverberant room. Thus the power per unit area could be found from the mean square sound pressure2, the reverberation time T, the area of the radiating surface S (twice the area of the plate), and the volume V of the room:
13.8v -2
p _
p.
a2 TSp ’
The measurements in water were conducted in Lake Ammersee in Southern Germany, which for our purposes behaved as an infinite medium. Therefore the radiated sound pressure was measured at several distances R from the centre of the plate and the power calculated according to the formula p=I_2xRZ p0a
2
PY S
(41)
a mean value being taken. In the measurements a 4 mm thick steel plate 2 m x 1 m was used. The “point” loads consisted of 0.5 kg steel masses glued at randomly selected points of the plate. The results are shown in Figures 3 and 4. Theoretical curves from formula (31) are also shown for comparison. The simplest case of g(K) = 0 has been adopted for this purpose, since no precise form for the correlation function was available for use in equation (31). It can be. seen that the measurements in water give quite good agreement with the theory, indicating that in this particular case correlation effects are not too important. In air the agreement is not so good (although both theory and experiment yield the same frequency dependence,
202
M. S. HOWE AND M. HECKL
Figure 2. Schematic representation of experimental rig. A, 4 mm steel plate; B, mass loadings; C, pick-up for measuring ;” ; D, shaker driven by one-third octave band noise.
b
H
-30
-
-40
-
-50
-
-60
-
-70
’ 31.5
P ; v 2 :: E ii t ,D 2 s I 63
I 125
I 250 Frequency,
Figure 3. Experimental results: water. Theoretical (-) plate is loaded with ten 05 kg masses; theoretical (-----) is a total of 30 masses.
z
-30
I 1000
I 500 f
I 2000
I 4000
I 6000
(Hz)
and experimental (M) curves when the and experimental (O----O) curves when there
-
0 :: 0 e
-4o-
2 P E
-50
-
; 0 ,D D
-60
-
s -701
31.5
I 63
I 125
I 250
I 500
Frequency,
Figure 4. Experimental results: air. Theoretical (is loaded with ten O-5 kg masses; theoretical (-----) total of 30 masses.
I 1000
f
I 2000
I 4000
6000
(Hz)
) and experimental (U) curves when the plate and experimental (O----O) curves when there is a
different from that in water!), probably because the masses were so heavy that they can no longer be considered to be a small perturbation of the mean density of the plate. This would not be the case in water since the masses are then small compared with the added effective mass of the water. At low frequencies the theory and experiment indicate rather small loss factors. Referring to formula (34) we see that in air this is due to the fact that ~ou/omo is rather small, and in water it is due to the small value of tan-‘(omo/2po a).
SOUND RADIATION PROM PLATES
203
The above results lead to the following general remarks regarding radiation loss factors of point loaded plates. When radiation loss factors are compared with the loss factors due to internal damping and energy transfer into supporting structures, one concludes that at low frequencies the radiation losses are very small, and at higher frequencies they are comparable with loss factors caused by other mechanisms. It seems, therefore, that radiation losses do not in general play an important role in the attenuation of bending waves in plates. However, it is often important to have a rather precise knowledge of the radiation loss factor since, in spite of the relative smallness of acoustic losses, quite small radiating efficiencies tot is the loss factor for all damping mechanisms) can (represented by the ratio e/clot, where ?? result in the generation of a significant acoustic field. We have seen that the theory presented above agrees rather well with the experimental results, especially in water. Actually that comparison is between experiment and the theory for an array of uncorrelated point loads. It appears likely, however, that more complicated structures encountered in engineering practice would be amenable to similar analyses involving non-trivial correlations. By constructing a stochastic model of such a structure it may be argued that the loss factor obtained by the above means would give a rather good overall estimate for the acoustic losses when the structure is subject to excitation under normal working conditions. Indeed such an estimate would generally tend to be better than one based on an analysis of a specific configuration of the structure and of the exciting mechanism. REFERENCES 1. 2. 3. 4.
