Wiener-Hopf Elastic Plate Solution

Introduction

We present here the Wiener-Hopf solution to the problem of a two semi-infinite Two-Dimensional Floating Elastic Plates. The solution method is based on the one presented by Chung and Fox 2002. This problem has been well studied and the first solution was by Evans and Davies 1968 but they did not actually develop the method sufficiently to be able to calculate the solution. A solution was also developed by Balmforth and Craster 1999 and by Tkacheva 2004.

The theory is described in Wiener-Hopf.

Elastic Plate

We imagine two semi-infinite Floating Elastic Plates of (possibly) different properties. The equations are the following

[math]\displaystyle{ \left( D_{j}\left( \frac{\partial^{2}}{\partial x^{2}}-k^{2}\right) ^{2}+\rho g-m_{j}\omega^{2}\right) \phi_{z}-\rho\omega^{2}\phi =0,\;j=1,2,\;z=0 }[/math] [math]\displaystyle{ \left( \frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial z^{2}}-k^{2}\right) \phi =0,\;-H\lt z\lt 0, }[/math] [math]\displaystyle{ \phi_{z} =0,\;\;z=-H. }[/math]

where [math]\displaystyle{ j=1 }[/math] is to the left and [math]\displaystyle{ j=2 }[/math] is to the right of [math]\displaystyle{ x=0. }[/math] We apply the Fourier transform to these equations in [math]\displaystyle{ x\lt 0 }[/math] and [math]\displaystyle{ x\gt 0 }[/math] and obtain algebraic expressions of the Fourier transform of [math]\displaystyle{ \phi\left( x,0\right) }[/math]. The Fourier transforms of [math]\displaystyle{ \phi\left( x,0\right) }[/math] in [math]\displaystyle{ x\lt 0 }[/math] and [math]\displaystyle{ x\gt 0 }[/math] are defined as

[math]\displaystyle{ \Phi^{-}\left( \alpha,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right) e^{\mathrm{i}\alpha x}dx }[/math]

and

[math]\displaystyle{ \Phi^{+}\left( \alpha,z\right) =\int_{0}^{\infty}\phi\left( x,z\right) e^{\mathrm{i}\alpha x}dx. }[/math]

Notice that the superscript `[math]\displaystyle{ + }[/math]' and `[math]\displaystyle{ - }[/math]' correspond to the integral domain. The Sommerfeld Radiation Conditions introduced in section 2.3 restrict the amplitude of [math]\displaystyle{ \phi\left( x,z\right) }[/math] to stay finite as [math]\displaystyle{ \left| x\right| \rightarrow \infty }[/math] because of the absence of dissipation. It follows that [math]\displaystyle{ \Phi ^{-}\left( \alpha,z\right) }[/math] and [math]\displaystyle{ \Phi^{+}\left( \alpha,z\right) }[/math] are regular in [math]\displaystyle{ \operatorname{Im}\alpha\lt 0 }[/math] and [math]\displaystyle{ \operatorname{Im}\alpha\gt 0 }[/math], respectively.

It is possible to find the inverse transform of the sum of functions [math]\displaystyle{ \Phi=\Phi^{-}+\Phi^{+} }[/math] using the inverse formula if the two functions share a strip of their analyticity in which a integral path [math]\displaystyle{ -\infty\lt \varepsilon\lt \infty }[/math] can be taken. The Wiener-Hopf technique usually involves the spliting of complex valued functions into a product of two regular functions in the lower and upper half planes and then the application of Liouville's theorem, which states that a function that is bounded and analytic in the whole plane is constant everywhere. A corollary of Liouville's theorem is that a function which is asymptotically [math]\displaystyle{ o\left( \alpha^{n+1}\right) }[/math] as [math]\displaystyle{ \left| \alpha\right| \rightarrow\infty }[/math] must be a polynomial of [math]\displaystyle{ n }[/math]'th order.

We will show two ways of solving the given boundary value problem. First we figure out the domains of regularity of the functions of complex variable defined by integrals, thus we are able to calculate the inverse that has the appropriate asymptotic behaviour. Secondly we find the asymptotic behaviour of the solution from the physical conditions, thus we already know the domains in which the Fourier transforms are regular and are able to calculate the inverse transform.

Weierstrass's factor theorem

As mentioned above, we will require splitting a ratio of two functions of a complex variable in [math]\displaystyle{ \alpha }[/math]-plane. We here remind ourselves of Weierstrass's factor theorem (Carrier, Krook and Pearson 1966 section 2.9) which can be proved using the Mittag-Leffler theorem.

