Interaction Theory for Infinite Arrays

Introduction

There are two approaches to solution for the Infinite Array, one is Infinite Array Green Function the other is based on Interaction Theory. We present here a solution based on the latter, using Kagemoto and Yue Interaction Theory to derive a system of equations for the infinite array. This is based on Peter, Meylan, and Linton 2006

System of equations

We start with the final system of equations of the Kagemoto and Yue Interaction Theory, namely

[math]\displaystyle{ A_{m\mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=1,j \neq l}^{N} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big], }[/math]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math], [math]\displaystyle{ l=1,\dots,N }[/math].

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by [math]\displaystyle{ R }[/math], we have [math]\displaystyle{ R_{jl} = |j-l| R }[/math] and

[math]\displaystyle{ \varphi_{n} = \begin{cases} \pi, & n\gt 0,\\ 0, & n\lt 0. \end{cases} }[/math]

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients [math]\displaystyle{ A_{m\mu}^l }[/math] can be written as [math]\displaystyle{ A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu} }[/math], where the phase factor [math]\displaystyle{ P_l }[/math] is given by

[math]\displaystyle{ \ P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi}, }[/math]

where [math]\displaystyle{ \chi }[/math] is the angle which the direction of the ambient waves makes with the [math]\displaystyle{ x }[/math]-axis. The same can be done for the coefficients of the ambient wave, i.e. [math]\displaystyle{ \tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu} }[/math].

Therefore, the system simplifies to

[math]\displaystyle{ A_{m\mu} = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu \sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_{\tau - \nu} (k_n |j|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big]. }[/math]

Introducing the constants

[math]\displaystyle{ \sigma^n_\nu = \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_\nu(k_n|j|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR), }[/math]

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

[math]\displaystyle{ A_{m\mu} = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu \sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big], }[/math]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math]. Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula [math]\displaystyle{ A_{m\mu}^l = P_l A_{m\mu} }[/math].

Far Field Waves

An algebraically exact solution to the problem of linear water-wave scattering by a periodic array of scatterers is presented in which the scatterers may be of arbitrary shape. The method of solution is based on an interaction theory

in which the incident wave on each body from all the other bodies in the array is expressed in the respective local cylindrical eigenfunction expansion. We show how to calculate the slowly convergent terms efficiently which arise in the formulation and how to calculate the scattered field far from the array. The application to the problem of linear acoustic scattering by cylinders with arbitrary cross-section is also discussed. Numerical calculations are presented to show that our results agree with previous calculations. We present some computations for the case of fixed, rigid and elastic floating bodies of negligible draft concentrating on presenting the amplitudes of the scattered waves as functions of the incident angle.

The far field

In this section, the far field is examined which describes the scattering far away from the array. The derivation is equivalent to that of Twersky 1962. First, we define the scattering angles which give the directions of propagation of plane scattered waves far away from the array. Letting [math]\displaystyle{ p=2\pi/R }[/math], define the scattering angles [math]\displaystyle{ \chi_m }[/math] by

[math]\displaystyle{ \chi_m = \cos^{-1} (\psi_m/k) \quad =where= \quad \psi_m = k \cos \chi + m p }[/math]

and write [math]\displaystyle{ \psi }[/math] for [math]\displaystyle{ \psi_0 }[/math]. Also note that [math]\displaystyle{ \chi_0 = \chi }[/math] by definition. If [math]\displaystyle{ \abs{\psi_m}\lt k }[/math], i.e.~if

[math]\displaystyle{ -1 \lt \cos \chi +\frac{mp}{k}\lt 1, }[/math]

we say that [math]\displaystyle{ m\in \mathcal{M} }[/math] and then [math]\displaystyle{ 0\lt \chi_m\lt \pi }[/math]. It turns out (see below) that these angles ([math]\displaystyle{ \pm \chi_m }[/math] for [math]\displaystyle{ m \in \mathcal{M} }[/math]) are the directions in which plane waves propagate away from the array. If [math]\displaystyle{ \abs{\psi_m}\gt k }[/math] then [math]\displaystyle{ \chi_m }[/math] is no longer real and the appropriate branch of the [math]\displaystyle{ \arccos }[/math] function is given by

