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Aspects of the reflexion and free wave properties of a composite panel under fluid loading
Journal of
Sound and Vibration (1979) &I(4), 467-474
ASPECTS OF THE REFLEXION PROPERTIES
AND FREE WAVE
OF A COMPOSITE
PANEL
UNDER FLUID LOADING D. G. CRIGHTON Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, England (Received 13 December 1978)
The response of a composite panel to external forcing, with inclusion of fluid loading effects, is considered. Of the two strata comprising the composite panel one, backed by a vacuum, is of the conventional thin elastic plate or membrane kind, while the other, in contact with the fluid, is more like an elastic solid and may suffer significant compression. The behaviour of the acoustic field close to grazing incidence is examined, this behaviour being determined by that of the plane wave reflexion coefficient. In the absence of the upper stratum it is well known that the reflexion coefficient has the value -1 around grazing incidence, so that direct and reflected fields from an external source cancel and preclude the propagation of a genuine acoustic field over the surface (a situation known in optics as the “Lloyd’s mirror” effect). It is shown that, at any given frequency, an impedance for the upper stratum can be prescribed which will lead to the value + 1 for the grazing incidence reflexion coefficient, and will thus obviate the severe power loss which would otherwise occur in directions close to the surface. Next the free subsonic surface waves which can exist in the coupled three-part (two-layer panel plus fluid) system are examined. It is shown analytically that, if the upper layer has low impedance controlled by stiffness forces, a new surface wave can exist in the system. This wave essentially involves the stiffness of the upper layer and the mass of the fluid, and has a wavenumber much higher than that of the surface wave in a single conventional panel. It is also shown from numerical studies that two subsonic surface waves will exist over quite a wide range of parameters, though not necessarily with the wide wavenumber separation of the low impedance case. A discussion is given of the possible importance of the high wavenumber mode in the case of excitation by high wavenumber boundary layer turbulence, and of the significance of two free subsonic surface wave modes in calculations of energy transmission over composite panels of the kind modelled here.
1. INTRODUCTION The studies described here were provoked by some results obtained, though as yet unpublished, on the transmission of free surface wave energy across ribs on composite panels under the influence of fluid loading. In the case of a conventional thin elastic plate type of panel it is a straightforward matter to calculate the transmission across a rib with specified impedance; the crucial point is that then there is, under all conditions, one and only one subsonic surface wave which can propagate in a coupled system formed by a compressible static fluid and a thin elastic plate [l]. The same property does not, however, hold if the thin elastic plate forms one stratum of a two-layer composite panel of the kind which can be modelled by the analysis of Maidanik and Tucker [2]. The author 467 0022-460X/79/120467+ 08 $02.00/O 0 1979Academic Press Inc. (London) Limited
468
D. G.
C‘RIGHTON
was, in fact, led at first to erroneous results because of failure to recognize that a coupled system comprising fluid and a two-layer panel can support more than one type of subsonic free surface wave capable of transferring energy across a rib or other discontinuity on a panel. When the possibility was realized, it was easy enough to produce analytical approximations to the two subsonic wavenumbers for a certain limiting case, and numerical studies then showed that the presence of two subsonic wave modes was fairly common throughout a wide parameter range. These aspects of the joint influence of two-layer panel structure and fluid loading are discussed in section 3 of this paper, and the importance of the phenomenon is mentioned briefly in section 4. A second unusual feature associated with the composite panel was found in examination of the acoustic field generated by a source in the presence of the panel. This acoustic field is the sum of the direct field from the source plus a reflected field from a specular image source with a reflexion coefficient dependent upon frequency, angle, and the various panel and fluid parameters. It is a well-known fact, especially in the context of the optical analogue known as the “Lloyd’s mirror” effect (and its implications for radar transmission over the Earth’s surface), that if the surface impedance remains finite for a grazing incidence wave then the reflexion coefficient must necessarily have the value - 1 at all angles “sufficiently close” to grazing (see e.g., reference [3]). In the case of a conventional thin elastic plate the reflexion coefficient is effectively equal to -1 over a surprisingly large range of angles from the surface, as is clear from the numerical examples presented by King [4]. This leads to a cancellation of the leading order power-carrying acoustic field over a large angular range from the surface-an obviously undesirable feature in certain practical situations, though doubtless attractive in others. The only way of alleviating this problem seems to involve an increase in the stiffness or mass of the plate, so that the plate impedance continues to dominate the fluid impedance except at the very smallest angles to the surface. In many cases, however, this could only be achieved by the addition of an unrealistically large mass to the plate. A new possibility is opened up if a more elaborate two-layer pane1 is considered. Here one layer of this composite-the one not in contact with the fluid whose infinite impedance for grazing incidence leads to the Lloyd’s mirror effect-is taken to be the usual kind of elastic plate, which will then be separated from the fluid by an upper stratum whose properties will, to some extent, include the compressibility aspect excluded from the thin plate model. This may be regarded as a simple representation of the compressibility properties of thick plates, the use of very complicated thick plate equations thus being avoided, or, alternatively, the upper stratum might actually be a layer of bubbly liquid or other fluid with compressibility different from that of the main body of fluid. The mode1 should indicate trends and mechanisms for both situations without necessarily being in direct correspondence with either. The essentials of the action of the composite panel envisaged lie in the fact that the two strata act as impedances in parallel. Consequently, if the sum of the impedances of the layers is made to vanish for a grazing incidence wave at any given frequency, then the joint impedance of the layers in parallel is infinite, and, as is shown in section 2, the reflexion coefficient is then equal to +l around grazing incidence. Thus an appropriately chosen upper layer has the desired effect of “stiffening” the lower plate to give it a larger impedance. This argument of course depends for its practical realization upon the design of an upper layer complying with the assumptions of the model; it is idealized in the model of Maidanik and Tucker [Z] as being unable to withstand any pressure jump across its faces, which differ only in their velocities. Any departure from such a model will lead to a finite surface impedance under all circumstances, and thus inevitably again to a reflexion coefficient of - 1 near to grazing. The author believes, however, that experimental study of
FLUID
LOADED
COMPOSITE
PANEL
469
RESPONSE
the degree to which a composite plate can be stiffened by the choice of an upper layer approximately resembling that of the model is justified by the results of section 2 below, and this is the principal recommendation of the conclusions in section 4.