Let [math]\displaystyle{ H\left( \alpha\right) }[/math] denote a function that is analytic in the whole [math]\displaystyle{ \alpha }[/math]-plane (except possibly at infinity) and has zeros of first order at [math]\displaystyle{ a_{0} }[/math], [math]\displaystyle{ a_{1} }[/math], [math]\displaystyle{ a_{2} }[/math], ..., and no zero is located at the origin. Consider the Mittag-Leffler expansion of the logarithmic derivative of [math]\displaystyle{ H\left( \alpha\right) }[/math], i.e.,

[math]\displaystyle{ \frac{d\log H\left( \alpha\right) }{d\alpha} =\frac{1}{H\left( \alpha\right) }\frac{dH\left( \alpha\right) }{d\alpha} =\frac{d\log H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[ \frac{1}{\alpha-a_{n}}+\frac{1}{a_{n}}\right] . }[/math]

Integrating both sides in [math]\displaystyle{ \left[ 0,\alpha\right] }[/math] we have

[math]\displaystyle{ \log H\left( \alpha\right) =\log H\left( 0\right) +\alpha\frac{d\log H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[ \log\left( 1-\frac{\alpha}{a_{n}}\right) +\frac{\alpha}{a_{n}}\right] . }[/math]

Therefore, the expression for [math]\displaystyle{ H\left( \alpha\right) }[/math] is

[math]\displaystyle{ H\left( \alpha\right) =H\left( 0\right) \exp\left[ \alpha\frac{d\log H\left( 0\right) }{d\alpha}\right] \prod_{n=0}^{\infty}\left( 1-\frac{\alpha}{a_{n}}\right) e^{\alpha/a_{n}}. }[/math]

If [math]\displaystyle{ H\left( \alpha\right) }[/math] is even, then [math]\displaystyle{ dH\left( 0\right) /d\alpha=0 }[/math] and [math]\displaystyle{ -a_{n} }[/math] is a zero if [math]\displaystyle{ a_{n} }[/math] is a zero. Then we have the simpler expression

[math]\displaystyle{ H\left( \alpha\right) =H\left( 0\right) \prod_{n=0}^{\infty}\left( 1-\frac{\alpha^{2}}{a_{n}^{2}}\right) . }[/math]

Derivation of the Wiener-Hopf equation

We derive algebraic expressions for [math]\displaystyle{ \Phi^{\pm}\left( \alpha,z\right) }[/math] using integral transforms of the equations which gives

[math]\displaystyle{ \left\{ \frac{\partial^{2}}{\partial z^{2}}-\left( \alpha^{2}+k^{2}\right) \right\} \Phi^{\pm}\left( \alpha,z\right) =\pm\left\{ \mathrm{i} \alpha\phi\left( 0,z\right) -\phi_{x}\left( 0,z\right) \right\} . }[/math]

Hence, the solutions of the above ordinary differential equations with the Fourier transform of condition ((4-45)),

[math]\displaystyle{ \Phi_{z}^{\pm}\left( \alpha,-H\right) =0, }[/math]

can be written as

[math]\displaystyle{ \Phi^{\pm}\left( \alpha,z\right) =\Phi^{\pm}\left( \alpha,0\right) \frac{\cosh\gamma\left( z+H\right) }{\cosh\gamma H}\pm g\left( \alpha,z\right) }[/math]

where [math]\displaystyle{ \gamma=\sqrt{\alpha^{2}+k^{2}} }[/math] and [math]\displaystyle{ g\left( \alpha,z\right) }[/math] is a function determined by [math]\displaystyle{ \left\{ \mathrm{i}\alpha\phi\left( 0,z\right) -\phi_{x}\left( 0,z\right) \right\} }[/math],

[math]\displaystyle{ g\left( \alpha,z\right) =\frac{h_{z}\left( \alpha,-H\right) }{\gamma }\left( \tanh\gamma H\cosh\gamma\left( z+H\right) -\sinh\gamma\left( z+H\right) \right) }[/math] [math]\displaystyle{ +h\left( \alpha,z\right) \left( 1-\frac{\cosh\gamma\left( z+H\right) }{\cosh\gamma H}\right) , }[/math] [math]\displaystyle{ h\left( \alpha,z\right) =\int^{z}\frac{\sinh\gamma\left( z-t\right) }{\gamma}\left\{ \phi_{x}\left( 0,t\right) -\mathrm{i}\alpha \phi\left( 0,t\right) \right\} dt. }[/math]

