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| | == Introduction == | | == Introduction == |
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| | <center><math>\Psi^\pm(\alpha)=\frac{G^\pm(\alpha)\pm J(\alpha)}{K^\pm(\alpha)}.</math></center> | | <center><math>\Psi^\pm(\alpha)=\frac{G^\pm(\alpha)\pm J(\alpha)}{K^\pm(\alpha)}.</math></center> |
| | <math>J(\alpha)</math> is usually limited by Liouville's theorem to being a polynomial whose coefficients are determined by applying some additional conditions on the <math>\Psi^\pm.</math> | | <math>J(\alpha)</math> is usually limited by Liouville's theorem to being a polynomial whose coefficients are determined by applying some additional conditions on the <math>\Psi^\pm.</math> |
| | + | |
| | + | Note that the main difficulty is to factorise the functions <math>K^-(\alpha)F(\alpha)</math> etc. |
| | | | |
| | == Solution of a semi-infinite integral equation == | | == Solution of a semi-infinite integral equation == |
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| | where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give | | where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give |
| | <center> | | <center> |
| − | <math>\,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F^+(\alpha)+K(\alpha)\Psi^+(\alpha),</math> | + | <math>\,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)+\bar{K}(\alpha)\Psi^+(\alpha),</math> |
| | </center> | | </center> |
| | where | | where |
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| | </math></center> | | </math></center> |
| | <center><math> | | <center><math> |
| − | \Psi^-(\alpha)=\int_{-\infty}^0\left(\int_0^\infty k(x-\xi)\psi(\xi)d\xi\right)e^{i\alpha x}\mathrm{d}x,</math> | + | \Psi^-(\alpha)=\int_{-\infty}^0 \psi(x) e^{i\alpha x}\mathrm{d}x, |
| − | </center> | + | </math></center> |
| | + | <center> |
| | + | <math>F(\alpha)=\int_{\infty}^\infty f(x)e^{i\alpha x}\mathrm{d}x, |
| | + | </math></center> |
| | and | | and |
| − | <center> | + | <center><math> |
| − | <math>F^+(\alpha)=\int_0^\infty f(x)e^{i\alpha x}\mathrm{d}x,\quad | + | \bar{K}(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x, |
| − | K(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,</math>
| + | </math></center> |
| − | </center> | |
| | In the above formulae, <math>\pm</math> superscripts denote functions that are in the upper or lower complex half planes, and the Fourier transform of the integral equation is clearly a Wiener-Hopf equation on the real line, as described in the previous section. | | In the above formulae, <math>\pm</math> superscripts denote functions that are in the upper or lower complex half planes, and the Fourier transform of the integral equation is clearly a Wiener-Hopf equation on the real line, as described in the previous section. |
| − |
| |
| − | == A fuller description of the theory ==
| |
| − |
| |
| − | The Wiener-Hopf technique is an extension of the [http://en.wikipedia.org/wiki/Fourier_transform 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>\alpha </math> is extended into the complex plane. The transform <math>\hat{\phi }\left( \alpha,z\right) </math> may then have singularities on the complex plane depending on the integrability of <math>\phi\left( x,z\right) </math>.
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| − |
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| − | Consider a function <math>\psi\left( x\right) </math> of <math>x\in\mathbb{R}</math> that is bounded except at a finite number of points and has the asymptotic property <math>\left| \psi\left( x\right) \right| \leq A\exp\left( \delta_{-}x\right) </math> as <math>x\rightarrow\infty</math> and <math>\left| \psi\left( x\right) \right| \leq B\exp\left( \delta_{+}x\right) </math> as <math>x\rightarrow-\infty</math>. If <math>\delta _{-}<\delta_{+}</math>, the Fourier transform of <math>\psi\left( x\right) \exp\left( -\delta x\right) </math> for <math>\delta_{-}<\delta<\delta_{+}</math> can be obtained by the following integration,
| |
| − | <center>
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| − | <math> F\left( \varepsilon\right) =\int_{-\infty}^{\infty}\psi\left( x\right) e^{-\delta x} e^{i \varepsilon x}\mathrm{d}x. </math>
| |
| − | </center>
| |
| − | Then, the integral above defines the Fourier transform in the complex plane and the function <math> \hat{\psi}\left( \alpha\right) </math> defined as
| |
| − | <center>
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| − | <math> \hat{\psi}\left( \alpha\right) =\int_{-\infty}^{\infty}\psi\left( x\right)
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| − | e^{i \alpha x}\mathrm{d}x </math>
| |
| − | </center>
| |
| − | is an analytic function of <math>\alpha=\varepsilon+ i\delta </math>, regular in <math> \delta_{-}<\delta<\delta_{+} </math>. Using the usual inverse transform, we have for <math> \alpha=\varepsilon+ i \delta </math> in <math> \delta_{-} <\delta<\delta_{+} </math>
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| − | <center>
| |
| − | <math>
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| − | \frac{1}{2\pi}\int_{-\infty+ i\delta}^{\infty + i\delta}\left\{ \int_{-\infty}^{\infty}\psi\left(
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| − | \xi\right) e^{ i \alpha\xi}\mathrm{d}\xi\right\} e^{- i \alpha x}\mathrm{d}\alpha
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| − | =\frac{1}{2\pi}e^{-\delta x}\int_{-\infty}^{\infty}\left\{ \int_{-\infty
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| − | }^{\infty}\left( \psi\left( \xi\right) e^{\delta\xi}\right)
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| − | e^{ i \varepsilon\xi}\mathrm{d}\xi\right\} e^{- i \varepsilon x}\mathrm{d}\varepsilon
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| − | =e^{-\delta x}\left( \psi\left( x\right) e^{\delta x}\right)
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| − | =\psi\left( x\right) .