2. GRAZING
INCIDENCE
AND THE LLOYD’S MIRROR
EFFECT
Let an infinite thin elastic plate occupy the plane xj = 0, with static compressible fluid of density p0 and sound speed co on the side xj > 0 and a vacuum in x3 < 0 (the necessary modifications to allow for fluid in x1 < 0 being easily made). On the upper surface of the plate rests a second layer whose thickness is regarded as negligible compared with the scales of motion of interest. A time factor exp(-iiot) is understood throughout, and the deflexions of the elastic plate and of the top surface of the upper layer in the positive x,-direction are denoted by @‘(x1,x2) and qc(xl,xZ), respectively. The bending stiffness and specific mass of the plate are denoted by B and m, and the upper layer is characterized by a local reaction impedance zc, as in the model of Maidanik and Tucker [2]. Then the equation of motion of the lower element of the composite panel is BV4up - mo2?P = 4(x1,x,) - p(xl,xZ,O),
(2.1)
while that of the upper is ~hx~,O)
=
-i~zcC~p-Vl(xl,xz)
(2.2)
the surface pressure p(x,,x,,O) being taken as constant through the upper layer, and denoting the upward externally applied normal stress on the plate. The fluid -_ motion is governed by the Helmholtz equation q(x,,x,)
(V2+k;)4
= Q
(2.3)
for the potential 4, with k, = o/co the acoustic wavenumber at frequency o and with io4 relating pressure to potential. A kinematic condition
p = p.
~(x,,x,,O) =-iw’(x,,x,)
(2.4)
3
relates the fluid velocity to the velocity of the top face of the composite panel. In equation (2.3), Q denotes an acoustic source distribution in the volume of the fluid. It is sufficient only to consider a point source, Q(x) = Qo6(x - x0), and to take the external plate stress q as zero. Then it is a straightforward matter to use integral transforms and the method of steepest descent (as in reference [3], for example) to find the distant acoustic field in the form P(X) -
- poiwQo
ex;;;/x’ {1 +)R,(B)}.
(2.5)
In this estimate surface wave fields are ignored, and also non-uniformities in the “far but not too far” field associated with the beaming properties of leaky waves (see reference [l]), though it fully describes the acoustic field at all angles at sufficiently great distances. In it, R,(8) is the reflexion coefficient for a plane wave incident at an angle 8 to the surface, and defined by R,(@ = C(~--~)/(z+z”)lt=kocos8,
(2.6)
470
D.
G. CRIGHTON
in terms of the joint impedance z of the upper and lower elements of the panel and the fluid acoustic impedance za. These in turn are defined by zP(k, w) = ; (Bk4 - mo’).
(2.7-2.9)
In these expressions, the magnitude k of the wavenumber vector (k,, k2) conjugate to (x1,x2) is to be set equal to k,cos 0 in order to obtain the reflexion coefficient R, of equation (2.6), 6 being the angle between the vector x and the surface x3 = 0. Equation (2.6) bears out the points made in section 1. If z remains finite as 8 -+ 0, then R, -+ - 1 and the leading term in the far field expansion for pressure vanishes at and near grazing incidence. Equation (2.8) shows, however, that the impedances zc and zp of the two elements of the composite are in parallel, and therefore that z -+ cc if zc+zP-+ 0. Suppose then that one chooses (zp+zC)(k,,,u) = 0, (2.10) a relation which, of course, can only hold, in general, at a particular specified frequency: then one finds from equation (2.6) that R,(8)
= 1 + O(S’)
(2.11)
for small values of 8, the error term O(0’) indicating that R* may be taken as + 1 over a large range of angles-perhaps up to 30°-around grazing. Thus the Lloyd’s mirror effect can be circumvented by the substitution of a composite plate whose upper layer, in contact with the fluid, satisfies equation (2.10) which amounts to a statement that the composite should, in the absence of fluid loading, be capable of supporting a free coupled wave of wavenumber k,-i.e., a wave with precisely sonic phase velocity. If the upper layer impedance comprises a “stiffness” K and a “mass” term Mo2, so that zc = (i/o)(K - Mo’),
(2.12)
then equation (2.10) reads K + Bk; = (M+m)o’,
(2.13)
which may be used to define the constants K, M in terms of given values of B, m. It is to be noted here that in equations (2.5H2.13) it is not necessary to assume that the upper layer is capable of only local reaction. The equations hold, in fact, for any homogeneous layer which can be characterized by an impedance operator zc in physical space, or equivalently, by an impedance zC(k, w) in wavenumber-frequency space. The surface rigidity, and hence pressure-doubling implied by equation (2.11), obtained by satisfying equation (2.10), depends crucially on the parallel-impedance relation (2.8) for the joint impedance z, and it may be felt that the simple characterization by a two-layer model of the action of practical composite panels may be inadequate, and too simplified, to correctly deal with the grazing wave problem. That issue is not addressed here; all the author wishes to point out is that within one currently accepted simple model of a composite panel the clear possibility exists of designing such a surface (actually, an infinity of surfaces for each frequency) with infinite impedance at the grazing incidence condition, and thereby of allowing acoustic propagation, with pressure doubling, over the surface.
