## Abstract

In this paper, we derive weakly nonlinear equations for the dynamics of a thin elastic plate of large extent under conditions of heavy fluid loading. Two situations are then considered. First, we consider the case in which transverse motion of the plate generates a weaker in-plane motion, which is in turn coupled back to the evolution of the transverse motion. This results in the familiar nonlinear Schrödinger equation for the amplitude of a transverse plane wave, and we show that solitary-wave solutions are possible over the range of (non-dimensional) frequencies *ω*>*ω*_{c}, which depends on the material properties. Dimensional values of *ω*_{c} are physically realizable for a typical composite material underwater. Second, we consider the case in which the amplitudes of the transverse and in-plane motion are of the same order of magnitude, possible at a single resonant frequency, which leads to an evolution equation of rather novel type. We find a range of travelling-wave solutions, including cases in which incident in-plane waves can generate localized regions of transverse displacement.

## 1. Introduction

The dynamic behaviour of thin elastic plates subjected to heavy fluid loading is a subject of considerable practical interest in naval engineering, and a great deal of fundamental research has been completed over a number of years. The key issue, we believe, concerns the full coupling between the fluid and solid motion, especially since, in practise, wavelengths are long so that a considerable added mass of water must be excited along with the structure.

In the classical linear theory of thin plates, the transverse flexural displacement of the plate is totally uncoupled from its in-plane tangential displacements. In the Hamiltonian formulation of the problem, this is manifested by neglecting the kinetic energy of in-plane motion and the potential energy of mid-surface membrane deformation. In the case of a statically pre-stressed plate, the in-plane membrane forces are treated as independent of the plate's actual deformation and therefore they contribute also only to the potential energy of bending. On the other hand, linear shell theory couples in-plane and transverse displacements in the formulation of the potential energy due to the curvature and twist of a shell element in its non-deformed configuration. Respectively, the kinetic energies of in-plane and transverse motions are regarded as being of the same order of magnitude.

In the weakly nonlinear theory of thin plates, which is used in this paper, the kinetic and potential energies are formulated as in the linear shell theory (e.g. Novozhilov 1964), but the curvature and twist of a plate element are produced exclusively by its deformation (see Timoshenko 1959 and also Vol'mir 1972). The components of in-plane strain contain products of the angles of rotation of perpendiculars to the neutral surface of the plate. This theory has been used intensively for analysis of nonlinear free and forced vibrations of plates of finite dimensions (see Vol'mir 1972 and the research paper references in this text), mostly under some further simplifying assumptions (e.g. those of a ‘shallow shell theory’, see Leissa 1973). To the best of the authors' knowledge, no work has yet been conducted on wave propagation in weakly nonlinear plates under heavy fluid loading. This issue is addressed in the present paper.

Linearized models of simple elastic plates with heavy fluid loading have been studied in great detail. For instance, the nature of waves allowed on infinitely long simple plates is described in Crighton (1979, 1989), with recent asymptotic work by Chapman & Sorokin (2005). An important extension to include the effects of the presence of mean flow was made by Crighton & Oswell (1991), which was in turn extended to cylindrical geometry by Peake (1997). The stability of more complicated plate and flow models is described in great detail in Carpenter & Garrad (1985, 1986). Work on *finite plates* has largely been numerical, see for instance Lucey (1998), although Peake (2004) has considered the case of a long, but finite, baffled plate asymptotically.

Turning now to work on nonlinear problems, Abrahams (1987, 1988) considered the response of a finite baffled plate to incident sound waves in conditions of light fluid loading. His plate equation includes the familiar nonlinear tension term (e.g. Dowell 1975), representing the additional tension induced in the plate by its deflection. Sorokin (2000) has extended this approach to the case of a baffled plate with heavy fluid loading, necessitating the inclusion of additional nonlinear terms in the Bernoulli integral for the fluid pressure and in the plate–fluid boundary condition. Peake (2001) and Sorokin & Chapman (2005) sought to consider the scenario in which the plate is sufficiently long that the effects of the plate–baffle junctions can be neglected, allowing the introduction of an additional nonlinearity in the plate equation corresponding to the exact representation of the plate curvature. In this approach, however, no attempt was made to make proper account of the in-plane motions of the plate. The static formulation of this theory has been presented in Timoshenko & Woinowsky-Kreiger (1959). This is a standard Kirchhoff Theory, with nonlinear geometry effects included. In the literature (e.g. Vol'mir 1972), nonlinear dynamics has been studied with in-plane inertia neglected. The originality of the present contribution lies in our analysis of the role of the in-plane inertia in the nonlinear dynamics of the plate. So, this paper presents an extended and consistent formulation of nonlinear Kirchhoff plate theory with heavy fluid loading.

