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 and by Tcakava

\author{Hyuck Chung \thanks{Acoustics Research Centre, School of Architecture, The University of Auckland, PB 92019 Auckland, New Zealand, hyuck@math.auckland.ac.nz}}

Wave-ice interaction using the Wiener-Hopf technique

The Wiener-Hopf technique

Theoretical background

The Wiener-Hopf technique is an extension of the Fourier transform method to semi-infinite domains of simple geometry, such as those with a straight or circular boundary. In the Wiener-Hopf technique the Fourier transform variable [math]\displaystyle{ \alpha }[/math] is extended into the complex plane. The transform [math]\displaystyle{ \hat{\phi }\left( \alpha,z\right) }[/math] may then have singularities on the complex plane depending on the integrability of [math]\displaystyle{ \phi\left( x,z\right) }[/math]. This page describes the Wiener-Hopf technique in the context of the flexural wave propagation in a floating-plate.

Consider a function [math]\displaystyle{ \psi\left( x\right) }[/math] of [math]\displaystyle{ x\in\mathbb{R} }[/math] that is bounded except at a finite number of points and has the asymptotic property [math]\displaystyle{ \left| \psi\left( x\right) \right| \leq A\exp\left( \delta_{-}x\right) }[/math] as [math]\displaystyle{ x\rightarrow\infty }[/math] and [math]\displaystyle{ \left| \psi\left( x\right) \right| \leq B\exp\left( \delta_{+}x\right) }[/math] as [math]\displaystyle{ x\rightarrow-\infty }[/math]. If [math]\displaystyle{ \delta _{-}\lt \delta_{+} }[/math], the Fourier transform of [math]\displaystyle{ \psi\left( x\right) \exp\left( -\delta x\right) }[/math] for [math]\displaystyle{ \delta_{-}\lt \delta\lt \delta_{+} }[/math] can be obtained by the following integration (for introduction on the Fourier transform, go to ),

[math]\displaystyle{ F\left( \varepsilon\right) =\int_{-\infty}^{\infty}\psi\left( x\right) e^{-\delta x}e^{\operatorname*{i}\varepsilon x}dx. }[/math]

Then, the integral above defines the Fourier transform in the complex plane and the function [math]\displaystyle{ \hat{\psi}\left( \alpha\right) }[/math] defined as

[math]\displaystyle{ \hat{\psi}\left( \alpha\right) =\int_{-\infty}^{\infty}\psi\left( x\right) e^{\operatorname*{i}\alpha x}dx (4-11) }[/math]

is an analytic function of [math]\displaystyle{ \alpha=\varepsilon+\operatorname*{i}\delta }[/math], regular in [math]\displaystyle{ \delta_{-}\lt \delta\lt \delta_{+} }[/math]. Using the usual inverse transform, we have for [math]\displaystyle{ \alpha=\varepsilon+\operatorname*{i}\delta }[/math] in [math]\displaystyle{ \delta_{-} \lt \delta\lt \delta_{+} }[/math]

[math]\displaystyle{ \begin{matrix} & \frac{1}{2\pi}\int_{-\infty+\operatorname*{i}\delta}^{\infty +\operatorname*{i}\delta}\left\{ \int_{-\infty}^{\infty}\psi\left( \xi\right) e^{\operatorname*{i}\alpha\xi}d\xi\right\} e^{-\operatorname*{i} \alpha x}d\alpha\\ & =\frac{1}{2\pi}e^{-\delta x}\int_{-\infty}^{\infty}\left\{ \int_{-\infty }^{\infty}\left( \psi\left( \xi\right) e^{\delta\xi}\right) e^{\operatorname*{i}\varepsilon\xi}d\xi\right\} e^{-\operatorname*{i} \varepsilon x}d\varepsilon\\ & =e^{-\delta x}\left( \psi\left( x\right) e^{\delta x}\right) =\psi\left( x\right) . \end{matrix} }[/math]

Note that in the second line we change the variable from [math]\displaystyle{ \alpha }[/math] to [math]\displaystyle{ \varepsilon }[/math]. Thus the inverse Fourier transform is obtained by

[math]\displaystyle{ \psi\left( x\right) =\frac{1}{2\pi}\int_{-\infty+\operatorname*{i}\delta }^{\infty+\operatorname*{i}\delta}\hat{\psi}\left( \alpha\right) e^{-\operatorname*{i}\alpha x}dx (4-7) }[/math]

where [math]\displaystyle{ \delta_{-}\lt \delta\lt \delta_{+} }[/math]. An immediate consequence of this is that if a function [math]\displaystyle{ \psi\left( x\right) }[/math] satisfies [math]\displaystyle{ \left| \psi\left( x\right) \right| \leq A\exp\left( \delta_{-}x\right) }[/math] as [math]\displaystyle{ x \rightarrow \infty }[/math] then the Fourier transform in the half space