L. CREMERand M. HECKL 1967 Kiirperschall. Berlin: Springer-Verlag. M. S. HOWE 1971 Journal of Fluid Mechanics 45, 769-783. Wave propagation in random media. E. C. TITCHMARSH1960The Theory of Functions. Oxford: Oxford University Press. Second edition. L. D. LANDAUand E. M. LIFSHITZ 1959 Statistical Physics. London: Pergamon.
SOUND
RADIATION
FROM
STIFFNESS
193-203
PLATES
WITH
DENSITY
AND
DISCONTINUITIES M. S.
HOWE
Department of Mathematics, Imperial College of Science and Technology, London S. W.7, England AND
M. HECKL Miiller-B.B.N., Munich, West Germany
(Received 28 October 1971)
This paper is concerned with the theory of sound radiation from infinite, plane plates due to the interaction of bending waves with density and stiffness fluctuations in the material of the plate. Because of the great variety of configurations encountered in practice the theory is worked out in the case in which the variations in density and/or bending stiffness may be regarded as random functions of position on the plate. This should give valuable insight into the order of magnitude of acoustic losses to be encountered in engineering practice. Cases where the plate may be regarded as loaded with randomly sited point masses, and randomly sited parallel line masses, are considered in detail. Experiments
conducted with a large steel plate in air show a large measure of agreement with the theory. 1. INTRODUCTION
over a homogeneous thin plate of uniform thickness and bending stiffness is effectively decoupled from an ambient atmosphere provided that the wavelength is sufficiently long [l, Cremer and Heck1 19671. The wave field developed in the fluid decays exponentially with distance from the plate. This means that the wave propagates over the plate without the attenuation of total plate-wave energy, in other words, without the transfer of energy from the plate into the surrounding medium in the form of sound waves. In practice no plate has exactly uniform thickness or bending stiffness. For example, in practical structures plates are often held in position by means of rivets or coplanar beams, and these sometimes tend to behave as concentrated mass loadings. In this paper we attempt to represent such loadings as fluctuations in the mass density of the plate. These fluctuations may be regarded as waves of zero frequency so that wave-wave interactions with a long bending wave might be expected to generate a whole spectrum of bending waves which are acoustically coupled to the ambient fluid. Equivalently, the mass loadings may also be regarded as secondary sources due to the fictitious forces which must be applied at the loads to overcome the extra inertia of the plate. Mass loadings can also induce variations in the effective bending stiffness of the plate. However it is shown here that such variations in the bending stiffness behave as secondary sources of higher order and consequently generally radiate less energy than mass density fluctuations. It is often important to have a measure of the magnitude of the sound radiated into the 193 A bending wave propagating
194
M. S. HOWE AND M. HECKL
fluid by these mechanisms. This is conveniently expressed in terms of a loss,factor, E, of the incident bending wave defined by E=
acoustic energy radiated per cycle per unit area of the plate 27~x reversible energy per unit area of the plate ’
(1)
A great simplification in the analysis ensues if it can be assumed that the fluctuations in the mass density and/or bending stiffness of the plate are random. Actually no array of rivets or ribs can be perfectly regular, and it seems desirable, therefore, to construct a stochastic model to describe the effect of such loadings in rather general terms. This should give valuable insight into the order of magnitude of the acoustic losses to be expected in typical situations. In this paper a theory is proposed along such lines to account for the acoustic attenuation of long bending waves due to the presence of random mass and stiffness fluctuations (see sections 2 and 3). In section 4 the statistical specification of point and beam (rib) loadings is discussed, and the results are applied to point loaded and beam loaded plates. Finally, section 5 presents the results of an experimental investigation in which a 4 mm steel plate was used, loaded with a random array of point masses, and vibrating in air and also in water.
2. BENDING
WAVES
ON AN INHOMOGENEOUS
FLUID
LOADED
PLATE
Consider an infinite plane plate of density cr per unit area immersed in an unbounded compressible fluid of density p,-,.In the undisturbed state the plate occupies the plane z = 0 of a rectangular coordinate system. If h denotes the thickness of the plate, E and y respectively the Young’s modulus and Poisson’s ratio of the material of the plate, then the bending stiffness, B, is given by
(2)
B = Eh3/[12(1 - r*)].