In this paper, we seek to extend the work of Peake (2001) and Sorokin & Chapman (2005) by now using a much more detailed and comprehensive plate model, in which the coupling between the three-dimensional in-plane and transverse plate motion is included. The Hamiltonian formulation of the plate equations is given in §2. In §3, we describe a scenario in which a primary bending wave induces a shear wave, which in turn modulates the bending wave as it propagates. This leads to a classical nonlinear Schrödinger amplitude equation, and the existence of solitary waves is predicted. In §4, we consider a resonant case, in which bending and shear waves are of comparable magnitude, leading to a new evolution equation. Some solutions of this equation are derived, one of which can be interpreted as being the nonlinear generation of a localized region of transverse motion by an incident wave of in-plane motion.

## 2. Hamiltonian formulation of governing equations

We first present a systematic derivation of our governing equations, in which we start by writing down a Hamiltonian which accounts for weakly nonlinear plate motion, and then use standard variational arguments to derive the plate equations and edge conditions. We consider a thin elastic plate of uniform thickness *h*^{*} (‘*’ denotes dimensional variables) and density , which in its undeformed state lies in the *x*^{*}*–y*^{*} plane with . Above the plate, in *z*^{*}>0, there is an ideal compressible fluid of undisturbed density , while below the plate, there is a vacuum. Denoting the unsteady deformation of the plate by , with *t*^{*} the time, the kinetic energy per unit area can be written as(2.1)where we have made the usual thin-plate assumption of the deflection being uniform through the plate thickness. The potential energy per unit area is well known, and takes the form(2.2)The first three terms in (2.2) are the membrane components of the potential energy (see Timoshenko 1959, p. 419), while the second three terms correspond to the strain energy in pure bending (see Timoshenko 1959, p. 47). Here, the membrane forces and *S*^{*} are given in terms of the strain * ϵ* in a standard way by(2.3)with a similar equation for , where

*E*

^{*}is Young's modulus and

*ν*is Poisson's ratio. The bending moment and the twisting moment

*H*

^{*}are given by(2.4)with a similar expression for . In writing down (2.3) and (2.4), we are of course assuming that the material is linear, and hence obeys Hooke's Law. The presence of the twist term

*H*

^{*}follows the theory of Novozhilov (1964, p. 48). The components of the deformation are formulated to include weakly nonlinear corrections in the form(2.5)as given by Timoshenko (1959, p. 416). For consistency, the curvatures

*need only be evaluated to linear order, however, since the corresponding nonlinear corrections are cubic in the displacements, and we therefore write(2.6)*

**κ**We can now write down our variational problem in the form(2.7)from which the usual Euler–Lagrange equations can be derived, one for each coordinate direction, together with end conditions to apply at the edges of the plate. The effect of the fluid on the plate motion is expressed as an (as yet unknown) external force in the *z*^{*} Euler–Lagrange equation. Before presenting the results of this, we will non-dimensionalize length by and time by(2.8)This choice of non-dimensionalization is particularly convenient, since as we will see it eliminates a number of parameters from the transverse-wave dispersion relationship. After some algebra, we then find from the *x* and *y* Euler–Lagrange equations the two longitudinal plate equations,(2.9)and(2.10)From the *z* Euler–Lagrange equation, and using (2.9) and (2.10), we find the transverse plate equation,(2.11)where *p*(* x*,

*t*) is the dimensionless fluid pressure (which we will describe later).