[math]\displaystyle{ \hat{\psi}^{+}\left( \alpha\right) =\int_{0}^{\infty}\psi\left( x\right) e^{\operatorname*{i}\alpha x}dx }[/math]

is an analytic function of [math]\displaystyle{ \alpha }[/math] and regular in [math]\displaystyle{ \delta_{-}\lt \delta }[/math]. Also the function can be recovered by

[math]\displaystyle{ \psi\left( x\right) =\frac{1}{2\pi}\int_{-\infty+\operatorname*{i}\delta }^{\infty+\operatorname*{i}\delta}\hat{\psi}^{+}\left( \alpha\right) e^{-\operatorname*{i}\alpha x}d\alpha }[/math]

as [math]\displaystyle{ x\rightarrow-\infty }[/math], where [math]\displaystyle{ \psi }[/math] is zero in [math]\displaystyle{ x\lt 0 }[/math]. The equivalent relation holds for [math]\displaystyle{ \psi }[/math] defined in [math]\displaystyle{ x\lt 0 }[/math] satisfying [math]\displaystyle{ \left| \psi\left( x\right) \right| \leq B\exp\left( \delta_{+}x\right) }[/math] as [math]\displaystyle{ x\rightarrow -\infty }[/math], then the Fourier transform [math]\displaystyle{ \hat{\psi}^{-} }[/math] is regular in [math]\displaystyle{ \delta\lt \delta_{+} }[/math].

Conversely, suppose that [math]\displaystyle{ \hat{\psi}\left( \alpha\right) }[/math] is regular in the strip defined by [math]\displaystyle{ \delta_{-}\lt \delta\lt \delta_{+} }[/math] and tends to zero uniformly as [math]\displaystyle{ \left| \alpha\right| \rightarrow\infty }[/math] in the strip. If [math]\displaystyle{ \hat{\psi} }[/math] is defined as a solution of the equation

[math]\displaystyle{ \psi\left( x\right) =\frac{1}{2\pi}\int_{-\infty+\operatorname*{i}\delta }^{\infty+\operatorname*{i}\delta}\hat{\psi}\left( \beta\right) e^{-\operatorname*{i}\beta x}d\beta (4-10) }[/math]

then for a given [math]\displaystyle{ \alpha=\varepsilon+\operatorname*{i}\delta }[/math], [math]\displaystyle{ \delta _{-}\lt c\lt \delta\lt d\lt \delta_{+} }[/math]

[math]\displaystyle{ \begin{matrix} I & =\frac{1}{2\pi}\int_{-\infty}^{\infty}\left\{ \int_{-\infty +\operatorname*{i}\delta}^{\infty+\operatorname*{i}\delta}\hat{\psi}\left( \beta\right) e^{-\operatorname*{i}\beta x}d\beta\right\} e^{\operatorname*{i}\alpha x}dx\\ & =\frac{1}{2\pi}\int_{-\infty}^{0}\left\{ \int_{-\infty+\operatorname*{i} c}^{\infty+\operatorname*{i}c}\hat{\psi}\left( \beta\right) e^{-\operatorname*{i}\beta x}d\beta\right\} e^{\operatorname*{i}\alpha x}dx+\frac{1}{2\pi}\int_{0}^{\infty}\left\{ \int_{-\infty+\operatorname*{i} d}^{\infty+\operatorname*{i}d}\hat{\psi}\left( \beta\right) e^{-\operatorname*{i}\beta x}d\alpha\right\} e^{\operatorname*{i}\alpha x}dx \end{matrix} }[/math]

since [math]\displaystyle{ \hat{\psi} }[/math] is regular in the strip and [math]\displaystyle{ \operatorname{Im}\left( \alpha-\beta\right) \lt 0\lt math\gt for }[/math]\delta<d[math]\displaystyle{ and }[/math]\operatorname{Im}\left( \alpha-\beta\right) >0[math]\displaystyle{ for }[/math]c<\delta</math>. Each split integral is convergent in the respective strip of analyticity. Cauchy's integral theorem and [math]\displaystyle{ \hat{\psi }\rightarrow0 }[/math] as [math]\displaystyle{ \left| \alpha\right| \rightarrow\infty }[/math] in the strip gives