[math]\displaystyle{ \arccos t = \begin{cases} \mathrm{i} \arccosh t, & t\gt 1,\\ \pi-\mathrm{i} \arccosh (-t) & t\lt -1, \end{cases} }[/math]

with [math]\displaystyle{ \arccosh t = \log \left(t+\sqrt{t^2-1}\right) }[/math] for [math]\displaystyle{ t\gt 1 }[/math].

For the total potential we have

[math]\displaystyle{ \begin{matrix} \notag \phi &=\phi^\mathrm{In}+ \sum_{m=0}^{\infty} f_m(z) \sum_{j=-\infty}^{\infty} P_j \sum_{\mu = -\infty}^{\infty} A_{m\mu} K_\mu(k_m r_j)\mathrm{e}^{\mathrm{i} \mu\theta_j} \\ &\sim \phi^\mathrm{In}+ \frac{\pi}{2} f_0(z) \sum_{j=-\infty}^{\infty} P_j \sum_{\mu = -\infty}^{\infty} A_{0\mu} \i^{\mu+1} H^{(1)}_\mu (kr_j) \mathrm{e}^{\mathrm{i} \mu\theta_j}, (26) \end{matrix} }[/math]

as [math]\displaystyle{ kr\to\infty }[/math], away from the array axis [math]\displaystyle{ y=0 }[/math], where we have used the identity \eqref{H_K}.

The far field can be determined as follows. If we insert the integral representation

[math]\displaystyle{ H^{(1)}_\mu (kr) \mathrm{e}^{\mathrm{i} \mu \theta}= \frac{(-\i)^{\mu+1}}{\pi} \int\limits_{-\infty}^{\infty} \frac{\mathrm{e}^{-k\gamma(t)\abs{y}}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}\,\mathrm{e}^{\i \mu \sgn(y)\arccos t} \,\mathrm{d} t, }[/math]

in which [math]\displaystyle{ x=r\cos\theta }[/math], [math]\displaystyle{ y=r\sin\theta }[/math] and [math]\displaystyle{ \gamma(t) }[/math] is defined for real [math]\displaystyle{ t }[/math] by

[math]\displaystyle{ \gamma(t) = \begin{cases} -\mathrm{i} \sqrt{1-t^2} & \abs{t} \leq 1 \\ \sqrt{t^2-1} & \abs{t}\gt 1, \end{cases} }[/math]

into (26) we get

[math]\displaystyle{ \begin{matrix} \phi & \sim\phi^\mathrm{In}+ \frac{1}{2} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \int\limits_{-\infty}^{\infty} \frac{\mathrm{e}^{-k \gamma(t)\abs{y}}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt} \,\mathrm{e}^{\i(\psi-kt) jR}\,\mathrm{e}^{\i \mu \sgn(y)\arccos t} \,\mathrm{d} t \\ & =\phi^\mathrm{In}+ \frac{\pi}{kR} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \frac{\mathrm{e}^{-k\gamma(\psi_j/k)\abs{y}}}{\gamma(\psi_j/k)} \,\mathrm{e}^{\mathrm{i} x\psi_j}\,\mathrm{e}^{\i \mu\sgn(y)\arccos \psi_j/k} \\ & =\phi^\mathrm{In}+ \frac{\pi\i}{kR} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \frac{1}{\sin\chi_j} \,\mathrm{e}^{\mathrm{i} kr\cos(\abs{\theta}-\chi_j)}\,\mathrm{e}^{\i \mu \sgn(\theta)\chi_j}, \end{matrix} }[/math]

in which we have used the Poisson summation formula,

[math]\displaystyle{ \sum_{m=-\infty}^\infty \int_{-\infty}^{\infty} f(u)\, \mathrm{e}^{-\mathrm{i} mu} \,\mathrm{d} u= 2\pi \sum_{m=-\infty}^\infty f(2m\pi). }[/math]