The final terms in the longitudinal equations (2.9) and (2.10) correspond to in-plane inertia. Note how these terms contain the factor , which is usually small (about 0.02 for steel and water), and one might therefore be tempted to ignore this effect altogether. However, in deriving the transverse plate equation (2.11), the two in-plane inertia terms reappear as the second-to-last and third-to-last terms on the left-hand side, and note now how these terms do not contain the small density-ratio factor. In principle, the in-plane inertia therefore has an effect on the solution for *w*, which is of comparable size to the other nonlinearities in (2.11). The inclusion of in-plane inertia in our model is entirely comparable to the difference between so-called ‘shallow shell’ theory for a weakly curved shell, in which in-plane inertia is neglected, and the more precise Donnell–Mushtari theory (see Leissa (1973, pp. 27 ff), in which in-plane inertia is retained. We believe that the inclusion of in-plane inertia is a key feature of our model.

The pressure in the fluid is given by the Bernoulli integral, and for (assumed) irrotational motion this takes the dimensionless form(2.12)Here is the unsteady fluid potential, which satisfies the wave equation,(2.13)where *c* is the dimensionless sound speed in the fluid. Finally, we need to specify the continuity condition of continuous velocity normal to the plate, which gives(2.14)Note how this boundary condition is applied on the mean plate position *z*=0, consistent with our small-deflection shell theory. The effect of fluid viscosity is to induce motion in the fluid due to purely in-plane plate displacement, but the inclusion of this is essential only when the plate is loaded by a thin fluid layer. In the present case, when the fluid loading is produced by an unbounded volume of fluid, this effect may be reliably ignored and the standard inviscid model of the acoustic medium is perfectly valid.

## 3. Expansion for primary transverse motion

In this section, we will consider the weakly nonlinear motion of the plate and fluid in the case in which the primary motion is that of a transverse (bending) wave, which excites weaker longitudinal motion in the plate through the nonlinear coupling terms present in (2.9)–(2.12). In §4, we will consider an alternative scenario, in which the in-plane and transverse plate motions are of comparable size.

The leading-order amplitude of the primary wave will be denoted , with *X*, *Y*, *T* slow space and time-scales (to be defined) and an ordering parameter. The primary wave has a given frequency *ω*. We begin by writing the expansions,(3.1)(3.2)(3.3)where and c.c. stands for complex conjugate. In what follows, and without loss of generality, we will assume that the *x*-axis is chosen to be aligned with the phase velocity of the primary wave, so that *l*≡0. This choice does not preclude the possibility of slow modulation in the spanwise direction, so that the *Y*-dependence of *A* is retained. Note that in (3.1)–(3.3), the leading term in the transverse deflection induces longitudinal deflection through the quadratic nonlinearity in the longitudinal equations (2.9) and (2.10). Our aim will be to explore the evolution of the primary wave *AE* over large distances and times, and to do this we consider an observer moving with the group velocity of the primary wave and then introduce the slow space scales and the slow time-scale as measured by that moving observer. Here, is the group velocity of the primary wave (given our choice *l*≡0, it turns out that the group velocity is parallel to the *x*-axis), and the scalings in *X*, *Y*, *T* have been chosen to lead to the correct subsequent balance between nonlinearity and modulation. Following the usual procedure in the Method of Multiple Scales, see Hinch (1991), inclusion of these slow scales leads to the replacements:(3.4)We now proceed by substituting the expansions (3.1)–(3.4) into the governing equations (2.9)–(2.14), and then equating powers of *ε*. The analysis proceeds as follows.

### (a) Terms at *O*(*ϵ*)

At leading order, the problem reduces to the linear propagation of a bending wave on a simple plate with fluid loading. For arbitrary spanwise wavenumber *l*, the dispersion relation is(3.5)Here, , where . We will only need to evaluate *γ* (*k*, 0, *ω*), and in that case the branch cuts connect the branch points to infinity through the upper and lower half planes, respectively, and we choose *γ* to be real and positive as along the positive real axis. In what follows we shall be concerned only with the subsonic surface wave solution of (3.5), so that *γ* is real and positive. From the dispersion relation (3.5), the group velocity of the primary wave can be calculated in a straightforward way, and we find that(3.6)