The (small) deflexion, q = v(x,y, t), of the surface from the undisturbed planar form then satisfies the bending wave equation a$
+ V*(BV*) q = Ap,
(3)
with V2 E 82/8x* + a*/+*. In this equation Ap denotes the net perturbation pressure of the fluid in contact with the plate: i.e., the difference between the fluid pressures on the negative and positive sides of the plate. Let us define the Fourier transformf(k, w) of a functionf (x, t) of x = (x,~) and the time t by means of the reciprocal relations
(4) f(x, t) = li f(k, w) ei(k.x-wt)dk do. -m
I
Then, by using the linearized, inviscid form of the Navier-Stokes equation it readily can be shown that the Fourier transform of Ap on the plate, z = 0, is related to that of r] by
2iw*PO77@, w) Ap@,w> = (&/a* - k *)I 12’
(5)
SOUND RADIATION
195
FROM PLATES
where a is the speed of sound in the fluid, and the branch of (w2/a2 - k2)1’2 having positive imaginary part for -ak < w < ak is taken. Consider now the effect on equation (3) of allowing the mass density u to contain small random fluctuations. To do this we denote the mean density by mo, and define a dimensionless random function f(x) by 0 = m0{1 + &)), (6) with the condition (c(x)) = 0. (Angle brackets and, where convenient, an overbar, denote averages taken over an ensemble of plates.) Then equation (3) may be written in the form
where L is a linear operator obtained with the aid of equation (5) and containing no random quantities. We now decompose the wave field rl into an ensemble average y and a random fluctuation about this average denoted by 7’: 7j = +j+ 17’.
(8)
The lowest order equations describing the evolution of these quantities have the forms (see, for example, [2, Howe 19711) L+j=-mo(l(x)?$),j 827
Lq’ = -m. S(x) p
Pa, b) . i
The wave problem associated with these equations is posed in the following way. An incident coherent wave +j propagates over the plate and generates a random field $, determined by equation (9b), by scattering off the random density fluctuations. These random waves are formed at the expense of the coherent wave whose amplitude therefore decreases. Actually equation (9b) is essentially a local Born approximation and is therefore unsuitable for determining 7’as such. However, because 9’appears in equation (9a) as a mean value in combination with the random fluctuations f(x), we see that equation (9b) is a good approximation for determining this mean value provided only that the Born approximation is valid over .distances of the order of the correlation scale of the fluctuations c(x). Now two cases must be distinguished. If the frequency of the coherent wave is sufficiently high, i.e. if the bending wave speed is higher than the speed of sound in the surrounding medium, the ambient fluid is forced to oscillate in an acoustic mode with consequent radiation of energy away from the plate. In this case the random fluctuations t(x) change the velocity field on the plate, but have hardly any effect on the sound radiated. If, however, the frequency is below a certain coincidence frequency the coherent wave would radiate no sound energy, and it is only the existence of the scattered waves that causes an energy transfer from the plate to the fluid. We shall concern ourselves only with this second class of low frequency, long bending waves. To analyse the properties of the mean wave field +jwe can first determine 7’ in terms of q from equation (9b) and substitute the result into the right-hand side of equation (9a) to obtain an equation for +j alone. This procedure may be executed formally by Fourier analysis. In terms of the definition (4) it is a simple matter to write the solution of equation ,(9b) in the form a0
SSSS [ [or> ‘2
-m
(X, T) e*(K.(x-X)-O(I-T))/L(K, J2) dXdTdK dQ
I
(10)
196
M. S. HOWE AND M. HECKL
In this result the radiation condition is automatically satisfied by taking the path of integration in the complex Q-plane parallel to the real axis but above all the singularities of the integrand: in other words above all the zeros of the dispersion function L(K,@, which are given by 2ipo Q* L(K, $2) E BK4 - m. ii” -
(Q21a2
_
K2)‘/2
=
”
(11)
By considering the mapping from the Q-plane to the Z-plane defined by Z = (BK4 - m. Q2) @*/a2 - K*)“* - zip0 a*,
(12)
it may be shown that when K is real, equation (11) has no roots in Im(SZ) > 0, a result which merely reflects the fact that the undisturbed planar form of the plate is one of stable equilibrium. Indeed, when D traverses a semicircular contour indented at the branch points Q = *aK and the real zeros, Q = iwo, say, [see Figure l(a)] in the anticlockwise sense, the
(a)
(b) Figure 1. Contours in the 9- and z-planes.