Note that [math]\displaystyle{ \operatorname{Re}\gamma\gt 0 }[/math] when [math]\displaystyle{ \operatorname{Re}\alpha\gt 0 }[/math] and [math]\displaystyle{ \operatorname{Re}\gamma\lt 0 }[/math] when [math]\displaystyle{ \operatorname{Re}\alpha\lt 0 }[/math]. We have, by differentiating both sides with respect to [math]\displaystyle{ z }[/math] at [math]\displaystyle{ z=0 }[/math]

[math]\displaystyle{ \Phi_{z}^{\pm}\left( \alpha,0\right) =\Phi^{\pm}\left( \alpha,0\right) \gamma\tanh\gamma H\pm g_{z}\left( \alpha,0\right) }[/math]

where [math]\displaystyle{ \Phi_{z}^{\pm}\left( \alpha,0\right) }[/math] denotes the [math]\displaystyle{ z }[/math]-derivative. We apply the integral transform to the free-surface conditions in [math]\displaystyle{ x\lt 0 }[/math] and [math]\displaystyle{ x\gt 0 }[/math],

[math]\displaystyle{ \left\{ D_{1}\gamma^{4}-m_{1}\omega^{2}+\rho g\right\} \Phi_{z}^{-}\left( \alpha,0\right) -\rho\omega^{2}\Phi^{-}\left( \alpha,0\right) +P_{1}\left( \alpha\right) =0, }[/math] [math]\displaystyle{ \left\{ D_{2}\gamma^{4}-m_{2}\omega^{2}+\rho g\right\} \Phi_{z}^{+}\left( \alpha,0\right) -\rho\omega^{2}\Phi^{+}\left( \alpha,0\right) -P_{2}\left( \alpha\right) =0, }[/math]

where

[math]\displaystyle{ P_{j}\left( \alpha\right) =D_{j}\left[ c_{3}^{j}-\mathrm{i}c_{2} ^{j}\alpha-\left( \alpha+2k^{2}\right) \left( c_{1}^{j}-\mathrm{i} c_{0}^{j}\alpha\right) \right] ,\;j=1,2, }[/math] [math]\displaystyle{ c_{\mathrm{i}}^{1}=\left. \left( \frac{\partial}{\partial x}\right) ^{i}\phi_{z}\right| _{x=0-,z=0},\;c_{\mathrm{i}} ^{2}=\left. \left( \frac{\partial}{\partial x}\right) ^{i }\phi_{z}\right| _{x=0+,z=0},\;i=0,1,2,3. }[/math]

We therefore have

[math]\displaystyle{ \begin{matrix} f_{1}\left( \gamma\right) \Phi_{z}^{-}\left( \alpha,0\right) +C_{1}\left( \alpha\right) & =0 \\ f_{2}\left( \gamma\right) \Phi_{z}^{+}\left( \alpha,0\right) +C_{2}\left( \alpha\right) & =0 \end{matrix} }[/math]

where

[math]\displaystyle{ f_{j}\left( \gamma\right) =D_{j}\gamma^{4}-m_{j}\omega^{2}+\rho g-\frac{\rho\omega^{2}}{\gamma\tanh\gamma H},\;j=1,2, }[/math] [math]\displaystyle{ C_{1}\left( \alpha\right) =-\frac{\rho\omega^{2}g_{z}\left( \alpha,0\right) }{\gamma\tanh\gamma H}+P_{1}\left( \alpha\right) ,\;C_{2}\left( \alpha\right) =\frac{\rho\omega^{2}g_{z}\left( \alpha,0\right) }{\gamma\tanh\gamma H}-P_{2}\left( \alpha\right) . }[/math]