Turning now to the unsteady velocity potential in the fluid, and insisting that the field is bounded far from the plate, we find(3.7)where all the functions decay as *z*→∞. There is an analogous expansion for the unsteady pressure, with coefficients , etc. At *O*(*ϵ*), it is easy to show that and(3.8)

### (b) Terms at *O*(*ϵ*^{2})

At *O*(*ϵ*^{2}), a number of nonlinear effects first appear, including the generation of second harmonics of the primary wave in the transverse deflection by the quadratic nonlinearities in the fluid pressure (2.12) and in the boundary condition (2.14), and the generation of longitudinal deflection by the quadratic nonlinearities in (2.9) and (2.10). We will consider first the equations corresponding to the fluid loading and the transverse motion. In the wave equation (2.13), at *O*(*ϵ*^{2}), we find from the terms proportional to *E* that(3.9)while from the terms proportional to *E*^{2}, we find(3.10)From the Bernoulli integral (2.12) at *O*(*ϵ*^{2}), and using (3.8), it follows that(3.11)Note that the non-zero term in (3.11) corresponds to a static pressure induced by the bending, which must be supported by inclusion of an extra static force term in the transverse plate equation. At *O*(*ϵ*^{2}) in the boundary condition (2.14),(3.12)Finally, taking the *E*^{2} terms in bending equation (2.11) at *O*(*ϵ*^{2}), and using (3.11) and (3.12), we find that the quadratic transverse deflection term is given by(3.13)It is easy to show that if with real wavenumbers and frequency, then is strictly negative (and definitely non-zero), so that there is no issue of quadratic resonance in (3.13). By considering the terms proportional to *E* in the *O*(*ϵ*^{2}) equations, it follows that without loss of generality we can set , while the values of and follow from (3.12) and (3.11). In fact, it will follow that we could take the term in *w* which is proportional to *E* to be zero at any higher order too, since that simply corresponds to rescaling the forcing wave amplitude *A*.

Turning now to the in-plane equations (2.9) and (2.10) at *O*(*ϵ*^{2}), the in-plane motion is driven by the terms quadratic in the primary transverse deflection. It follows that , and(3.14)Here,(3.15)and is the dispersion relation for purely longitudinal linear wave motion in the *x*-direction, i.e. longitudinal displacement parallel to phase velocity. In this section, we assume that , so that the wavenumbers of the transverse wave, which satisfy , do not match the wavenumbers of allowed longitudinal modes at a given frequency. Note that the dispersion relation for purely longitudinal linear wave motion in the *y*-direction, i.e. longitudinal displacement normal to phase velocity, is . The function is given by (3.15), but with the extra factor (1−*ν*)/2 multiplying the final term. We have also assumed that , allowing us to set in (3.14).

The only possible remaining terms from the longitudinal plate equations are mean displacements , say, which are time-independent. However, when the primary transverse wave deflection *AE*+c.c. is substituted into the nonlinear terms in (2.9) and (2.10), the terms in *E*^{0} disappear, demonstrating that there is no time-independent forcing by the primary wave. Without loss of generality we therefore set . As would be expected, we therefore find zero longitudinal deflection in the direction normal to the primary wave direction (i.e. *v*=0).

### (c) Terms at *O*(*ϵ*^{3})

At *O*(*ϵ*^{3}), we will find the evolution equation satisfied by the primary-wave amplitude *A*(*X*, *Y*, *T*). This will contain cubically nonlinear terms as well as single derivatives with respect to the slow time variable *T* and double derivatives with respect to the slow space variable *X*. Since this will involve a considerable amount of algebra, we will simply summarize key intermediate results below.

Turning first to the transverse plate equation (2.11) and taking terms proportional to *ϵ*^{3}*E*, we find that(3.16)where(3.17)

(3.18)Note that in (3.17), the first term, which does not depend on the in-plane motion, is precisely the same term as would be found from a one-dimensional nonlinear beam equation. The third term in (3.17) corresponds to the contribution from the in-plane inertia term discussed in §2. Note how the in-plane inertia appears at the same asymptotic order as the ordinary beam term, confirming the importance of including it within our theory. This result confirms that it is dangerous to ignore in-plane inertial effects in a fluid-loaded plate, and therefore it proves the relevance of our suggested theoretical model.