image point Z describes a contour of the form illustrated in Figure l(b). Clearly the variation in the argument of Z over a complete circuit of the contour vanishes, and hence expression (1 I) can have no zeros inside the contour of Figure l(a), and hence in the upper complex Q-plane [3, Titchmarsh 1960, seep. 1161.It follows that it is sufficient to evaluate the integrals in equation (10) in the limit Im(Q) -+ +O. 3. THE ACOUSTIC
LOSS FACTOR
We now make the assumption that the random density fluctuations t(x) constitute a stationary random process, and define the correlation function R by
R(x - Y) = (Kx) 5(y)).
(13)
The spectrum function of the fluctuations is then given by means of the definition m
+m=& JR(x) e-lH.xdx. -0J
(14)
SOUND RADIATION FROM PLATES
197
Hence, if the formal solution (10) is substituted into the mean wave equation (9a), and the Fourier transform of the resulting equation is taken, we obtain L(k, co) = rni u4
m+(k-K)dK s Jw,~) -m
05)
’
where the integral on the right is to be evaluated in the limit Im(w) -+ $0 for real w. This is the dispersion relation for the mean wave field. Consider next a monochromatic wave propagating over the plate. More precisely, consider a trapped bending wave whose frequency wo, say, is less than the coincidence frequency, i.e., satisfies k > Iwolla, where k is the wavenumber. In the absence of random density fluctuations k and w. would be related by L(k, wo) = 0. (16) When account is taken of the presence of the random fluctuations the frequency w and the wavenumber k satisfy the mean wave dispersion relation (15), which is an O(p) non-secular correction to the homogeneous dispersion relation (16). The result may be interpreted as a small change in the frequency or in the wavenumber from the homogeneous case of relation (16). It is convenient here to regard it as producing a small change Au, say, in the frequency, the wavenumber being held fixed and real. We therefore set w = w. + Aw,
(17)
where the correction term Aw is O(p) and is to be calculated from equation (15). In fact correct to terms of O(F) we find -sgn(wo) rnz w$ 2
-=
(k
>
I~Ol/a>.
08)
/( --m
The lossfactor of the coherent field may be obtained from the imaginary part of Aw/lwo). The damping is due to two causes. The first comes from that part of the integral in expression
(18) taken around a small semicircular indentation avoiding the real zero of L(K,w,). This corresponds to the scattering of energy out of the coherent wave into trapped bending waves of the same frequency and wavelength, but of arbitrary direction of propagation. The second cause is due to a continuum of scattered waves contributing to that part of the integral over which L(K, wo) is complex. These waves satisfy K < Iwol/a, and are acoustically coupled to the fluid. They attenuate the wave field of the plate by radiating energy into the fluid in the form of sound waves. Now the acoustic loss factor Efor a monochromatic bending wave (k, o) defined by equation (1) may equivalently be expressed in the following analytic form : E = -2Im(Aw)/(wl,
in which the right-hand side is obtained by restricting the range of integration in equation (18) to those wavenumbers K satisfying K < lwl/a (the sufSx zero has been dropped): i.e.,
E =
13
sgn(w) miw2
m. +
* (19) )I s J%,w)
po(2k2 - w2/a2) ImX<‘w”a #C - K) dK j&/a2 - k2(3/2
198
M. S. HOWE AND M. HECKL
Note that the integrand in expression (19) is regular in the range of integration and that there is now no need to give w a small positive imaginary part. The explicit form of equation (19) is
We may briefly quote the corresponding results for the case in which the scattering is not due to the presence of random density fluctuations, but rather to random fluctuations in the bending stiflness, B. By analogy with equation (6) we set B = BoU + P(X))
(21)
and this leads to the following dispersion relation for the mean wave field : L(k, 0) = B,zk4
s
mK4&k - K)dK Jw,4 ’
(22)
where t/t(K)is the spectrum function of the stiffness fluctuations defined in a manner analogous to that for #II) in equation (14). If we now isolate the scattered waves responsible for the acoustic attenuation we obtain a loss factor eB, say, given by
(23) This result is considered further in section 5. 4. ACOUSTIC LOSS FACTORS FOR POINT AND BEAM LOADED
PLATES
We now apply the above theory to plates loaded with a random array of point masses and of parallel line masses (beams or ribs). It will be assumed that the loads do not alter the stiffness of the plate. In order to apply the formula (20) giving the acoustic loss factor we must fkst determine the appropriate spectrum functions, #K), for these cases. Consider then a uniform plate of density M per unit area. In the first instance let point loads each of mass m be randomly distributed at points x, on the plate, so that the effective plate density u has the form a=M+L:m&x-x,). (24) I If there are on average n loads per unit area of the plate, then clearly mo=(a>=M+nm.