The only terms which contribute to the far field are those for which [math]\displaystyle{ \abs{\psi_m}\lt k }[/math]. Thus, as [math]\displaystyle{ y\to\pm\infty }[/math], the far field consists of a set of plane waves propagating in the directions [math]\displaystyle{ \theta=\pm\chi_m }[/math]:

[math]\displaystyle{ \phi\sim \phi^\mathrm{In}+ \frac{\pi \i}{kR} f_0(z) \sum_{m\in\mathcal{M}} \frac{1}{\sin\chi_m} \,\mathrm{e}^{\mathrm{i} kr\cos(\theta\mp\chi_m)} \sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}. (30) }[/math]

>From \eqref{eqn:inffar} the amplitudes of the scattered waves for each scattering angle [math]\displaystyle{ \pm \chi_m }[/math] are given in terms of the coefficients [math]\displaystyle{ A_{0\mu} }[/math] by

[math]\displaystyle{ (31) A^\pm_m = \frac{\pi \i}{kR} \frac{1}{\sin\chi_m} \sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}. }[/math]

Note that the primary reflection and transmission coefficients are recovered by [math]\displaystyle{ A^-_0 }[/math] and [math]\displaystyle{ 1 + A^+_0 }[/math], respectively.

It is implicit in all the above that [math]\displaystyle{ \sin\chi_m\neq 0 }[/math] for any [math]\displaystyle{ m }[/math]. If [math]\displaystyle{ \sin\chi_m=0 }[/math] then we have the situation where one of the scattered plane waves propagates along the array. We will not consider this resonant case here except for stating that then, the scattered field is dominated by waves travelling along the array, either towards </math>x = \infty</math> (if [math]\displaystyle{ \chi_m = 0 }[/math]) or towards [math]\displaystyle{ x=-\infty }[/math] (if [math]\displaystyle{ \chi_m = \pi }[/math]). Also, we will not consider the excitation of Rayleigh-Bloch waves, which are waves which travel along the array with a phase difference between adjacent bodies greater than [math]\displaystyle{ Rk }[/math] (include refs). Both the resonant and Rayleigh-Bloch case are important but beyond the scope of the present work.

The efficient computation of the [math]\displaystyle{ \sigma_{\nu}^0 }[/math]

The constants [math]\displaystyle{ \sigma_{\nu}^0 }[/math] (cf.~\eqref{eq_op_sigma}) appearing in the system of equations for the coefficients of the scattered wavefield of the bodies cannot be computed straightforwardly. This is due to the slow decay of the modified Bessel function of the second kind for large imaginary argument as was discussed in \S 14. First, note that

[math]\displaystyle{ \sigma_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) K_{\nu} (-\mathrm{i} k j R) = \frac{\pi \i^{\nu+1}}{2} \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR), }[/math]

where we have used \eqref{H_K}. Therefore, it suffices to discuss the computation of the constants [math]\displaystyle{ \tilde{\sigma}^0_\nu }[/math] defined via

[math]\displaystyle{ \tilde{\sigma}_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR) }[/math]

as the [math]\displaystyle{ \sigma^0_\nu }[/math] are then determined by </math>\sigma^0_\nu = \pi/2 \,\, \i^{\nu+1} \, \tilde{\sigma}^0_\nu[math]\displaystyle{ . An efficient way of computing the \lt math\gt \tilde{\sigma}_{\nu}^0 }[/math] is given in linton98 and the results are briefly outlined in our notation. Noting that [math]\displaystyle{ H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu} H^{(1)}_{\nu} (\,\cdot\,) }[/math], it suffices to discuss the computation of the [math]\displaystyle{ \sigma_{\nu}^0 }[/math] for non-negative [math]\displaystyle{ \nu }[/math].