The unsteady pressure on the right-hand side of (3.16) follows from (2.12), from which we find that(3.19)Here, the nonlinear term is given by(3.20)and we recall that . Now in the boundary condition (2.14), terms proportional to *ϵ*^{3}*E* yield(3.21)where(3.22)In order to complete the solution, we require a relationship between and on *z*=0, and this will be provided by the wave equation (2.13). Taking terms in *ϵ*^{3}*E* from (2.13) results in an ODE in *z*, with a non-zero right-hand side containing terms involving derivatives of *A* with respect to *X*, *Y*, *T*. By solving this equation and then setting *z*=0, we find that(3.23)The quantities *a* and *b* are given by(3.24)and have arisen from the terms on the right of the above ODE.

We can now form the evolution equation for *A*(*X*, *Y*, *T*). We determine from (3.21) and (3.23), and substituting into (3.19) and then into (3.16), we find the two-dimensional nonlinear Schrödinger equation in the form(3.25)where the coefficients of the linear terms are(3.26)and the coefficient of the nonlinear term is(3.27)It should be emphasized that the coefficients of the linear terms in (3.25) are exactly what one would expect from expansion of the dispersion equation (3.5), having accounted for the Galilean transformation to a frame moving with the group velocity of the primary wave. Note that the coefficients in (3.25) are all real. It is easy to show analytically that is strictly positive and *c*_{2} is strictly negative for all *ω*>0. The coefficient *c*_{1} can be either positive or negative (see figure 1), and it is easy to show numerically that when for a critical frequency *ω*_{c}. The value of *ω*_{c} depends on the fluid and solid properties through the dimensionless sound speed *c*. For definiteness, we consider two representative test cases: one being steel in water, for which we take , , , *ν*=0.33; and the second being a typical composite material in water with , , , *ν*=0.33. In both cases, we take and . For steel in water, we find that , leading to the dimensional result , which is an unrealistically high frequency. However, for the composite in water, we find , leading to the much more physically realizable dimensional result . We therefore conclude that the case *c*_{1}<0 is the one which is relevant for steel in water, but for the lighter composite material it seems that *c*_{1} can be either positive or negative in practise depending on the excitation frequency. It can be shown numerically that the nonlinear coefficient *c*_{3} turns out to be negative for *ω*<*ω*_{d} and positive for *ω*>*ω*_{d}. The value of *ω*_{d} depends on the density ratio ; for steel in water, we have the unrealistically high value , but for the composite in water we find the practically attainable value .

The type of solution of (3.25) depends crucially on the signs of the coefficients, and full details are given in Craik (1985) and Infeld & Rowlands (1990). Consider first the one-dimensional case by setting *A*_{YY}=0. Then it follows (Craik 1985, pp. 199 ff) that if (i.e. ), then (3.25) takes the so-called ‘focusing’ form and possesses a family of soliton solutions (Zakharov & Shabat 1972) of the form(3.28)where *α* and *β* are arbitrary positive constants. The height to width ratio of the soliton is proportional to , and results are shown in figure 2—note that the height-to-width ratio is quite small, except when when it rises steeply ( in this case). This indicates that the solitary waves which may be observed in practise would be of relatively wide spatial extent. Alternatively, if (i.e. *ω*<*ω*_{c} or *ω*<*ω*_{d}), then (3.25) takes the so-called ‘defocusing’ form, no solitons are present and localized solutions will typically decay in time. However, even in this case waves of permanent form are still possible, and, in particular, a shock-wave solution exists (Infeld & Rowlands 1990, p. 151) between states of uniform non-zero |*A*|.