3. FREE COUPLED SUBSONIC SURFACE WAVES Consider now a free wave in the coupled three-part system. The frequency is prescribed, and one seeks the possible wavenumbers k such that the system (2.1H2.4) is satisfied, with
FLUID
LOADED
COMPOSITE
PANEL
RESPONSE
471
4 = Q = 0, by a surface deflexion proportional to exp(ik, x1 +ik,x,). It is clearly sufficient to consider only a plane wave propagating along the positive x,-axis, SO that ki = k, k, = 0, say. The significance of complex values of k needs careful attention (see reference [1]), and consideration is restricted here to waves with subsonic phase velocity and to fluid:upper 1ayer:lower layer systems with no damping. These conditions ensure that the values of k under consideration are real, positive, and greater than the acoustic wavenumber k,. For simplicity, attention here is also restricted to impedances zc of local reaction only, so that zCis a function of frequency alone. The dispersion relation for the free wavenumbers is easily obtained in the form l/P + l/z’ + l/z” = 0,
(3.1)
or, with the definitions k, = (rr~u~/B)“~,
p = ip,ojzC,
P = bblm,
y = +(k2-k;)lj2,
(3.2) (3.3)
in the form (k4 - k;)(y - /I) - pk;
= 0.
(3.4)
Suppose now that /? + + cc for fixed values of CL,k, and k,. This means that the upper layer is rather soft (zc small), with an impedance in which the positive “stiffness” term dominates the negative “mass” term. It is then easy to show that equation (3.4) has a large, real, positive root k, _ p + k;/2fl - k;/Sp3 + pk;/b4
+ ..-,
(3.5)
as well as a root close to k,, the free wavenumber of the plate in the absence of both fluid loading and the upper layer. This root is given by k, N k, - pk,/4P + . . . .
(3.6)
and is real if k, > k, (i.e., if the vacuum free wavenumber k, corresponds to a subsonic phase velocity o/k,), but has a small positive imaginary part if k, < k,; in that case the low impedance upper stratum provides a weak coupling with the fluid through which the energy of the supersonic free wave in the plate can slowIy “leak” as radiation to infinity. All other roots of equation (3.4), besides expressions (3.5) and (3.6), have substantial imaginary parts; their number and nature depend on a precise specification of the branch cuts for the function y(k) and, as discussed at length in reference [l], these roots have no 1 physical significance. Thus the upper layer with low impedance (with /I > 0) has the effect of (i) decoupling the plate from the fluid loading and leading to a wavenumber k, very close to that in the plate in the absence of both coating and fluid loading, and (ii) introducing a new high wavenumber mode (3.5) in which-as is quite clear from the fact that the plate properties enter into expression (3.5) only in the fourth term-the balance lies between the mass of the fluid and the stiffness of the upper layer. Feature (ii) distinguishes the composite plate from the conventional; for the latter it was proved in reference [l] that there is under all circumstances a single subsonic free surface wave (within the context of thin plate theory). It might be thought that the existence of the k, mode would be restricted to large values of /I. It appears, however, from numerical studies of equation (3.4) that two subsonic surface waves are to be found over quite a wide range of parameter values, though in all cases so far in which the author has found two roots the parameter b has been positioe. Also, of course, if /I is not large the two roots may be fairly close together and it will not be possible to interpret them in the clear-cut way given above.
472
D. G. CRIGHTON
To provide some examples, some studies have been made of the simpler case in which the plate is replaced by a membrane under tension T, with k, defined as k, = (mo2/T)1’2.
(3.7)
The free wave dispersion relation is then (k2-k,$(y-/I)
- pk;
= 0
(3.8)
k, N p + k;/2/3 + pk;/jI= - k:/8/33 + . . . .
(3.9)
and the roots of interest for large positive B are
k, N k,{l
-(p/2/?)-(p/2/Y2)[(k;
- k,$)“‘+p/4]
+ ++.}.
(3.10)
Equation (3.8) has been studied numerically, in the form (02 - 1) ((a2 - sz)l’2 - (1 - Q)“2(52/8,)3’2} - &/52”2 = 0,
(3.11)
with Q = k/k,,, and the three parameters, 8, Sz, and E equivalent to the three ratios of the four wavenumbers k,, ko, B and ~1.The reasons for using the form (3.11) are irrelevant here, though it is clear that the values of k,/k,, /?/km and p/k, are uniquely determined by those of 8, G!,and E. Limited results of these studies are given in Table 1, from which one must conclude that a check should always be made to see if two free modes are possible in given circumstances. It should also be noted that when there are two modes it is by no means necessarily the case that they are at very different wavenumbers. Indeed, Table 1 shows that the wavenumbers may lie less than an octave apart on either side of the vacuum wavenumber k = k, (a = 1). Attempts have been made to produce a simple analytical condition on the various parameters at which the number of subsonic free waves increases from 1 to 2, but so far without success.