(25)
Accordingly the random function f(x) of equation (6) becomes mo&x)=o-
(u)=m[$B(x-xj)-n).
(26)
The correlation function, R(x - y), is now given by R(x - Y) =
,2 (m”H.
K(s(x-x~)~(Y-x3>-nz).
(27)
SOUND RADIATION FROM PLATES
199
When the loads are randomly distributed on the plate the probability that the jth mass occupies an element of area dx, is proportional to dx,. The probability that the ith mass occupies dxl, given that thejth mass occupies dx,, is then proportional tof(x‘ - x,)dx,dx,, wheref (z) is a joint probability distribution function which is an even function of its argument and tends to zero as ]z] + 0, and to unity when z is large. In terms of this function it is known that 1<6(x - Xj) S(y - Xi)) = n 6(x - y) -I-nZf(x - y) (28) 1.1 (see, for example [4, Landau and Lifshitz 19591): i.e., that R(x - Y) = n(Mr~o)~ @(x - Y) - ng(x - Y)}.
(29)
In this final expression g(x - y) = 1 -f(x - y) is the “packing function”, which is essentially equal to unity over the region “occupied” by a point mass situated at x = y and vanishes elsewhere. It is related to the mean square fluctuation number density of the point masses, ((AZV)2), say, by the formula m
g(z)dz=
1
_
(30)
n
-CC
([4], see Section 115), and this result may be used as a starting point for determining the form of the packing function in practical situations. Using the form (29) we are now in a position to determine the acoustic loss factor for a trapped bending wave (k,w). Equation (20) gives this in terms of the spectrum function C+(K)of the density fluctuations. From equation (29) we have in fact
(31) where g(K) is the Fourier transform of the packing function. We deduce, therefore, the following integral expression for the acoustic loss factor: (w2/a2- K2)“2 [ 1/b2 - ngQ -
K)]dK (32)
This result shows that the largest loss factors are expected in cases where the relative positions of the point loads are uncorreluted, for, in general, g(K) > 0 (but suthciently small, of course, to ensure that d(K) is always positive), so that the value of the integral in equation (32) is normally less than that obtaining when g(K) = 0. In practice such correlations will always exist. It is of interest, however, to note the form of the result in this singular case of g(K) s 0. Equation (32) then becomes
Now for long waves satisfying k2 < moa2/B, we have, in the integrand of this result, K2 < &/a2 N Bk4/mo a2 < k2
200
M. S. HOWE AND M. HECKL
and hence lwl/a
s
nm2PO
’ 21 n(mo + 2po/k) o
(w2/a2 - K2)1’2 KdK
{mi(w2/u2 - K2) + 4~3 ’
i.e.,
nm2p0 1 - J$$ ’ N nami(m + 2po/k)
tan-i (2))
(34)
which should be compared with the result of Cremer and Heck1 ([l], see Chapter VI). They are concerned with the acoustic power radiated into the fluid by a point force applied to an infinite plate. Our result shows that in the present approximation (g z 0), the interaction between the mean wave and the randomly sited point loads produces an acoustic field equivalent to one that would be generated by a random distribution of point forces each of intensity proportional to the inertia of the mass load. Indeed each point mass m experiences an acceleration w2v so that the equivalent force is just F= mw2q. Cremer and Heck1 show that the acoustic energy generated per cycle by such a force is equal to AE= “1”_‘i’ l - $itan-i awmO (
(za)),