Dispersion Relation for a Floating Elastic Plate

Functions [math]\displaystyle{ f_{1} }[/math] and [math]\displaystyle{ f_{2} }[/math] are the Dispersion Relation for a Floating Elastic Plate and the zeros of these functions are the primary tools in our method of deriving the solutions. Functions [math]\displaystyle{ \Phi_{z}^{-}\left( \alpha,0\right) }[/math], and [math]\displaystyle{ \Phi_{z}^{+}\left( \alpha,0\right) }[/math] are defined in [math]\displaystyle{ \operatorname{Im}\alpha\lt 0 }[/math] and [math]\displaystyle{ \operatorname{Im}\alpha\gt 0 }[/math], respectively. However they can be extended in the whole plane defined via analytic continuation. This show that the singularities of [math]\displaystyle{ \Phi_{z}^{-} }[/math] and [math]\displaystyle{ \Phi_{z}^{+} }[/math] are determined by the positions of the zeros of [math]\displaystyle{ f_{1} }[/math]\ and [math]\displaystyle{ f_{2} }[/math], since [math]\displaystyle{ g_{z}\left( \alpha,0\right) }[/math] is bounded and zeros of [math]\displaystyle{ \gamma\tanh\gamma H }[/math] are not the singularities of [math]\displaystyle{ \Phi_{z}^{\pm} }[/math]. We denote sets of singularities corresponding to zeros of [math]\displaystyle{ f_{1} }[/math] and [math]\displaystyle{ f_{2} }[/math] by [math]\displaystyle{ \mathcal{K}_{1} }[/math] and [math]\displaystyle{ \mathcal{K}_{2} }[/math] respectively

[math]\displaystyle{ \mathcal{K}_{j}=\left\{ \alpha\in\mathbb{C}\mid f_{j}\left( \gamma\right) =0,\;\alpha=\sqrt{\gamma^{2}-k^{2}},\, \operatorname{Im}(\alpha)\gt 0\,\,\,\mathrm{or}\,\,\, \alpha\gt 0\,\,\,\mathrm{for}\, \alpha\in\mathbb{R}\right\} . }[/math]

We avoid numbering the roots with this notation, but for numerical purposes this is important and we order them with increasing size.

Solution of the W-H Equation

Using the Mittag-Leffler theorem (Carrier, Krook and Pearson 1966 section 2.9), functions [math]\displaystyle{ \Phi_{z}^{\pm} }[/math] can be expressed by a series of fractional functions that contribute to the solutions. Thus, we have series expansions of [math]\displaystyle{ \Phi_{z}^{-} }[/math] and [math]\displaystyle{ \Phi_{z}^{+} }[/math]

[math]\displaystyle{ \Phi_{z}^{-}\left( \alpha,0\right) =\frac{Q_{1}\left( -\lambda\right) }{\alpha+\lambda}+\sum_{q\in\mathcal{K}_{1}}\frac{Q_{1}\left( q\right) }{\alpha-q},\;\Phi_{z}^{+}\left( \alpha,0\right) =\sum_{q\in\mathcal{K}_{2} }\frac{Q_{2}\left( q\right) }{\alpha+q}, }[/math]

where [math]\displaystyle{ \lambda\ }[/math]is a positive real singularity of [math]\displaystyle{ \Phi_{z}^{-} }[/math] and [math]\displaystyle{ Q_{1} }[/math], [math]\displaystyle{ Q_{2} }[/math] are coefficient functions yet to be determined. Note that [math]\displaystyle{ \Phi _{z}^{-}\left( \alpha,0\right) }[/math]\ has an additional term corresponding to [math]\displaystyle{ -\lambda }[/math]\ because of the incident wave. The solution [math]\displaystyle{ \phi\left( x,0\right) }[/math], [math]\displaystyle{ x\lt 0 }[/math] is then obtained using the inverse Fourier transform taken over the line shown in Fig.~((roots5)a)

[math]\displaystyle{ \phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty-\mathrm{i} \tau}^{\infty-\mathrm{i}\tau}\Phi_{z}^{-}e^{-\mathrm{i}\alpha x}d\alpha=\mathrm{i}Q_{1}\left( -\lambda\right) e^{\mathrm{i} \lambda x}+\sum\limits_{q\in\mathcal{K}_{1}}\mathrm{i}Q_{1}\left( q\right) e^{-\mathrm{i}qx} (4-51) }[/math]

where [math]\displaystyle{ \tau }[/math]\ is an infinitesimally small positive real number. Note that [math]\displaystyle{ k=\lambda\sin\theta }[/math]. Similarly, we obtain [math]\displaystyle{ \phi\left( x,0\right) }[/math] for [math]\displaystyle{ x\gt 0 }[/math] by taking the integration path shown in Fig.~((roots5)b), then we have

[math]\displaystyle{ \phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty+\mathrm{i} \tau}^{\infty+\mathrm{i}\tau}\Phi_{z}^{+}e^{-\mathrm{i}\alpha x}d\alpha=-\sum\limits_{q\in\mathcal{K}_{2}}\mathrm{i}Q_{2}\left( q\right) e^{\mathrm{i}qx}. }[/math]