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| − | </math>
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| − | </center>
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| − | Note that in the second line we change the variable from <math>\alpha </math> to <math>\varepsilon</math>. Thus the inverse Fourier transform is obtained by
| |
| − | <center>
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| − | <math>
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| − | \psi\left( x\right) =\frac{1}{2\pi}\int_{-\infty+ i\delta
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| − | }^{\infty+ i \delta}\hat{\psi}\left( \alpha\right) e^{- i \alpha x}\mathrm{d}\alpha</math>
| |
| − | </center>
| |
| − | where <math>\delta_{-}<\delta<\delta_{+} </math>. An immediate consequence of this is that if a function <math> \psi\left( x\right) </math> satisfies <math>\left| \psi\left( x\right) \right| \leq A\exp\left( \delta_{-}x\right) </math> as <math>x \rightarrow \infty</math> then the Fourier transform in the half space
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| − | <center>
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| − | <math>
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| − | \hat{\psi}^{+}\left( \alpha\right) =\int_{0}^{\infty}\psi\left( x\right)
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| − | e^{ i \alpha x}\mathrm{d}x
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| − | </math>
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| − | </center>
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| − | is an analytic function of <math>\alpha </math> and regular in <math>\delta_{-}<\delta </math>. Also the function can be recovered by
| |
| − | <center>
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| − | <math>
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| − | \psi\left( x\right) =\frac{1}{2\pi}\int_{-\infty+ i\delta
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| − | }^{\infty+ i\delta}\hat{\psi}^{+}\left( \alpha\right)
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| − | e^{- i \alpha x}\mathrm{d}\alpha
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| − | </math>
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| − | </center>
| |
| − | as <math>x\rightarrow-\infty</math>, where <math>\psi</math> is zero in <math>x<0</math>. The equivalent relation holds for <math>\psi</math> defined in <math>x<0</math> satisfying <math>\left| \psi\left( x\right) \right| \leq B\exp\left( \delta_{+}x\right) </math> as <math>x\rightarrow -\infty</math>, then the Fourier transform
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| − | <math>\hat{\psi}^{-}</math>
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| − | is regular in <math>\delta<\delta_{+}</math>.
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| − |
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| − | Conversely, suppose that <math>\hat{\psi}\left( \alpha\right) </math> is regular in the strip defined by <math>\delta_{-}<\delta<\delta_{+}</math> and tends to zero uniformly as <math>\left| \alpha\right| \rightarrow\infty</math> in the strip. If <math>\hat{\psi}</math> is defined as a solution of the equation
| |
| − | <center>
| |
| − | <math>
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| − | \psi\left( x\right) =\frac{1}{2\pi}\int_{-\infty+ i \delta
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| − | }^{\infty+ i \delta}\hat{\psi}\left( \beta\right)
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| − | e^{- i\beta x}\mathrm{d}\beta</math>
| |
| − | </center>
| |
| − | then for a given <math>\alpha=\varepsilon+ i\delta</math>, <math>\delta
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| − | _{-}<c<\delta<d<\delta_{+}</math>
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| − | <center>
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| − | <math>
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| − | I =\frac{1}{2\pi}\int_{-\infty}^{\infty}\left\{ \int_{-\infty + i\delta}^{\infty+ i\delta}\hat{\psi}\left( \beta\right) e^{- i \beta x}\mathrm{d}\beta\right\}
| |
| − | e^{ i \alpha x}\mathrm{d}x =\frac{1}{2\pi}\int_{-\infty}^{0}\left\{ \int_{-\infty+ i c}^{\infty+ i c}\hat{\psi}\left( \beta\right) e^{- i \beta x}\mathrm{d}\beta\right\} e^{ i \alpha x}\mathrm{d}x+\frac{1}{2\pi}\int_{0}^{\infty}\left\{ \int_{-\infty+ i }^{\infty+ i d}\hat{\psi}\left( \beta\right) e^{- i \beta x}\mathrm{d}\alpha\right\} e^{ i \alpha x}\mathrm{d}x </math>
| |
| − | </center>
| |
| − | since <math>\hat{\psi}</math> is regular in the strip and <math>\mathbf{Im}\left(
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| − | \alpha-\beta\right) <0</math> for <math>\delta<d</math> and <math>\mathbf{Im}\left(
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| − | \alpha-\beta\right) >0</math> for <math>c<\delta</math>. Each split integral
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| − | is convergent in the respective strip of analyticity. Cauchy's integral theorem and <math>\hat{\psi }\rightarrow0 </math> as <math>\left| \alpha\right| \rightarrow\infty</math> in the strip gives
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| − | <center>