provided that the frequency is well below the coincidence frequency. Hence, noting that the mechanical energy of the plate is equal to +(mo + 2po/k)02j712 and using the definition (1) of the loss factor E we obtain a result identical with equation (34) when there are n masses per unit area. The case of parallel beam or rib loading may be considered in a similar fashion. Suppose that, as a particular example, the beams are parallel to the y-axis and are distributed almost periodically at a distance 1 apart. Then the appropriate form for the correlation function, R(x - X), is given by
R(x-X)=f(; 0)2(s(*-X)-fH(l-2~x-X~)) where His the Heaviside unit function. This ensures that ((AN)2) vanishes in the case of an almost periodic structure. Note that n = l/l is the number of beams per unit length in the x-direction, and m is the mass per unit length of beam. Hence, l-
(36)
where k = (k,, k,), K = (K,, K,.). Substituting this result into (20) gives the acoustic loss factor. It is clear that the presence of the S-function implies that there are no acoustic losses when 1k,, 1 > 1w 1/a, In the important case of long waves (k2 < moa2/B) considered above, the result simplifies to the following expression for the acoustic loss factor: -PO
m2
’ = 2rni I(mo + 2po/k)
1 _ sin($klcos 8) fkl cos 6’
](l +[1 +$($-k2sin2B)r1’2),
(37)
where 0 is the angle between the x-direction and the wavenumber vector k: i.e., k = k(cos8, sin 0). 5. EXPERIMENTAL
INVESTIGATION
OF A POINT
LOADED
PLATE
Before describing the experimental arrangements it is of interest to compare in orders of magnitude the relative importance to the acoustic radiation problem of the random per-
SOUND RADIATION
FROM PLATES
201
turbations being in the bending stiffness B or, as discussed in detail in the previous sections, in the mass density. Referring to equations (20) and (23) makes it clear that the ratio of the acoustic loss factors E and +, due, respectively, to mass and stiffness fluctuations, is given in order of magnitude Ed B,2k4K4 -Np<-N E - mgio4
B;k4 mia4-
(38)
where w, is the coincidence frequency. The final term on the right of expression (38) is therefore small provided that the frequency is well below the coincidence frequency. In the experiments described below we shall assume that the effect of loading the plate with an array of “point” masses does not have an appreciable effect on the bending stiffness of the plate, which effect is therefore neglected in calculating the radiated sound power. In practice bending waves on plates are attenuated not only by the radiation of sound, but also by internal losses, by viscous losses at the plate-fluid interface, and by energy transfer into supports. Since some of these mechanisms give rise to loss factors of the same order of magnitude as the E of equation (32), it would have been impossible to measure the radiation damping by observing the (time or space) decay of bending waves. For this reason a different method was used which is based on the definition of the loss factor given by equation (1). Thus the radiation loss factor E can be expressed in terms of the reversible energy m,, w2q of the plate and the radiated sound power P, say (all quantities being per unit area of the plate). Since the energy lost per cycle as sound is 27rP/w we obtain
?? =P/co3mo~7- -P/t_Omov72,
(39)
where 3 is the mean square plate velocity. The experimental arrangements are illustrated in Figure 2. The mean square plate velocity was measured by means of a pick-up at five different points on the plate. The sound power, P, was measured in two different ways. In air it was possible to place the experimental rig in a fairly reverberant room. Thus the power per unit area could be found from the mean square sound pressure2, the reverberation time T, the area of the radiating surface S (twice the area of the plate), and the volume V of the room:
13.8v -2
p _
p.