The Wiener-Hopf technique enables us to calculate coefficients [math]\displaystyle{ Q_{1} }[/math] and [math]\displaystyle{ Q_{2} }[/math] without knowing functions [math]\displaystyle{ C_{1} }[/math], [math]\displaystyle{ C_{2} }[/math], or [math]\displaystyle{ \left\{ \phi _{x}\left( 0,z\right) -\mathrm{i}\alpha\phi\left( 0,z\right) \right\} }[/math]. It requires the domains of analyticity of Eqn.~((4-46)) and Eqn.~((4-47)) to have a common strip of analyticity which they do not have right now. We create such a strip by shifting a singularity of [math]\displaystyle{ \Phi_{z}^{-} }[/math] in Eqn.~((4-46)) to [math]\displaystyle{ \Phi_{z}^{+} }[/math] in Eqn.~((4-47)) (we can also create a strip by moving a singularity of [math]\displaystyle{ \Phi_{z}^{+} }[/math], and more than one of the singularities can be moved). Here, we shift [math]\displaystyle{ -\lambda }[/math] as shown in Fig.~((roots5)a), so that the common strip of analyticity denoted by [math]\displaystyle{ \mathcal{D} }[/math] is created on the real axis, which passes above the two negative real singularities and below the two positive real singularities. We denote the domain above and including [math]\displaystyle{ \mathcal{D} }[/math] by [math]\displaystyle{ \mathcal{D}_{+} }[/math]\ and below and including [math]\displaystyle{ \mathcal{D} }[/math] by [math]\displaystyle{ \mathcal{D}_{-} }[/math]. Hence, the zeros of [math]\displaystyle{ f_{1} }[/math] and [math]\displaystyle{ f_{2} }[/math] belong to either [math]\displaystyle{ \mathcal{D}_{+} }[/math] or [math]\displaystyle{ \mathcal{D}_{-} }[/math].

Let [math]\displaystyle{ \Psi_{z}^{-} }[/math] be a function created by subtracting a singularity from function [math]\displaystyle{ \Phi_{z}^{-} }[/math]. Then\ [math]\displaystyle{ \Psi_{z}^{-}\left( \alpha,0\right) }[/math] is regular in [math]\displaystyle{ \mathcal{D}_{-} }[/math].\ Since the removed singularity term makes no contribution to the solution,\ from Eqn.~((4-46)), [math]\displaystyle{ \Psi_{z}^{-} }[/math] satisfies

[math]\displaystyle{ f_{1}\left( \gamma\right) \Psi_{z}^{-}\left( \alpha,0\right) +C_{1}\left( \alpha\right) =0. (100) }[/math]

Eqn.~((4-47)) becomes, as a result of modifying function [math]\displaystyle{ \Phi_{z}^{+} }[/math] to a function denoted by [math]\displaystyle{ \Psi_{z}^{+} }[/math] with an additional singularity term,

[math]\displaystyle{ f_{2}\left( \gamma\right) \Psi_{z}^{+}\left( \alpha,0\right) -\frac {f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right) } {\alpha+\lambda}+C_{2}\left( \alpha\right) =0. (4-48) }[/math]

Our aim now is to find a formula for

[math]\displaystyle{ \Psi_{z}\left( \alpha,0\right) =\Psi_{z}^{-}\left( \alpha,0\right) +\Psi_{z}^{+}\left( \alpha,0\right) }[/math]

in [math]\displaystyle{ \alpha\in\mathcal{D} }[/math] so that its inverse Fourier transform can be calculated.

Adding both sides of Eqn.~((100)) and Eqn.~((4-48)) gives the Wiener-Hopf equation

[math]\displaystyle{ f_{1}\left( \gamma\right) \Psi_{z}^{-}\left( \alpha,0\right) +f_{2}\left( \gamma\right) \Psi_{z}^{+}\left( \alpha,0\right) -\frac{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda }+C\left( \alpha\right) =0 (4-41) }[/math]

where [math]\displaystyle{ C\left( \alpha\right) =C_{1}\left( \alpha\right) -C_{2}\left( \alpha\right) }[/math]. This equation can alternatively be written as