Turning now to the two-dimensional case, and considering modulation only along a direction making an angle *ψ* with the *X*-axis, it follows (Craik 1985, pp. 199 ff) that the soliton (3.28) can exist if(3.29)If *ω*<*ω*_{c}, then (since *c*_{2} is always negative), and so condition (3.29) is not satisfied for any *ψ*. If , then we require in order for (3.29) to be satisfied. Finally, if *ω*>*ω*_{d}, then we require in order for (3.29) to be satisfied. We can therefore conclude that if *ω*<*ω*_{c}, then we have the defocusing form of the nonlinear Schrödinger equation in all modulation directions, and the solitary wave (3.28) is not possible. When *ω*>*ω*_{c} the solitary waves are present; for *inside*, and for *ω*>*ω*_{d} *outside*, the wedge of semi-angle aligned with the *X*-axis. This is to be paralleled with the work of Hui & Hamilton (1979), who studied gravity waves in deep water and found solitons inside a cone aligned with the direction of motion of the wave-maker. Given the comments we have already made about the dimensional values of *ω*_{c,d}, it follows that solitons will not be seen for the steel in water case, but are possible at realistic (higher) frequencies for the composite plate in water.

Simplification of our analysis is possible in the limit of small *ω*. If we consider the bending wavelength of the plate in vacuum, then it turns out that the ratio of the mass of fluid within one bending wavelength of the plate to the mass per unit length of the plate is proportional to , so that the limit of small *ω* corresponds to the limit of very heavy fluid loading. In this limit, the solution of the dispersion relation (3.5) is , and to leading order in *ω* we find that(3.30)while(3.31)In this limit note that fluid compressibility does not feature (given the absence of *c* from the above results), and that the nonlinear coefficient *c*_{3} is dominated by . By tracing the origin of the terms in the asymptotic form of , it becomes clear that the controlling nonlinearities in this limit are the terms and *w*_{xx}*u*_{x} in the transverse plate equation (2.11), which now dominate the in-plane inertia term. The longitudinal deflection *u* is given in this limit by the balance between the tension term *u*_{xx} and the nonlinearity *w*_{x}*w*_{xx} in (2.9). It may at first sight appear surprising that the plate nonlinearity dominates over the fluid nonlinearities in the regime of very heavy fluid loading. However, this can be explained by noting that the dominant dynamic balance is between the fluid pressure and the plate bending stiffness (hence ), leading to a situation in which the non-dimensional frequency is much smaller than the non-dimensional wavenumber. This balance occurs at linear order, and the nonlinear terms just provide small corrections to this. The correction to the fluid pressure depends on unsteady velocities, which at low frequency are very small, while the correction to the plate bending (e.g. the term *w*_{xx}*w*_{x}) depends on wavelength, and therefore dominate.

Alternatively, when (i.e. very light fluid loading), we find that the subsonic surface wave has and (i.e. is only just subsonic relative to the fluid sound speed). After some algebra we find that , while(3.32)so that the nonlinear terms associated with the fluid motion and the boundary condition dominate the plate terms, and leading to *c*_{3}=*O*(*ω*^{12}) with a positive coefficient. The linear coefficients in (3.25) behave like *c*_{1}=*O*(*ω*^{2}) and *c*_{2}=*O*(*ω*^{8}) with *c*_{2} still negative, so that interestingly in this limit the dispersion in the direction transverse to the wave direction (i.e. the *Y*-direction) is significantly larger than the *X*-ward dispersion. Note that at these large frequencies and with a very long wavelength into the fluid (*γ*≈0), the fluid pressure required to bend the plate by a given amount is large (see (3.8)), and this explains why the nonlinear terms in the fluid are larger than the plate nonlinearities in this regime. The two fluid nonlinearities occur in the Bernoulli integral for the pressure (i.e. ) and in the boundary condition (i.e. ), and act so as to tend to focus and defocus solitary waves, respectively—as can be seen in (3.32), the effect of the fluid pressure term dominates, so that the nonlinear Schrödinger equation takes its focusing from in the *Y*-direction. We have already noted that solitary waves can occur when *ω*>*ω*_{d} outside a wedge aligned with the *X*-axis. In the limit of large *ω*, the corresponding asymptotic result shows that the solitons can be found for modulation directions outside the wedge of semi-angle . This means that at high frequency the solitary waves are found in all modulation directions away from the *X*-axis.