a2 TSp ’
The measurements in water were conducted in Lake Ammersee in Southern Germany, which for our purposes behaved as an infinite medium. Therefore the radiated sound pressure was measured at several distances R from the centre of the plate and the power calculated according to the formula p=I_2xRZ p0a
2
PY S
(41)
a mean value being taken. In the measurements a 4 mm thick steel plate 2 m x 1 m was used. The “point” loads consisted of 0.5 kg steel masses glued at randomly selected points of the plate. The results are shown in Figures 3 and 4. Theoretical curves from formula (31) are also shown for comparison. The simplest case of g(K) = 0 has been adopted for this purpose, since no precise form for the correlation function was available for use in equation (31). It can be. seen that the measurements in water give quite good agreement with the theory, indicating that in this particular case correlation effects are not too important. In air the agreement is not so good (although both theory and experiment yield the same frequency dependence,
202
M. S. HOWE AND M. HECKL
Figure 2. Schematic representation of experimental rig. A, 4 mm steel plate; B, mass loadings; C, pick-up for measuring ;” ; D, shaker driven by one-third octave band noise.
b
H
-30
-
-40
-
-50
-
-60
-
-70
’ 31.5
P ; v 2 :: E ii t ,D 2 s I 63
I 125
I 250 Frequency,
Figure 3. Experimental results: water. Theoretical (-) plate is loaded with ten 05 kg masses; theoretical (-----) is a total of 30 masses.
z
-30
I 1000
I 500 f
I 2000
I 4000
I 6000
(Hz)
and experimental (M) curves when the and experimental (O----O) curves when there
-
0 :: 0 e
-4o-
2 P E
-50
-
; 0 ,D D
-60
-
s -701
31.5
I 63
I 125
I 250
I 500
Frequency,
Figure 4. Experimental results: air. Theoretical (is loaded with ten O-5 kg masses; theoretical (-----) total of 30 masses.
I 1000
f
I 2000
I 4000
6000
(Hz)
) and experimental (U) curves when the plate and experimental (O----O) curves when there is a
different from that in water!), probably because the masses were so heavy that they can no longer be considered to be a small perturbation of the mean density of the plate. This would not be the case in water since the masses are then small compared with the added effective mass of the water. At low frequencies the theory and experiment indicate rather small loss factors. Referring to formula (34) we see that in air this is due to the fact that ~ou/omo is rather small, and in water it is due to the small value of tan-‘(omo/2po a).
SOUND RADIATION PROM PLATES
203
The above results lead to the following general remarks regarding radiation loss factors of point loaded plates. When radiation loss factors are compared with the loss factors due to internal damping and energy transfer into supporting structures, one concludes that at low frequencies the radiation losses are very small, and at higher frequencies they are comparable with loss factors caused by other mechanisms. It seems, therefore, that radiation losses do not in general play an important role in the attenuation of bending waves in plates. However, it is often important to have a rather precise knowledge of the radiation loss factor since, in spite of the relative smallness of acoustic losses, quite small radiating efficiencies tot is the loss factor for all damping mechanisms) can (represented by the ratio e/clot, where ?? result in the generation of a significant acoustic field. We have seen that the theory presented above agrees rather well with the experimental results, especially in water. Actually that comparison is between experiment and the theory for an array of uncorrelated point loads. It appears likely, however, that more complicated structures encountered in engineering practice would be amenable to similar analyses involving non-trivial correlations. By constructing a stochastic model of such a structure it may be argued that the loss factor obtained by the above means would give a rather good overall estimate for the acoustic losses when the structure is subject to excitation under normal working conditions. Indeed such an estimate would generally tend to be better than one based on an analysis of a specific configuration of the structure and of the exciting mechanism. REFERENCES 1. 2. 3. 4.
L. CREMERand M. HECKL 1967 Kiirperschall. Berlin: Springer-Verlag. M. S. HOWE 1971 Journal of Fluid Mechanics 45, 769-783. Wave propagation in random media. E. C. TITCHMARSH1960The Theory of Functions. Oxford: Oxford University Press. Second edition. L. D. LANDAUand E. M. LIFSHITZ 1959 Statistical Physics. London: Pergamon.