4. DISCUSSION Little more needs to be said here about the results of section 2 on the grazing incidence field, except to emphasize again that these results depend very much on the parallelimpedance concept for the joint action of the two layers of the composite. There is, therefore, a need for experimental investigation of the degree to which this model of a composite panel is valid, as well as for theoretical study of the possibilities for forming the upper layer from’a layer of bubbly fluid adjacent to the plate, or from a suitably contained layer of pure gas. These theoretical studies are now being conducted, and it is hoped to report on them in due course. On the question of multiple (in fact, double) subsonic free wavenumbers for the composite plate, there are at least two areas in which this property could be important. The first, referred to in section 1, concerns the transmission of free surface wave energy across a rib, or other inhomogeneity, on a panel, a common application of which is to the problem of the isolation of some parts of a structure in air or water from other parts in which intense vibrational energy is generated. In calculations of the energy transfer across an inhomogeneity such as a stiffener one must take into account the fact that free wave energy may be incident upon the inhomogeneity in either or both of the two subsonic wave modes, and the inhomogeneity will act then as a wavenumber converter, ensuring that energy will be transmitted in both modes even in the unlikely event of its incidence in a single mode. In simple cases this amounts to no more than a piece of book-keeping, though in more complicated cases, involving several inhomogeneities and the associated
473
FLUID LOADED COMPOSITE PANEL RESPONSE TABLE
1
The real positive roots ol, c2 of equation (3.11) for CJ= k/k,; the parameter Sz = ki/ki takes values from 0.1 to 1.0 (“coincidence”), while the fluid loading parameter E has the values 0.3, 0.1, O-03 and the parameter 0, characterizing the stiffness of the upper layer has the values 0.25 and 0.5
E = 0.3, s2, = 0.25
E = 0.3, a, = 0.5
r s2
aI
0.1 0.2 0.3 0.4 0.5
1.363 1.397 1.592 2.011 2.619
0.6 0.7 0.8 0.9 1.0
3.349 4.164 5.051 6,000 7.006
a2
al
02
0.6094 0.7925 0.8827
1.331 1.286 1.287 1.316 1.377
-
0.9259 0.9490 0.9629 0.9719
1.477 1.623 1.814 2.041 2.299
E = 0.1, 8, = 0.25 I
\
E =
0.7758 0.8371
-
0.1,
0,
=
0.5
r
,
a
a1
a2
0.1 0.2
1.163 1.204
0.5406
1.143 1.124
-
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.425 1.919 2.574 3.324 4.150 5.042 5.994 7.002
0.8128 0.9239 0.9599 0.9751 0.9830 0.9876 0.9906 -
1.130 1.158 1.221 1.339 1.515 1.735 1.985 2.258
0.5517 0.6694 0.7844 0.8704 0.9209 0.9486 0.9650 -
E = 0.03, 9, = 0.25 \ n
a1
02
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.058 1.087 1.328 1.881 2.557 3.316 4.145 5.039 5,992 7GOl
0.3355 0.6994 0.9272 0.9762 0.9879 0.9925 0.9949 0.9963 0.9972 0.9978
a1
02
E = 0.03, Q, = 0.5 a1
1.049 1.043 1.047 1.065 1.122 1.261 1.465 1.703 1.963 2,243
a2
0.4570 0.5993 0.7475 0.8785 0.948 1 0.9731 0.9835 0.9889 -
474
D. G. CRIGHTON
resonance phenomena (as with two ribs on a panel, for example), the presence of two subsonic wave modes may lead to greater difficulties. Study of the energy transmission problem in detail is now in progress, with the multiple wave modes firmly in mind when dealing with composite panels. The second application of section 3 may arise in the case p --++ co, for which it was shown that, for virtually any lower layer of the plate or membrane type, a high wavenumber mode with k - j? exists, almost independently of the plate properties. It may then be possible that in some circumstances k, may be comparable with the characteristic wavenumber o/U, of intense boundary layer pressure fluctuations, U, being the convection velocity of the surface pressure field (generally around 0.6 of the velocity in the mainstream of a flow over the surface). There is then the possibility that relatively large levels of surface vibration may be generated on composite panels at forward speed in air or water through the familiar spatial resonance mechanism between the surface waves with k = k, and the turbulent excitation concentrated around k = w/U,. Before such a possibility can be quantified further, however, a fourth element, associated with the mean flow past the surface, must be added to the three-part coupled system studied here. The author hopes to discuss the effects of mean flow on coupled wave-bearing systems of the plate: fluid kind in a future paper.
ACKNOWLEDGMENT
This work was supported by the Ofice of Naval Research, Code 222, Acoustic Technology Program, under Grant N00014-77-G-0072. Some of the results were presented at the 95th Meeting of the Acoustical Society of America, Providence, Rhode Island, May 1978. The assistance of Dr J. F. Scott in the preparation of Table 1 is acknowledged with thanks.
REFERENCES D. G. CRIGHTON1979Journal of Sound and Vibration 63,225-236. The free and forced waves on a fluid-loaded elastic plate. G. MAIDANIK and A. J. TUCKER 1974Journal of Sound and Vibration 34, 519-550. Acoustic properties of coated panels immersed in fluid media. D. G. CRIGHTON1971 Journal of Fluid Mechanics 47, 625-638. Acoustic beaming and reflexion from wave-bearing surfaces. W. F. KING III 1973 Journal of Sound and Vibration 30,279-288. The influence of fluid loading on acoustic propagation near surfaces.