[math]\displaystyle{ \begin{matrix} [c]{c} f_{2}\left( \gamma\right) \left[ f\left( \gamma\right) \Psi_{z} ^{+}\left( \alpha,0\right) -\frac{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda}+C\left( \alpha\right) \right] \\ =-f_{1}\left( \gamma\right) \left[ f\left( \gamma\right) \Psi_{z} ^{-}\left( \alpha,0\right) +\frac{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda}-C\left( \alpha\right) \right] \end{matrix} (eq:WH2) }[/math]

where [math]\displaystyle{ f\left( \gamma\right) =f_{2}\left( \gamma\right) -f_{1}\left( \gamma\right) . }[/math]

We now modify Eqn.~((eq:WH2)) so that the right and left hand sides of the equation become regular in [math]\displaystyle{ \mathcal{D}_{-} }[/math] and [math]\displaystyle{ \mathcal{D}_{+} }[/math] respectively. Using Weierstrass's factor theorem given in the previous subsection, the ratio [math]\displaystyle{ f_{2}/f_{1} }[/math] can be factorized into infinite products of polynomials [math]\displaystyle{ \left( 1-\alpha/q\right) }[/math], [math]\displaystyle{ q\in\mathcal{K}_{1} }[/math] and [math]\displaystyle{ \mathcal{K}_{2} }[/math]. Hence, using a regular non-zero function [math]\displaystyle{ K\left( \alpha\right) }[/math] in [math]\displaystyle{ \mathcal{D}_{+} }[/math],

[math]\displaystyle{ K\left( \alpha\right) =\left( \prod\limits_{q\in\mathcal{K}_{1}} \frac{q^{\prime}}{q+\alpha}\right) \left( \prod\limits_{q\in\mathcal{K}_{2} }\frac{q+\alpha}{q^{\prime}}\right) (eq:K) }[/math]

where [math]\displaystyle{ q^{\prime}=\sqrt{q^{2}+k^{2}} }[/math], then we have

[math]\displaystyle{ \frac{f_{2}}{f_{1}}=K\left( \alpha\right) K\left( -\alpha\right) . }[/math]

Note that the factorization is done in the [math]\displaystyle{ \alpha }[/math]-plane, hence functions [math]\displaystyle{ f_{1} }[/math] and [math]\displaystyle{ f_{2} }[/math] are here seen as functions of [math]\displaystyle{ \alpha }[/math] and we are actually factorizing

[math]\displaystyle{ \frac{f_{2}\left( \gamma\right) \gamma\sinh\gamma H}{f_{1}\left( \gamma\right) \gamma\sinh\gamma H} }[/math]

in order to satisfy the conditions given in the previous subsection. Then Eqn.~((eq:WH2)) can be rewritten as

[math]\displaystyle{ \begin{matrix} [c]{c} K\left( \alpha\right) \left[ f\left( \gamma\right) \Psi_{z}^{+}+C\right] -\left( K\left( \alpha\right) -\frac{1}{K\left( \lambda\right) }\right) \frac{f_{2}\left( \lambda^{\prime}\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda}\\ =-\frac{1}{K\left( -\alpha\right) }\left[ f\left( \gamma\right) \Psi _{z}^{-}-C\right] -\left( \frac{1}{K\left( -\alpha\right) }-\frac {1}{K\left( \lambda\right) }\right) \frac{f_{2}\left( \lambda^{\prime }\right) Q_{1}\left( -\lambda\right) }{\alpha+\lambda}. \end{matrix} (4-26) }[/math]

Note that the infinite products in Eqn.~((eq:K)) converge in the order of [math]\displaystyle{ q^{-5} }[/math] as [math]\displaystyle{ \left| q\right| }[/math] becomes large, thus numerical computation of [math]\displaystyle{ K\left( \alpha\right) }[/math] does not pose any difficulties.