Referring to Linton 1998, the constants [math]\displaystyle{ \tilde{\sigma}_{\nu}^0 }[/math] can be written as

[math]\displaystyle{ \tilde{\sigma}_{0}^0 &= -1 -\frac{2\i}{\pi} \left( C + \log \frac{k}{2p} \right) + \frac{2}{R k \sin \chi} - \frac{2 \mathrm{i} (k^2 + 2 \psi^2)}{p^3 R} \zeta(3)\\ &\quad + \frac{2}{R} \sum_{m=1}^\infty \left( \frac{1}{k \sin \chi_{-m}} + \frac{1}{k \sin \chi_m} + \frac{2 \i}{p m} + \frac{\mathrm{i} (k^2 + 2 \psi^2)}{p^3 m^3} \right) }[/math]

where [math]\displaystyle{ C \approx 0.5772 }[/math] is Euler's constant and [math]\displaystyle{ \zeta }[/math] is the Riemann zeta function and the terms in the sum converge like [math]\displaystyle{ O(m^{-4}) }[/math] as [math]\displaystyle{ m\rightarrow\infty }[/math] (by which we mean that the error in the sum is proportional to [math]\displaystyle{ m^{-4} }[/math] for large values of [math]\displaystyle{ m }[/math]) as well as

[math]\displaystyle{ \begin{matrix} &\quad \tilde{\sigma}_{2\nu}^0 &= 2 (-1)^{\nu} \left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi} }{R k \sin \chi} - \frac{\i}{\pi} \left( \frac{k}{2 p} \right)^{2\nu} \zeta(2\nu +1) \right) + \frac{\i}{\nu \pi} \\ &\quad + 2 (-1)^\nu \sum_{m=1}^\infty \left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi_m}}{R k \sin \chi_{m}} + \frac{\mathrm{e}^{-2 \mathrm{i} \nu \chi_{-m}}}{R k \sin \chi_{-m}} + \frac{\i}{m\pi} \left( \frac{k}{2 m p} \right)^{2\nu} \right)\\ &\quad + \frac{\i}{\pi} \sum_{m=1}^\nu \frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m)! (\nu-m)!} \left( \frac{p}{k} \right)^{2m} B_{2m}(\psi/p), \\ & \tilde{\sigma}_{2\nu-1}^0 &= - 2 (-1)^\nu \left( \frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1) \chi}}{R k \sin \chi} - \frac{ \psi R \nu}{\pi^2} \left( \frac{k}{2 p} \right)^{2\nu-1} \zeta(2\nu +1) \right)\\ &\quad - 2 (-1)^\nu \sum_{m=1}^\infty \left(\frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)\chi_m} }{R k \sin \chi_m} + \frac{\mathrm{i} \mathrm{e}^{-\mathrm{i} (2\nu-1) \chi_{-m}}}{R k \sin \chi_{-m}} + \frac{\psi R \nu}{m^2\pi^2} \left( \frac{k}{2 m p} \right)^{2\nu-1} \right)\\ &\quad - \frac{2}{\pi} \sum_{m=0}^{\nu-1} \frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m+1)! (\nu-m-1)!} \left( \frac{p}{k} \right)^{2m+1} B_{2m+1}(\psi/p), \end{matrix} }[/math]

for [math]\displaystyle{ \nu\gt 0 }[/math] where [math]\displaystyle{ B_m }[/math] is the [math]\displaystyle{ m }[/math]th Bernoulli polynomial. The slowest convergence in this representation occurs in [math]\displaystyle{ \tilde{\sigma}^0_1 }[/math] and [math]\displaystyle{ \tilde{\sigma}^0_2 }[/math] in which the terms converge like [math]\displaystyle{ O(m^{-5}) }[/math] as [math]\displaystyle{ m\rightarrow\infty }[/math].