Results for arbitrary *ω* are given in figure 3. Note that for *ω* small and *ω* large, the various curves asymptote to straight lines, with gradient consistent with the results given above. Note also that the deep cusps in the curves for correspond to the points where *c*_{1} and *c*_{3} change sign.

## 4. Comparable in-plane and transverse motion

In §3, we considered the case in which the transverse plate deflection is much larger than the longitudinal deflection, but we now move on to consider the case in which the two are of comparable size. We again choose the *x*-direction to be aligned with the carrier-wave phase speed, so that *l*=0, and in the first instance consider the longitudinal primary wave with deflection parallel to the *x*-axis. It is then easy to show that there is a single resonant frequency, *ω*_{0}, and a corresponding wavenumber, *k*_{0} (taken to be positive without loss of generality), such that , with *ω*_{0} the single positive real root of(4.1)For this to be possible, we have assumed that the square-root in the final term is real, which on reintroducing dimensional quantities yields the condition . This means that for (4.1) to have a real root, *ω*_{0}, we require the longitudinal wave speed to be subsonic relative to the fluid sound speed. It should be noted that this is not the case for the steel plate and composite plate in water studied in §3, but certainly does occur for softer plate material. For instance, for soft rubber, , *ν*=0.5 and (see Howe 1998), we have , so that the longitudinal plate wave is very slow. Taking underwater yields the resonant frequency for soft rubber.

Our next observation is that if , then . This means that a wave satisfying the longitudinal dispersion relation can directly couple with the sub-harmonic wave satisfying the transverse dispersion relation through the term quadratic in *w* in the in-plane plate equation (2.9). This leads us to make the expansion(4.2)where now and the amplitudes depend on the slow scales . Choosing implies that *v*=0 to leading order.

At *O*(*ϵ*), the longitudinal equations (2.9) and (2.10) are automatically satisfied, while at *O*(*ϵ*^{2}) we find the evolution equation,(4.3)which arises from the forcing of the longitudinal motion by terms quadratic in the transverse motion.

At *O*(*ϵ*), the transverse plate and fluid equations are automatically satisfied, with the fluid potential and pressure again given by (3.8). At *O*(*ϵ*^{2}) rather more manipulation is required, but it follows in a relatively straightforward way that the terms linear in *E* yield the second evolution equation,(4.4)The nonlinear terms which contribute here are the nonlinear bending term *w*_{xx}*u*_{x} and the in-plane inertia term *u*_{tt}*w*_{x}. Note that neither of the evolution equations (4.3) and (4.4) contains a *Y* derivative, due to the fact that the *l* derivatives of both and vanish when *l*=0. It therefore follows that spanwise modulation does not occur in this case. Also note that the nonlinearities in (4.4) have arisen from the nonlinear bending term *w*_{xx}*u*_{x} and the in-plane inertia term *u*_{tt}*w*_{x} in the plate equation (2.11) only, so that the nonlinear terms in the fluid pressure and the boundary condition do not feature here. (Of course, the fluid loading is still important, but appears through the dependence of the coefficients in (4.4) on the linear dispersion function and on *ω*_{0}.)

Introducing the characteristic coordinates(4.5)and eliminating *B* between (4.3) and (4.4), we find the evolution equation,(4.6)with *B* given by(4.7)and the real constants *d*_{1,2} by(4.8)It is easy to show that . We are not aware of the evolution equation (4.6) having been previously studied in another context.

In order to seek a solution of (4.6), we follow the procedure of Dr N. J. Berloff (2005, personal communication) and write(4.9)where *f*, *ϕ*, *ψ* are as yet unknown real functions. Substituting into (4.6) and equating real and imaginary parts, we find that(4.10)

(4.11)From (4.11) it follows that there are two possible families of solutions, which are described in §4*a*,*b*.