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| − | <math>
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| − | I =-\frac{1}{ i 2\pi}\int_{-\infty+ i d}^{\infty+ i d}\frac{\hat{\psi}\left( \beta\right) }
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| − | {\beta-\alpha}\mathrm{d}\beta+\frac{1}{ i 2\pi}\int_{-\infty + i c}^{\infty+ i c}\frac{\hat{\psi}\left( \beta\right) }{\beta-\alpha}\mathrm{d}\beta =\frac{1}{ i 2\pi}\int_{C}\frac{\hat{\psi}\left(
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| − | \beta\right) }{\beta-\alpha}\mathrm{d}\beta=\hat{\psi}\left( \alpha\right)
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| − | </math>
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| − | </center>
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| − | where <math>C</math> is a rectangular contour formed by four points <math>\left( \pm
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| − | \infty+ i c\right) </math> and <math>\left( \pm\infty+ i d\right) </math>. Therefore, <math>\hat{\psi}</math> can be obtained.
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| − |
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| − | Detailed discussion of the analyticity of complex valued functions that are
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| − | defined by integral transforms can be found in sections 1.3 and 1.4 of [[Noble 1958]]
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| − | and chapter 7 of [[Carrier, Krook and Pearson 1966]]).
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| − |
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| − | We apply the Fourier transform to Eqn.~(\ref{4-22}) and Eqn.~(\ref{4-27}) in
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| − | <math>x<0</math> and <math>x>0</math> and obtain algebraic expressions of the Fourier transform of
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| − | <math>\phi\left( x,0\right) </math>. The Fourier transforms of <math>\phi\left( x,0\right)
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| − | </math> in <math>x<0</math> and <math>x>0</math> are defined as
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| − | <center>
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| − | <math>
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| − | \Phi^{-}\left( \alpha,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right)
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| − | e^{ i \alpha x}\mathrm{d}x </math> and <math>\Phi^{+}\left( \alpha,z\right)
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| − | =\int_{0}^{\infty}\phi\left( x,z\right) e^{ i \alpha x}\mathrm{d}x.</math>
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| − | </center>
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| − | Notice that the superscript `<math>+</math>' and `<math>-</math>' correspond to the integral domain.
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| − | The radiation conditions introduced in section 2.3 restrict the amplitude of
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| − | <math>\phi\left( x,z\right) </math> to stay finite as <math>\left| x\right| \rightarrow
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| − | \infty</math> because of the absence of the dissipation. It follows that <math>\Phi
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| − | ^{-}\left( \alpha,z\right) </math> and <math>\Phi^{+}\left( \alpha,z\right) </math> are
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| − | regular in <math>\mathbf{Im}\alpha<0</math> and <math>\mathbf{Im}\alpha>0</math>, respectively.
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| − |
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| − | It is possible to find the inverse transform of the sum of functions
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| − | <math>\Phi=\Phi^{-}+\Phi^{+}</math> using the inverse formula (\ref{4-7}) if the two
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| − | functions share a strip of their analyticity in which a integral path
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| − | <math>-\infty<\varepsilon<\infty</math> can be taken. The Wiener-Hopf technique usually
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| − | involves the splitting of complex valued functions into a product of two
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| − | regular functions in the lower and upper half planes and then the application
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| − | of Liouville's theorem, which states that ''a function that is bounded and
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| − | analytic in the whole plane is constant everywhere''. A corollary of
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| − | Liouville's theorem is that a function which is asymptotically <math>o\left(
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| − | \alpha^{n+1}\right) </math> as <math>\left| \alpha\right| \rightarrow\infty</math> must be a
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| − | polynomial of <math>n</math>'th order.