Sound and Vibration (1979) &I(4), 467-474
ASPECTS OF THE REFLEXION PROPERTIES
AND FREE WAVE
OF A COMPOSITE
PANEL
UNDER FLUID LOADING D. G. CRIGHTON Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, England (Received 13 December 1978)
The response of a composite panel to external forcing, with inclusion of fluid loading effects, is considered. Of the two strata comprising the composite panel one, backed by a vacuum, is of the conventional thin elastic plate or membrane kind, while the other, in contact with the fluid, is more like an elastic solid and may suffer significant compression. The behaviour of the acoustic field close to grazing incidence is examined, this behaviour being determined by that of the plane wave reflexion coefficient. In the absence of the upper stratum it is well known that the reflexion coefficient has the value -1 around grazing incidence, so that direct and reflected fields from an external source cancel and preclude the propagation of a genuine acoustic field over the surface (a situation known in optics as the “Lloyd’s mirror” effect). It is shown that, at any given frequency, an impedance for the upper stratum can be prescribed which will lead to the value + 1 for the grazing incidence reflexion coefficient, and will thus obviate the severe power loss which would otherwise occur in directions close to the surface. Next the free subsonic surface waves which can exist in the coupled three-part (two-layer panel plus fluid) system are examined. It is shown analytically that, if the upper layer has low impedance controlled by stiffness forces, a new surface wave can exist in the system. This wave essentially involves the stiffness of the upper layer and the mass of the fluid, and has a wavenumber much higher than that of the surface wave in a single conventional panel. It is also shown from numerical studies that two subsonic surface waves will exist over quite a wide range of parameters, though not necessarily with the wide wavenumber separation of the low impedance case. A discussion is given of the possible importance of the high wavenumber mode in the case of excitation by high wavenumber boundary layer turbulence, and of the significance of two free subsonic surface wave modes in calculations of energy transmission over composite panels of the kind modelled here.
1. INTRODUCTION The studies described here were provoked by some results obtained, though as yet unpublished, on the transmission of free surface wave energy across ribs on composite panels under the influence of fluid loading. In the case of a conventional thin elastic plate type of panel it is a straightforward matter to calculate the transmission across a rib with specified impedance; the crucial point is that then there is, under all conditions, one and only one subsonic surface wave which can propagate in a coupled system formed by a compressible static fluid and a thin elastic plate [l]. The same property does not, however, hold if the thin elastic plate forms one stratum of a two-layer composite panel of the kind which can be modelled by the analysis of Maidanik and Tucker [2]. The author 467 0022-460X/79/120467+ 08 $02.00/O 0 1979Academic Press Inc. (London) Limited
468
D. G.
C‘RIGHTON
was, in fact, led at first to erroneous results because of failure to recognize that a coupled system comprising fluid and a two-layer panel can support more than one type of subsonic free surface wave capable of transferring energy across a rib or other discontinuity on a panel. When the possibility was realized, it was easy enough to produce analytical approximations to the two subsonic wavenumbers for a certain limiting case, and numerical studies then showed that the presence of two subsonic wave modes was fairly common throughout a wide parameter range. These aspects of the joint influence of two-layer panel structure and fluid loading are discussed in section 3 of this paper, and the importance of the phenomenon is mentioned briefly in section 4. A second unusual feature associated with the composite panel was found in examination of the acoustic field generated by a source in the presence of the panel. This acoustic field is the sum of the direct field from the source plus a reflected field from a specular image source with a reflexion coefficient dependent upon frequency, angle, and the various panel and fluid parameters. It is a well-known fact, especially in the context of the optical analogue known as the “Lloyd’s mirror” effect (and its implications for radar transmission over the Earth’s surface), that if the surface impedance remains finite for a grazing incidence wave then the reflexion coefficient must necessarily have the value - 1 at all angles “sufficiently close” to grazing (see e.g., reference [3]). In the case of a conventional thin elastic plate the reflexion coefficient is effectively equal to -1 over a surprisingly large range of angles from the surface, as is clear from the numerical examples presented by King [4]. This leads to a cancellation of the leading order power-carrying acoustic field over a large angular range from the surface-an obviously undesirable feature in certain practical situations, though doubtless attractive in others. The only way of alleviating this problem seems to involve an increase in the stiffness or mass of the plate, so that the plate impedance continues to dominate the fluid impedance except at the very smallest angles to the surface. In many cases, however, this could only be achieved by the addition of an unrealistically large mass to the plate. A new possibility is opened up if a more elaborate two-layer pane1 is considered. Here one layer of this composite-the one not in contact with the fluid whose infinite impedance for grazing incidence leads to the Lloyd’s mirror effect-is taken to be the usual kind of elastic plate, which will then be separated from the fluid by an upper stratum whose properties will, to some extent, include the compressibility aspect excluded from the thin plate model. This may be regarded as a simple representation of the compressibility properties of thick plates, the use of very complicated thick plate equations thus being avoided, or, alternatively, the upper stratum might actually be a layer of bubbly liquid or other fluid with compressibility different from that of the main body of fluid. The mode1 should indicate trends and mechanisms for both situations without necessarily being in direct correspondence with either. The essentials of the action of the composite panel envisaged lie in the fact that the two strata act as impedances in parallel. Consequently, if the sum of the impedances of the layers is made to vanish for a grazing incidence wave at any given frequency, then the joint impedance of the layers in parallel is infinite, and, as is shown in section 2, the reflexion coefficient is then equal to +l around grazing incidence. Thus an appropriately chosen upper layer has the desired effect of “stiffening” the lower plate to give it a larger impedance. This argument of course depends for its practical realization upon the design of an upper layer complying with the assumptions of the model; it is idealized in the model of Maidanik and Tucker [Z] as being unable to withstand any pressure jump across its faces, which differ only in their velocities. Any departure from such a model will lead to a finite surface impedance under all circumstances, and thus inevitably again to a reflexion coefficient of - 1 near to grazing. The author believes, however, that experimental study of
FLUID
LOADED
COMPOSITE
PANEL
469
RESPONSE
the degree to which a composite plate can be stiffened by the choice of an upper layer approximately resembling that of the model is justified by the results of section 2 below, and this is the principal recommendation of the conclusions in section 4.