### (a) Family A

If *ψ*′≡0, so that *ψ*=*ψ*_{0} a constant, then (4.10) yields(4.12)and by seeking a travelling-wave solution of the form *f*(*ζ*), with , we find after some algebra that(4.13)for any given negative *V*. Without loss of generality an arbitrary multiplicative constant has been set equal to unity (this can be achieved by a simple multiplicative rescaling of *f*, *ξ* and *η* in (4.12)). Note that the phase function *φ*(*ξ*) is arbitrary. The solution for the wave amplitudes is(4.14)These solutions correspond to travelling waves in which the transverse motion decays as , while the longitudinal motion has non-zero amplitude right out to infinity. The propagation direction of the wave envelope is in the positive *ξ*-direction (recall *V*<0). However, in physical space, the envelope can propagate in either the positive or negative *X*-directions, depending on the choice of the arbitrary negative *V*; *ζ* is proportional to , with(4.15)and since the dispersion-relation derivatives in the numerator/denominator are negative/positive, respectively, while *V* is negative, it follows that can take either sign. In the simple case , we can interpret the solution as an incident plane wave of in-plane motion which induces a localized distribution of transverse motion, before being transmitted (without reflection) with a phase shift of *π*. The characteristic width of the distribution of transverse motion is . The presence of the arbitrary phase factor *ϕ*, which depends on the characteristic variable *ξ* associated with the linear longitudinal wave speed, corresponds to the existence of a whole series of such solutions for different incident longitudinal waves, but for which the envelope amplitude of the transverse motion is always of the same ‘sech’ form.

### (b) Family B

In (4.11), if , then (4.10) implies that(4.16)and since *f* depends only on *η* in this case, we necessarily have with constant *ϕ*_{0}. The phase function *ψ*(*η*) is then any continuously differentiable function, such that , and the wave amplitudes take the form(4.17)

(4.18)This family of solutions correspond to harmonic waves with longitudinal motion of uniform amplitude, but with a variable phase 2*ψ* depending on *η*, the characteristic variable associated with the transverse linear wave speed. This phase variation induces a sub-harmonic transverse deflection of varying amplitude, dependent on *ψ*′, and a very wide range of spatial distributions of *A* are possible for different choices of *ψ*. For instance, if we choose , then we can generate a localized transverse disturbance with the same ‘sech’ amplitude as in case (*a*) above.

As a final point in this section, we note that we can also investigate the alternative scenario, in which there is a resonance between the transverse wavenumber satisfying and the in-plane motion normal to the phase velocity, i.e. satisfying . We now have that *u* is zero to leading order, while *v* is given by an expansion of the form (4.2). However, in this case no nonlinear coupling terms arise between the transverse and longitudinal motion at leading order, essentially because we have chosen (without loss of generality) the *y* wavenumber to be zero. The evolution equations for the amplitudes *A* and *B* are now simply given by the linearized versions of (4.4) and (4.3), so that the waves simply propagate with constant group velocity and no interaction takes place.

## 5. Concluding remark

In this paper, we have presented a model for the nonlinear dynamics of a long elastic plate under conditions of heavy fluid loading. In particular, we have used a consistent Hamiltonian formulation to include nonlinear effects associated with both the transverse and longitudinal motions of the plate. We have then investigated two quite different regimes. In the first case (§3), we considered the case in which a primary transverse plate motion generates in-plane plate motion, which in turn couples back into the long-range modulation of the primary wave. This results in the evolution being described by a conventional nonlinear Schrödinger equation, and for a range of frequencies the existence of solitary waves is predicted. These frequencies are physically realizable for a typical composite material in water, but not for steel. In the second case (§4), we show that for a particular excitation frequency at which it is possible to have transverse and in-plane plate motions of comparable size, leading to a quite different evolution equation for the wave amplitudes. Two possible families of solutions are identified. The so-called family A waves can be interpreted as an incident plane wave of in-plane motion, which due to nonlinear action induces a localized transverse response before being transmitted (without reflection). The so-called family B waves correspond to in-plane motions of spatially uniform amplitude, but with a varying phase, which induces sub-harmonic transverse motion with spatially varying amplitude.

## Acknowledgments

The authors are very grateful to Dr N. J. Berloff for considerable assistance with the material in §4, and for funding provided by the EPSRC of UK under grant GR/R65657/01.

## Footnotes

- Received September 19, 2005.
- Accepted January 11, 2006.

- © 2006 The Royal Society