The left hand side of Eqn.~((4-26)) is regular in [math]\displaystyle{ \mathcal{D}_{+} }[/math] and the right hand side is regular in [math]\displaystyle{ \mathcal{D}_{-} }[/math]. Notice that a function is added to both sides of the equation to make the right hand side of the equation regular in [math]\displaystyle{ \mathcal{D}_{-} }[/math]. The left hand side of Eqn.~((4-26)) is [math]\displaystyle{ o\left( \alpha^{4}\right) }[/math] as [math]\displaystyle{ \left| \alpha\right| \rightarrow \infty }[/math] in [math]\displaystyle{ \mathcal{D}_{+} }[/math], since [math]\displaystyle{ \Psi_{z}^{+}\rightarrow0 }[/math] and [math]\displaystyle{ K\left( \alpha\right) =O\left( 1\right) }[/math]\ as [math]\displaystyle{ \left| \alpha\right| \rightarrow\infty }[/math] in [math]\displaystyle{ \mathcal{D}_{+} }[/math]. The right hand side of Eqn.~((4-26)) has the equivalent analytic properties in [math]\displaystyle{ \mathcal{D}_{-} }[/math]. Liouville's theorem (Carrier, Krook and Pearson carrier section 2.4) tells us that there exists a function, which we denote [math]\displaystyle{ J\left( \alpha\right) }[/math], uniquely defined by Eqn.~((4-26)), and function [math]\displaystyle{ J\left( \alpha\right) }[/math] is a polynomial of degree three in the whole plane. Hence

[math]\displaystyle{ J\left( \alpha\right) =d_{0}+d_{1}\alpha+d_{2}\alpha^{2}+d_{3}\alpha^{3}. }[/math]

Equating Eqn.~((4-26)) for [math]\displaystyle{ \Psi_{z} }[/math] gives

[math]\displaystyle{ \Psi_{z}\left( \alpha,0\right) =\frac{-F\left( \alpha\right) }{K\left( \alpha\right) f_{1}\left( \gamma\right) }\;=or= \;-\frac{K\left( -\alpha\right) F\left( \alpha\right) }{f_{2}\left( \gamma\right) } (4-50) }[/math]

where

[math]\displaystyle{ F\left( \alpha\right) =J\left( \alpha\right) -\frac{Q_{1}\left( -\lambda\right) f_{2}\left( \lambda^{\prime}\right) }{\left( \alpha+\lambda\right) K\left( \lambda\right) }. }[/math]

Notice that procedure from Eqn.~((eq:WH2)) to Eqn.~((4-26)) eliminates the need for calculating constant [math]\displaystyle{ C }[/math] in Eqn.~((4-26)).

For [math]\displaystyle{ x\lt 0 }[/math] we close the integral contour in [math]\displaystyle{ \mathcal{D}_{+} }[/math], and put the incident wave back, then we have

[math]\displaystyle{ \phi_{z}\left( x,0\right) =\mathrm{i}Q_{1}\left( -\lambda\right) e^{\mathrm{i}\lambda x}-\sum\limits_{q\in\mathcal{K}_{1}} \frac{\mathrm{i}F\left( q\right) q^{\prime}R_{1}\left( q^{\prime }\right) }{qK\left( q\right) }e^{-\mathrm{i}qx}, (eq:solution1) }[/math]

where [math]\displaystyle{ R_{1}\left( q^{\prime}\right) }[/math] is a residue of [math]\displaystyle{ \left[ f_{1}\left( \gamma\right) \right] ^{-1}\lt math\gt at }[/math]\gamma=q^{\prime}</math>

[math]\displaystyle{ \begin{matrix} R_{1}\left( q^{\prime}\right) & =\left( \left. \frac{df_{1}\left( \gamma\right) }{d\gamma}\right| _{\gamma=q^{\prime}}\right) ^{-1} \\ & =\left\{ 5D_{1}q^{\prime3}+\frac{b_{1}}{q^{\prime}}+\frac{H}{q^{\prime} }\left( \frac{\left( D_{1}q^{\prime5}+b_{1}q^{\prime}\right) ^{2}-\left( \rho\omega^{2}\right) ^{2}}{\rho\omega^{2}}\right) \right\} ^{-1}. (R) \end{matrix} }[/math]

We used [math]\displaystyle{ b_{1}=-m_{1}\omega^{2}+\rho g }[/math] and [math]\displaystyle{ f_{1}\left( q^{\prime}\right) =0 }[/math] to simplify the formula. Displacement [math]\displaystyle{ w\left( x\right) }[/math] can be obtained by multiplying Eqn.~((eq:solution1)) by [math]\displaystyle{ -\mathrm{i} /\omega }[/math]. Notice that the formula for the residue is again expressed by a polynomial using the dispersion equation as shown in section (sec:3), which gives us a stable numerical computation of the solutions.