2. GRAZING
INCIDENCE
AND THE LLOYD’S MIRROR
EFFECT
Let an infinite thin elastic plate occupy the plane xj = 0, with static compressible fluid of density p0 and sound speed co on the side xj > 0 and a vacuum in x3 < 0 (the necessary modifications to allow for fluid in x1 < 0 being easily made). On the upper surface of the plate rests a second layer whose thickness is regarded as negligible compared with the scales of motion of interest. A time factor exp(-iiot) is understood throughout, and the deflexions of the elastic plate and of the top surface of the upper layer in the positive x,-direction are denoted by @‘(x1,x2) and qc(xl,xZ), respectively. The bending stiffness and specific mass of the plate are denoted by B and m, and the upper layer is characterized by a local reaction impedance zc, as in the model of Maidanik and Tucker [2]. Then the equation of motion of the lower element of the composite panel is BV4up - mo2?P = 4(x1,x,) - p(xl,xZ,O),
(2.1)
while that of the upper is ~hx~,O)
=
-i~zcC~p-Vl(xl,xz)
(2.2)
the surface pressure p(x,,x,,O) being taken as constant through the upper layer, and denoting the upward externally applied normal stress on the plate. The fluid -_ motion is governed by the Helmholtz equation q(x,,x,)
(V2+k;)4
= Q
(2.3)
for the potential 4, with k, = o/co the acoustic wavenumber at frequency o and with io4 relating pressure to potential. A kinematic condition
p = p.
~(x,,x,,O) =-iw’(x,,x,)
(2.4)
3
relates the fluid velocity to the velocity of the top face of the composite panel. In equation (2.3), Q denotes an acoustic source distribution in the volume of the fluid. It is sufficient only to consider a point source, Q(x) = Qo6(x - x0), and to take the external plate stress q as zero. Then it is a straightforward matter to use integral transforms and the method of steepest descent (as in reference [3], for example) to find the distant acoustic field in the form P(X) -
- poiwQo
ex;;;/x’ {1 +)R,(B)}.
(2.5)
In this estimate surface wave fields are ignored, and also non-uniformities in the “far but not too far” field associated with the beaming properties of leaky waves (see reference [l]), though it fully describes the acoustic field at all angles at sufficiently great distances. In it, R,(8) is the reflexion coefficient for a plane wave incident at an angle 8 to the surface, and defined by R,(@ = C(~--~)/(z+z”)lt=kocos8,
(2.6)
470
D.
G. CRIGHTON
in terms of the joint impedance z of the upper and lower elements of the panel and the fluid acoustic impedance za. These in turn are defined by zP(k, w) = ; (Bk4 - mo’).
(2.7-2.9)
In these expressions, the magnitude k of the wavenumber vector (k,, k2) conjugate to (x1,x2) is to be set equal to k,cos 0 in order to obtain the reflexion coefficient R, of equation (2.6), 6 being the angle between the vector x and the surface x3 = 0. Equation (2.6) bears out the points made in section 1. If z remains finite as 8 -+ 0, then R, -+ - 1 and the leading term in the far field expansion for pressure vanishes at and near grazing incidence. Equation (2.8) shows, however, that the impedances zc and zp of the two elements of the composite are in parallel, and therefore that z -+ cc if zc+zP-+ 0. Suppose then that one chooses (zp+zC)(k,,,u) = 0, (2.10) a relation which, of course, can only hold, in general, at a particular specified frequency: then one finds from equation (2.6) that R,(8)
= 1 + O(S’)
(2.11)
for small values of 8, the error term O(0’) indicating that R* may be taken as + 1 over a large range of angles-perhaps up to 30°-around grazing. Thus the Lloyd’s mirror effect can be circumvented by the substitution of a composite plate whose upper layer, in contact with the fluid, satisfies equation (2.10) which amounts to a statement that the composite should, in the absence of fluid loading, be capable of supporting a free coupled wave of wavenumber k,-i.e., a wave with precisely sonic phase velocity. If the upper layer impedance comprises a “stiffness” K and a “mass” term Mo2, so that zc = (i/o)(K - Mo’),
(2.12)
then equation (2.10) reads K + Bk; = (M+m)o’,
(2.13)
which may be used to define the constants K, M in terms of given values of B, m. It is to be noted here that in equations (2.5H2.13) it is not necessary to assume that the upper layer is capable of only local reaction. The equations hold, in fact, for any homogeneous layer which can be characterized by an impedance operator zc in physical space, or equivalently, by an impedance zC(k, w) in wavenumber-frequency space. The surface rigidity, and hence pressure-doubling implied by equation (2.11), obtained by satisfying equation (2.10), depends crucially on the parallel-impedance relation (2.8) for the joint impedance z, and it may be felt that the simple characterization by a two-layer model of the action of practical composite panels may be inadequate, and too simplified, to correctly deal with the grazing wave problem. That issue is not addressed here; all the author wishes to point out is that within one currently accepted simple model of a composite panel the clear possibility exists of designing such a surface (actually, an infinity of surfaces for each frequency) with infinite impedance at the grazing incidence condition, and thereby of allowing acoustic propagation, with pressure doubling, over the surface.