[math]\displaystyle{ \begin{matrix} I & =-\frac{1}{\operatorname*{i}2\pi}\int_{-\infty+\operatorname*{i} d}^{\infty+\operatorname*{i}d}\frac{\hat{\psi}\left( \beta\right) } {\beta-\alpha}d\beta+\frac{1}{\operatorname*{i}2\pi}\int_{-\infty +\operatorname*{i}c}^{\infty+\operatorname*{i}c}\frac{\hat{\psi}\left( \beta\right) }{\beta-\alpha}d\beta\\ & =\frac{1}{\operatorname*{i}2\pi}\int_{C}\frac{\hat{\psi}\left( \beta\right) }{\beta-\alpha}d\beta=\hat{\psi}\left( \alpha\right) \end{matrix} }[/math]

where [math]\displaystyle{ C }[/math] is a rectangular contour formed by four points [math]\displaystyle{ \left( \pm \infty+\operatorname*{i}c\right) \lt math\gt and }[/math]\left( \pm\infty+\operatorname*{i} d\right) [math]\displaystyle{ . Therefore, }[/math]\hat{\psi}</math> can be obtained using Eqn.~((4-11)).

Detailed discussion of the analyticity of complex valued functions that are defined by integral transforms can be found in sections 1.3 and 1.4 of (Noble noble) and chapter 7 of (Carrier, Krook and Pearson carrier).

We apply the Fourier transform to Eqn.~((4-22)) and Eqn.~((4-27)) 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) \lt math\gt in }[/math]x<0[math]\displaystyle{ and }[/math]x>0</math> are defined as

[math]\displaystyle{ \Phi^{-}\left( \alpha,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right) e^{\operatorname*{i}\alpha x}dx\;=and= \;\Phi^{+}\left( \alpha,z\right) =\int_{0}^{\infty}\phi\left( x,z\right) e^{\operatorname*{i}\alpha x}dx. (4-9) }[/math]

Notice that the superscript `[math]\displaystyle{ + }[/math]' and `[math]\displaystyle{ - }[/math]' correspond to the integral domain. The 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\lt math\gt because of the absence of the dissipation. It follows that }[/math]\Phi ^{-}\left( \alpha,z\right) [math]\displaystyle{ \ and }[/math]\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 ((4-7)) 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 \emph{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) \lt math\gt as }[/math]\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 problems in this chapter. First in this section, we figure out the domains of regularity of the functions of complex variable defined by integrals ((4-9)), thus we are able to calculate the inverse that has the appropriate asymptotic behaviour. Secondly in section 4.7, 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 (sec:4-1)

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 section 2.9) which can be proved using the Mittag-Leffler theorem described in section 3.2.

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 (Eqn.~((4-9))) on Eqn.~((4-22)) and Eqn.~((4-27)). The Fourier transforms of Eqn.~((4-27)) according to the definition given by Eqn.~((4-9)) 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\{ \operatorname*{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) (4-44) }[/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\{ \operatorname*{i}\alpha\phi\left( 0,z\right) -\phi_{x}\left( 0,z\right) \right\} }[/math],

[math]\displaystyle{ \begin{matrix} 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) \\ & +h\left( \alpha,z\right) \left( 1-\frac{\cosh\gamma\left( z+H\right) }{\cosh\gamma H}\right) ,\\ h\left( \alpha,z\right) & =\int^{z}\frac{\sinh\gamma\left( z-t\right) }{\gamma}\left\{ \phi_{x}\left( 0,t\right) -\operatorname*{i}\alpha \phi\left( 0,t\right) \right\} dt. \end{matrix} }[/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 of Eqn.~((4-44)) 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) (eq:4) }[/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 Eqn.~((4-22)) in [math]\displaystyle{ x\lt 0 }[/math] and [math]\displaystyle{ x\gt 0 }[/math],

[math]\displaystyle{ \begin{matrix} \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, (4-23)\\ \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, (4-24) \end{matrix} }[/math]

where

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

From Eqn.~((eq:4)), Eqn.~((4-23)) and Eqn.~((4-24)) we have

[math]\displaystyle{ \begin{matrix} f_{1}\left( \gamma\right) \Phi_{z}^{-}\left( \alpha,0\right) +C_{1}\left( \alpha\right) & =0 (4-46)\\ f_{2}\left( \gamma\right) \Phi_{z}^{+}\left( \alpha,0\right) +C_{2}\left( \alpha\right) & =0 (4-47) \end{matrix} }[/math]

where

[math]\displaystyle{ \begin{matrix} 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,\\ 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) . \end{matrix} }[/math]

As we have seen in chapter 3, functions [math]\displaystyle{ f_{1} }[/math] and [math]\displaystyle{ f_{2} }[/math] are called dispersion functions and the zeros of these functions are the primary tools in our method of deriving the solutions. Notice that the dispersion functions have the same form as the one given in chapter 3 and the reason for this is given in section 3.5.2 with the general scaling consideration.