The velocity potential [math]\displaystyle{ \phi\left( x,z\right) }[/math] can be obtained using Eqn.~((4-44)) and Eqn.~((eq:4)),

[math]\displaystyle{ \phi\left( x,z\right) =\frac{\mathrm{i}Q_{1}\left( -\lambda\right) \cosh\lambda^{\prime}\left( z+H\right) }{\lambda^{\prime}\sinh \lambda^{\prime}H}e^{\mathrm{i}\lambda x}-\sum\limits_{q\in \mathcal{K}_{1}}\frac{\mathrm{i}F\left( q\right) R_{1}\left( q^{\prime}\right) \cosh q^{\prime}\left( z+H\right) }{qK\left( q\right) \sinh q^{\prime}H}e^{-\mathrm{i}qx} }[/math]

where [math]\displaystyle{ \lambda^{\prime}=\sqrt{\lambda^{2}+k^{2}} }[/math].

For [math]\displaystyle{ x\gt 0 }[/math], the functions [math]\displaystyle{ \phi_{z}\left( x,0\right) }[/math] and [math]\displaystyle{ \phi\left( x,z\right) }[/math]\ are obtained by closing the integral contour in [math]\displaystyle{ \mathcal{D} _{-} }[/math],

[math]\displaystyle{ \begin{matrix} \phi_{z}\left( x,0\right) & =-\sum\limits_{q\in\mathcal{K}_{2}} \frac{\mathrm{i}K\left( q\right) F\left( -q\right) q^{\prime} R_{2}\left( q^{\prime}\right) }{q}e^{\mathrm{i}qx}, (4-28)\\ \phi\left( x,z\right) & =-\sum\limits_{q\in\mathcal{K}_{2}}\frac {\mathrm{i}K\left( q\right) F\left( -q\right) R_{2}\left( q^{\prime}\right) \cosh q^{\prime}\left( z+H\right) }{q\sinh q^{\prime} H}e^{\mathrm{i}qx}, \end{matrix} }[/math]

where [math]\displaystyle{ R_{2} }[/math] is a residue of [math]\displaystyle{ \left[ f_{2}\left( \gamma\right) \right] ^{-1} }[/math]\ and its formula can be obtained by replacing the subscript [math]\displaystyle{ 1 }[/math] with [math]\displaystyle{ 2 }[/math] in Eqn.~((R)). Notice that since [math]\displaystyle{ R_{j}\sim O\left( q^{-9}\right) }[/math], [math]\displaystyle{ j=1,2 }[/math], the coefficients of [math]\displaystyle{ \phi_{z} }[/math] of Eqn.~((4-28)) decay as [math]\displaystyle{ O\left( q^{-6}\right) }[/math] as [math]\displaystyle{ \left| q\right| }[/math] becomes large, so the displacement is bounded up to the fourth [math]\displaystyle{ x }[/math]-derivatives. In a physical sense, the biharmonic term of the plate equation for the vertical displacement is associated with the strain energy due to bending of the plate as explained in chapter 2. Hence, up to fourth derivative of the displacement function should be bounded, as has been confirmed. The coefficients of [math]\displaystyle{ \phi }[/math], have an extra [math]\displaystyle{ 1/q^{\prime}\tanh q^{\prime}H }[/math] term which is [math]\displaystyle{ O\left( q^{4}\right) }[/math], hence the coefficients decay as [math]\displaystyle{ O\left( q^{-2}\right) }[/math] as [math]\displaystyle{ \left| q\right| }[/math] becomes large. Therefore, [math]\displaystyle{ \phi }[/math] is bounded everywhere including at [math]\displaystyle{ x=0 }[/math].

Shifting a singularity of one function to the other is equivalent to subtracting an incident wave from both functions then solving the boundary value problem for the scattered field as in Balmforth and Craster 1999. As mentioned, any one of the singularities can be shifted as long as it creates a common strip of analyticity for the newly created functions. We chose [math]\displaystyle{ -\lambda }[/math] because of the convenience of the symmetry in locations of the singularities. The method of subtracting either incoming or transmitting wave requires the Fourier transform be performed twice, first to express the solution with a series expansion, and second to solve the system of equations for the newly created functions. Thus, we find the method of shifting a singularity shown here is advantageous to other methods since it needs the Fourier transform only once to obtain the Wiener-Hopf equation.

The polynomial [math]\displaystyle{ J\left( \alpha\right) }[/math] is yet to be determined. In the following section the coefficients of [math]\displaystyle{ J\left( \alpha\right) }[/math] will be determined from conditions at [math]\displaystyle{ x=0\pm }[/math], [math]\displaystyle{ -\infty\lt y\lt \infty }[/math], [math]\displaystyle{ z=0 }[/math].