Note that since [math]\displaystyle{ \sin \chi_m }[/math] is purely imaginary for </math>m \notin \mathcal{M}[math]\displaystyle{ , the computation of the real part of \lt math\gt \tilde{\sigma}_{2\nu}^0 }[/math] and the imaginary part of [math]\displaystyle{ \tilde{\sigma}_{2\nu-1}^0 }[/math] is particularly simple. For [math]\displaystyle{ \nu \geq 0 }[/math], they are given by

[math]\displaystyle{ \begin{matrix} \Re \tilde{\sigma}_{2\nu}^0 &= -\delta_{\nu 0} + 2(-1)^\nu \sum_{m\in\mathcal{M}} \frac{\cos 2 \nu \chi_m}{R k \sin \chi_m}, \\ \Im \tilde{\sigma}_{2\nu+1}^0 &= 2\mathrm{i} (-1)^\nu \sum_{m\in\mathcal{M}} \frac{\cos (2 \nu-1) \chi_m}{R k \sin \chi_m}, \end{matrix} }[/math]

where [math]\displaystyle{ \delta_{mn} }[/math] is the Kronecker delta.

Acoustic scattering by an infinite array of identical generalized cylinders

The theory above has so far been developed for water-wave scattering of a plane wave by an infinite array of identical arbitrary bodies. It can easily be adjusted to the (simpler) two-dimensional problem of acoustic scattering. Namely, we consider the problem that arises when a plane sound wave is incident upon an infinite array of identical generalized cylinders (i.e.~bodies which have arbitrary cross-section in the [math]\displaystyle{ (x,y) }[/math]-plane but the cross-sections at any height are identical) in an acoustic medium.

For this problem, the [math]\displaystyle{ z }[/math]-dependence can be omitted and the above theory applies with the following modifications:

1. The dispersion relation \eqref{eq_k} is replaced by </math>k=\omega /

c[math]\displaystyle{ where }[/math]c[math]\displaystyle{ is the speed of sound in the medium under consideration and the dispersion relation is \eqref{eq_k_m} omitted. #All factors \lt math\gt \cos k_m(z+d) }[/math], [math]\displaystyle{ \cos k_m(c+d) }[/math], [math]\displaystyle{ \cos k_m d }[/math] and [math]\displaystyle{ f_0 }[/math] are replaced by 1.

1. The factor [math]\displaystyle{ N_0 }[/math] in \eqref{green_d} is [math]\displaystyle{ k/\pi }[/math].

Note that point [math]\displaystyle{ (a) }[/math] implies that there are no evanescent modes in this problem, i.e.~the sums over [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] in the eigenfunction expansions \eqref{basisrep_out_d} and \eqref{basisrep_in_d}, respectively, only contain the terms for [math]\displaystyle{ m=0 }[/math] and [math]\displaystyle{ n=0 }[/math]. Moreover, we have </math>k_0 = - \mathrm{i} \, \omega /c[math]\displaystyle{ . For circular cylinders, i.e.~cylinders which have a circular cross-section, this problem has been considered by [[linton93]]. In \S 52 we numerically compare our results for this problem with theirs. \section{Wave forcing of a fixed, rigid and flexible body of shallow draft}(37) The theory which has been developed so far has been for arbitrary bodies. No assumption has been made about the body geometry or its equations of motion. However, we want use this theory to make calculations for the specific case of bodies of shallow draft which may be fixed (which we shall refer to as a dock), rigid, or elastic (modelled as a thin plate). In the formulation, we concentrate on the elastic case of which the other two situations are subcases. This allows us to present a range of results while focusing on the geophysical problem which motivates our work, namely the wave scattering by a field of ice floes. ==Mathematical model for an elastic plate.== We briefly describe the mathematical model of a floating elastic plate. A more detailed account can be found in [[JGR02,JFM04]]. We assume that the elastic plate is sufficiently thin that we may apply the shallow-draft approximation, which essentially applies the boundary conditions underneath the plate at the water surface. Assuming the elastic plate to be in contact with the water surface at all times, its displacement \lt math\gt W }[/math] is that of the water surface and [math]\displaystyle{ W }[/math] is required to satisfy the linear plate equation in the area occupied by the elastic plate [math]\displaystyle{ \Delta }[/math]. In analogy to \eqref{time}, denoting the time-independent surface displacement (with the same radian frequency as the water velocity potential due to linearity) by [math]\displaystyle{ w }[/math] ([math]\displaystyle{ W=\Re\{w \exp(-\mathrm{i}\omega t)\} }[/math]), the plate equation becomes