3. FREE COUPLED SUBSONIC SURFACE WAVES Consider now a free wave in the coupled three-part system. The frequency is prescribed, and one seeks the possible wavenumbers k such that the system (2.1H2.4) is satisfied, with
FLUID
LOADED
COMPOSITE
PANEL
RESPONSE
471
4 = Q = 0, by a surface deflexion proportional to exp(ik, x1 +ik,x,). It is clearly sufficient to consider only a plane wave propagating along the positive x,-axis, SO that ki = k, k, = 0, say. The significance of complex values of k needs careful attention (see reference [1]), and consideration is restricted here to waves with subsonic phase velocity and to fluid:upper 1ayer:lower layer systems with no damping. These conditions ensure that the values of k under consideration are real, positive, and greater than the acoustic wavenumber k,. For simplicity, attention here is also restricted to impedances zc of local reaction only, so that zCis a function of frequency alone. The dispersion relation for the free wavenumbers is easily obtained in the form l/P + l/z’ + l/z” = 0,
(3.1)
or, with the definitions k, = (rr~u~/B)“~,
p = ip,ojzC,
P = bblm,
y = +(k2-k;)lj2,
(3.2) (3.3)
in the form (k4 - k;)(y - /I) - pk;
= 0.
(3.4)
Suppose now that /? + + cc for fixed values of CL,k, and k,. This means that the upper layer is rather soft (zc small), with an impedance in which the positive “stiffness” term dominates the negative “mass” term. It is then easy to show that equation (3.4) has a large, real, positive root k, _ p + k;/2fl - k;/Sp3 + pk;/b4
+ ..-,
(3.5)
as well as a root close to k,, the free wavenumber of the plate in the absence of both fluid loading and the upper layer. This root is given by k, N k, - pk,/4P + . . . .
(3.6)
and is real if k, > k, (i.e., if the vacuum free wavenumber k, corresponds to a subsonic phase velocity o/k,), but has a small positive imaginary part if k, < k,; in that case the low impedance upper stratum provides a weak coupling with the fluid through which the energy of the supersonic free wave in the plate can slowIy “leak” as radiation to infinity. All other roots of equation (3.4), besides expressions (3.5) and (3.6), have substantial imaginary parts; their number and nature depend on a precise specification of the branch cuts for the function y(k) and, as discussed at length in reference [l], these roots have no 1 physical significance. Thus the upper layer with low impedance (with /I > 0) has the effect of (i) decoupling the plate from the fluid loading and leading to a wavenumber k, very close to that in the plate in the absence of both coating and fluid loading, and (ii) introducing a new high wavenumber mode (3.5) in which-as is quite clear from the fact that the plate properties enter into expression (3.5) only in the fourth term-the balance lies between the mass of the fluid and the stiffness of the upper layer. Feature (ii) distinguishes the composite plate from the conventional; for the latter it was proved in reference [l] that there is under all circumstances a single subsonic free surface wave (within the context of thin plate theory). It might be thought that the existence of the k, mode would be restricted to large values of /I. It appears, however, from numerical studies of equation (3.4) that two subsonic surface waves are to be found over quite a wide range of parameter values, though in all cases so far in which the author has found two roots the parameter b has been positioe. Also, of course, if /I is not large the two roots may be fairly close together and it will not be possible to interpret them in the clear-cut way given above.
472
D. G. CRIGHTON
To provide some examples, some studies have been made of the simpler case in which the plate is replaced by a membrane under tension T, with k, defined as k, = (mo2/T)1’2.
(3.7)
The free wave dispersion relation is then (k2-k,$(y-/I)
- pk;
= 0
(3.8)
k, N p + k;/2/3 + pk;/jI= - k:/8/33 + . . . .
(3.9)
and the roots of interest for large positive B are
k, N k,{l
-(p/2/?)-(p/2/Y2)[(k;
- k,$)“‘+p/4]
+ ++.}.
(3.10)
Equation (3.8) has been studied numerically, in the form (02 - 1) ((a2 - sz)l’2 - (1 - Q)“2(52/8,)3’2} - &/52”2 = 0,
(3.11)
with Q = k/k,,, and the three parameters, 8, Sz, and E equivalent to the three ratios of the four wavenumbers k,, ko, B and ~1.The reasons for using the form (3.11) are irrelevant here, though it is clear that the values of k,/k,, /?/km and p/k, are uniquely determined by those of 8, G!,and E. Limited results of these studies are given in Table 1, from which one must conclude that a check should always be made to see if two free modes are possible in given circumstances. It should also be noted that when there are two modes it is by no means necessarily the case that they are at very different wavenumbers. Indeed, Table 1 shows that the wavenumbers may lie less than an octave apart on either side of the vacuum wavenumber k = k, (a = 1). Attempts have been made to produce a simple analytical condition on the various parameters at which the number of subsonic free waves increases from 1 to 2, but so far without success.