Functions [math]\displaystyle{ \Phi_{z}^{-}\left( \alpha,0\right) }[/math], and [math]\displaystyle{ \Phi_{z}^{+}\left( \alpha,0\right) \lt math\gt are defined in }[/math]\operatorname{Im}\alpha<0</math> and [math]\displaystyle{ \operatorname{Im}\alpha\gt 0 }[/math], respectively. However they can be extended in the whole plane defined by Eqn.~((4-46)) and Eqn.~((4-47)) via analytic continuation.\ Eqn.~((4-46)) and Eqn.~((4-47)) 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) \lt math\gt is bounded and zeros of }[/math]\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}}= either = \operatorname{Im} \alpha\gt 0= or = \alpha\gt 0= for = \alpha\in\mathbb{R}\right\} . }[/math]

Fig.~((roots5)a, b) show the relative positions of the singularities.

\begin{figure}[tbh]\begin{center} \includegraphics[height=4.547cm,width=12.6987cm]{roots5.eps} \caption{Locations (not to scale) of the singularities which determine [math]\displaystyle{ \Phi_{z}^{-} }[/math] (figure (a)) and [math]\displaystyle{ \Phi_{z}^{+} }[/math] (figure(b)). Thick arrow at [math]\displaystyle{ -i\tau }[/math] in (a) and at [math]\displaystyle{ i\tau }[/math] in (b) shows the integral path for the inverse Fourier transform. Figures (a) and (b) illustrate how the negative real singularity [math]\displaystyle{ -\lambda }[/math] of [math]\displaystyle{ \Phi_{z}^{-} }[/math]\ is moved to become a singularity of [math]\displaystyle{ \Phi_{z}^{+} }[/math].} (roots5) \end{center} \end{figure}

From Eqn.~((4-46)) and Eqn.~((4-47)) and using the Mittag-Leffler theorem ( carrier 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-\operatorname*{i} \tau}^{\infty-\operatorname*{i}\tau}\Phi_{z}^{-}e^{-\operatorname*{i}\alpha x}d\alpha=\operatorname*{i}Q_{1}\left( -\lambda\right) e^{\operatorname*{i} \lambda x}+\sum\limits_{q\in\mathcal{K}_{1}}\operatorname*{i}Q_{1}\left( q\right) e^{-\operatorname*{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+\operatorname*{i} \tau}^{\infty+\operatorname*{i}\tau}\Phi_{z}^{+}e^{-\operatorname*{i}\alpha x}d\alpha=-\sum\limits_{q\in\mathcal{K}_{2}}\operatorname*{i}Q_{2}\left( q\right) e^{\operatorname*{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) -\operatorname*{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) =\operatorname*{i}Q_{1}\left( -\lambda\right) e^{\operatorname*{i}\lambda x}-\sum\limits_{q\in\mathcal{K}_{1}} \frac{\operatorname*{i}F\left( q\right) q^{\prime}R_{1}\left( q^{\prime }\right) }{qK\left( q\right) }e^{-\operatorname*{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} \nonumber\\ & =\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\lt math\gt to simplify the formula. Displacement }[/math]w\left( x\right) </math> can be obtained by multiplying Eqn.~((eq:solution1)) by [math]\displaystyle{ -\operatorname*{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{\operatorname*{i}Q_{1}\left( -\lambda\right) \cosh\lambda^{\prime}\left( z+H\right) }{\lambda^{\prime}\sinh \lambda^{\prime}H}e^{\operatorname*{i}\lambda x}-\sum\limits_{q\in \mathcal{K}_{1}}\frac{\operatorname*{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^{-\operatorname*{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) \lt math\gt \ are obtained by closing the integral contour in }[/math]\mathcal{D} _{-}</math>,

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

where [math]\displaystyle{ R_{2} }[/math] is a residue of [math]\displaystyle{ \left[ f_{2}\left( \gamma\right) \right] ^{-1}\lt math\gt \ and its formula can be obtained by replacing the subscript }[/math]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 Balmforth99. 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].