[math]\displaystyle{ (38) D \, \nabla^4 w - \omega^2 \, \rho_\Delta \, h \, w = \mathrm{i} \, \omega \, \rho \, \phi - \rho \, g \, w, \quad {\mathbf{x}} \in \Delta, }[/math]

with the density of the water [math]\displaystyle{ \rho }[/math], the modulus of rigidity of the elastic plate [math]\displaystyle{ D }[/math], its density [math]\displaystyle{ \rho_\Delta }[/math] and its thickness [math]\displaystyle{ h }[/math]. The right-hand side of \eqref{plate_non} arises from the linearized Bernoulli equation. It needs to be recalled that [math]\displaystyle{ \mathbf{x} }[/math] always denotes a point of the undisturbed water surface. Free-edge boundary conditions apply, namely

[math]\displaystyle{ (39) \left[ \nabla^2 - (1-\nu) \left(\frac{\partial^2}{\partial s^2} + \kappa(s) \frac{\partial}{\partial n} \right) \right] w = 0, }[/math] [math]\displaystyle{ (40) \left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu) \frac{\partial}{\partial s} \left( \frac{\partial}{\partial n} \frac{\partial}{\partial s} -\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, }[/math]

where [math]\displaystyle{ \nu }[/math] is Poisson's ratio and \begin{gather} \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2} + \kappa(s) \frac{\partial}{\partial n}. \end{gather} Here, [math]\displaystyle{ \kappa(s) }[/math] is the curvature of the boundary, [math]\displaystyle{ \partial \Delta }[/math], as a function of arclength [math]\displaystyle{ s }[/math] along [math]\displaystyle{ \partial \Delta }[/math]; [math]\displaystyle{ \partial/\partial s }[/math] and [math]\displaystyle{ \partial/\partial n }[/math] represent derivatives tangential and normal to the boundary [math]\displaystyle{ \partial \Delta }[/math], respectively.

Non-dimensional variables (denoted with an overbar) are introduced,

[math]\displaystyle{ (\bar{x},\bar{y},\bar{z}) = \frac{1}{L} (x,y,z), \quad \bar{w} = \frac{w}{L}, \quad \bar{\alpha} = L\, \alpha, \quad \bar{\omega} = \omega \sqrt{\frac{L}{g}} \quad =and= \quad \bar{\phi} = \frac{\phi}{L \sqrt{L g}}, }[/math]

where [math]\displaystyle{ L }[/math] is a length parameter associated with the plate. In non-dimensional variables, the equation for the elastic plate \eqref{plate_non} reduces to

[math]\displaystyle{ (41) \beta \nabla^4 \bar{w} - \bar{\alpha} \gamma \bar{w} = \i \sqrt{\bar{\alpha}} \bar{\phi} - \bar{w}, \quad \bar{\mathbf{x}} \in \bar{\Delta}, }[/math]

with

[math]\displaystyle{ \beta = \frac{D}{g \rho L^4} \quad =and= \quad \gamma = \frac{\rho_\Delta h}{ \rho L}. }[/math]

The constants [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] represent the stiffness and the mass of the plate, respectively. For convenience, the overbars are dropped and non-dimensional variables are assumed in what follows.