4. DISCUSSION Little more needs to be said here about the results of section 2 on the grazing incidence field, except to emphasize again that these results depend very much on the parallelimpedance concept for the joint action of the two layers of the composite. There is, therefore, a need for experimental investigation of the degree to which this model of a composite panel is valid, as well as for theoretical study of the possibilities for forming the upper layer from’a layer of bubbly fluid adjacent to the plate, or from a suitably contained layer of pure gas. These theoretical studies are now being conducted, and it is hoped to report on them in due course. On the question of multiple (in fact, double) subsonic free wavenumbers for the composite plate, there are at least two areas in which this property could be important. The first, referred to in section 1, concerns the transmission of free surface wave energy across a rib, or other inhomogeneity, on a panel, a common application of which is to the problem of the isolation of some parts of a structure in air or water from other parts in which intense vibrational energy is generated. In calculations of the energy transfer across an inhomogeneity such as a stiffener one must take into account the fact that free wave energy may be incident upon the inhomogeneity in either or both of the two subsonic wave modes, and the inhomogeneity will act then as a wavenumber converter, ensuring that energy will be transmitted in both modes even in the unlikely event of its incidence in a single mode. In simple cases this amounts to no more than a piece of book-keeping, though in more complicated cases, involving several inhomogeneities and the associated
473
FLUID LOADED COMPOSITE PANEL RESPONSE TABLE
1
The real positive roots ol, c2 of equation (3.11) for CJ= k/k,; the parameter Sz = ki/ki takes values from 0.1 to 1.0 (“coincidence”), while the fluid loading parameter E has the values 0.3, 0.1, O-03 and the parameter 0, characterizing the stiffness of the upper layer has the values 0.25 and 0.5
E = 0.3, s2, = 0.25
E = 0.3, a, = 0.5
r s2
aI
0.1 0.2 0.3 0.4 0.5
1.363 1.397 1.592 2.011 2.619
0.6 0.7 0.8 0.9 1.0
3.349 4.164 5.051 6,000 7.006
a2
al
02
0.6094 0.7925 0.8827
1.331 1.286 1.287 1.316 1.377
-
0.9259 0.9490 0.9629 0.9719
1.477 1.623 1.814 2.041 2.299
E = 0.1, 8, = 0.25 I
\
E =
0.7758 0.8371
-
0.1,
0,
=
0.5
r
,
a
a1
a2
0.1 0.2
1.163 1.204
0.5406
1.143 1.124
-
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.425 1.919 2.574 3.324 4.150 5.042 5.994 7.002
0.8128 0.9239 0.9599 0.9751 0.9830 0.9876 0.9906 -
1.130 1.158 1.221 1.339 1.515 1.735 1.985 2.258
0.5517 0.6694 0.7844 0.8704 0.9209 0.9486 0.9650 -
E = 0.03, 9, = 0.25 \ n
a1
02
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.058 1.087 1.328 1.881 2.557 3.316 4.145 5.039 5,992 7GOl
0.3355 0.6994 0.9272 0.9762 0.9879 0.9925 0.9949 0.9963 0.9972 0.9978
a1
02
E = 0.03, Q, = 0.5 a1
1.049 1.043 1.047 1.065 1.122 1.261 1.465 1.703 1.963 2,243
a2
0.4570 0.5993 0.7475 0.8785 0.948 1 0.9731 0.9835 0.9889 -
474
D. G. CRIGHTON
resonance phenomena (as with two ribs on a panel, for example), the presence of two subsonic wave modes may lead to greater difficulties. Study of the energy transmission problem in detail is now in progress, with the multiple wave modes firmly in mind when dealing with composite panels. The second application of section 3 may arise in the case p --++ co, for which it was shown that, for virtually any lower layer of the plate or membrane type, a high wavenumber mode with k - j? exists, almost independently of the plate properties. It may then be possible that in some circumstances k, may be comparable with the characteristic wavenumber o/U, of intense boundary layer pressure fluctuations, U, being the convection velocity of the surface pressure field (generally around 0.6 of the velocity in the mainstream of a flow over the surface). There is then the possibility that relatively large levels of surface vibration may be generated on composite panels at forward speed in air or water through the familiar spatial resonance mechanism between the surface waves with k = k, and the turbulent excitation concentrated around k = w/U,. Before such a possibility can be quantified further, however, a fourth element, associated with the mean flow past the surface, must be added to the three-part coupled system studied here. The author hopes to discuss the effects of mean flow on coupled wave-bearing systems of the plate: fluid kind in a future paper.
ACKNOWLEDGMENT
This work was supported by the Ofice of Naval Research, Code 222, Acoustic Technology Program, under Grant N00014-77-G-0072. Some of the results were presented at the 95th Meeting of the Acoustical Society of America, Providence, Rhode Island, May 1978. The assistance of Dr J. F. Scott in the preparation of Table 1 is acknowledged with thanks.
REFERENCES D. G. CRIGHTON1979Journal of Sound and Vibration 63,225-236. The free and forced waves on a fluid-loaded elastic plate. G. MAIDANIK and A. J. TUCKER 1974Journal of Sound and Vibration 34, 519-550. Acoustic properties of coated panels immersed in fluid media. D. G. CRIGHTON1971 Journal of Fluid Mechanics 47, 625-638. Acoustic beaming and reflexion from wave-bearing surfaces. W. F. KING III 1973 Journal of Sound and Vibration 30,279-288. The influence of fluid loading on acoustic propagation near surfaces.