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| − |
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| − |
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| − | [[Category:Numerical Methods]]
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| − | [[Category:Linear Water-Wave Theory]]
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Introduction
A method for solving equations in which properties of the function in the complex plane are
used to find the solution. It was developed initially for integral equations over
a semi-infinite interval and it is used in water waves for linear problems in which
there is a semi-infinite region. The theory is described in Noble 1958 and
Carrier, Krook and Pearson 1966 and is mentioned
in Linton and McIver 2001.
The Wiener-Hopf method is extremely powerful and it furnishes what is virtually an explicit
solution to very complicated problems which at first sight seem tractable only by numerical
methods. It is far from simply to apply and it relies very much on a simple situation so
that the method often does not generalise or is not applicable to even slightly more complicated
problems.
It has been applied to several problems in linear water wave theory such as scattering by
an infinitely thin semi-infinite breakwater or a semi-infinite
Floating Elastic Plate (Wiener-Hopf Elastic Plate Solution).
Basic problem
The Basic Wiener-Hopf problem is to find a pair of unknown functions
[math]\displaystyle{ \Psi^\pm(\alpha) }[/math] given an equation the form
[math]\displaystyle{ K(\alpha)\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)\,\! }[/math]
which is satisfied on a line in the complex plane (usually the real line, but not necessarily), and where the functions [math]\displaystyle{ K }[/math] and [math]\displaystyle{ F }[/math] are known. The function [math]\displaystyle{ \Psi^+(\alpha) }[/math] is analytic on one side of the line (we shall call this side the "left") and [math]\displaystyle{ \Psi^-(\alpha) }[/math] is analytic on the other side (the "right"), and all other functions with [math]\displaystyle{ \pm }[/math] superscripts will have be analytic in the same regions as the [math]\displaystyle{ \Psi^\pm }[/math] functions (respectively).
The first step in the solution is to write [math]\displaystyle{ K(\alpha)=K^+(\alpha)/K^-(\alpha) }[/math], making the original equation become
[math]\displaystyle{ \,\!K^+(\alpha)\Psi^+(\alpha)+K^-(\alpha)\Psi^-(\alpha)=K^-(\alpha)F(\alpha);
}[/math]
the next step is to write [math]\displaystyle{ K^-(\alpha)F(\alpha)=G^+(\alpha)+G^-(\alpha) }[/math],
which implies
[math]\displaystyle{ \,\!K^+(\alpha)\Psi^+(\alpha)-G^+(\alpha)=G^-(\alpha)-K^-(\alpha)\Psi^-(\alpha).
}[/math]
Now, the left hand side of the above equation will be analytic to the left of the line that it holds on and the right hand side will be analytic to its left. From the Riemann principle they are consequently the analytic continuations of each other across the line and are both equal to a single entire function [math]\displaystyle{ J(\alpha). }[/math]
Hence
[math]\displaystyle{ \Psi^\pm(\alpha)=\frac{G^\pm(\alpha)\pm J(\alpha)}{K^\pm(\alpha)}. }[/math]
[math]\displaystyle{ J(\alpha) }[/math] is usually limited by Liouville's theorem to being a polynomial whose coefficients are determined by applying some additional conditions on the [math]\displaystyle{ \Psi^\pm. }[/math]
Note that the main difficulty is to factorise the functions [math]\displaystyle{ K^-(\alpha)F(\alpha) }[/math] etc.
Solution of a semi-infinite integral equation
As mentioned above, the Wiener-Hopf method was originally devised to solve semi-infinite integral equations of the form
[math]\displaystyle{ \psi(x)=f(x)+\int_0^\infty k(x-\xi)\psi(\xi)\mathrm{d}\xi, }[/math]
where we shall assume that [math]\displaystyle{ \psi }[/math] is integrable. To demonstrate the method we take the Fourier transform of the above equation to give
[math]\displaystyle{ \,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)+\bar{K}(\alpha)\Psi^+(\alpha), }[/math]
where
[math]\displaystyle{ \Psi^+(\alpha)=\int_0^\infty\psi(x)e^{i\alpha x}\mathrm{d}x,
}[/math]
[math]\displaystyle{
\Psi^-(\alpha)=\int_{-\infty}^0 \psi(x) e^{i\alpha x}\mathrm{d}x,
}[/math]
[math]\displaystyle{ F(\alpha)=\int_{\infty}^\infty f(x)e^{i\alpha x}\mathrm{d}x,
}[/math]
and
[math]\displaystyle{
\bar{K}(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,
}[/math]
In the above formulae, [math]\displaystyle{ \pm }[/math] superscripts denote functions that are in the upper or lower complex half planes, and the Fourier transform of the integral equation is clearly a Wiener-Hopf equation on the real line, as described in the previous section.
