Infinite Array Green Function

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We present here the solution to the Infinite Array based on an infinite image system of Free-Surface Green Functions

Problem Formulation

We begin by formulating the problem. Cartesian coordinates [math]\displaystyle{ (x,y,z) }[/math] are chosen with [math]\displaystyle{ z }[/math] vertically upwards such that [math]\displaystyle{ z=0 }[/math] coincides with the mean free surface of the water. An infinite array of identical bodies are periodically spaced along the [math]\displaystyle{ y }[/math]-axis with uniform separation [math]\displaystyle{ l }[/math]. The problem is to determine the motion of the water and the bodies when plane waves are obliquely-incident from [math]\displaystyle{ x=-\infty }[/math] upon the periodic array of bodies.

The bodies occupy [math]\displaystyle{ \Delta_m }[/math], [math]\displaystyle{ -\infty \lt m \lt \infty }[/math]. Periodicity implies that if [math]\displaystyle{ (x,y) \in \Delta_0 }[/math], then [math]\displaystyle{ (x,y+ml) \in \Delta_m }[/math], [math]\displaystyle{ -\infty \lt m \lt \infty }[/math].

We assume that we have the Standard Linear Wave Scattering Problem. The incident wave potential given by

[math]\displaystyle{ \phi^{{\rm in}} = \frac{A}{k} e^{ik (x\cos\theta+y\sin\theta)}\,e^{kz}, }[/math]

where [math]\displaystyle{ A }[/math] is the dimensionless amplitude and [math]\displaystyle{ \theta }[/math] is the direction of propagation of the wave (with [math]\displaystyle{ \theta = 0 }[/math] corresponding to normal incidence.

Transformation to an Integral Equation

We now Floquet's theorem (Scott 1998) (also called the assumption of periodicity in the water wave context) which states the displacement from adjacent plates differ only by a phase factor. If the potential under the central plate [math]\displaystyle{ \Delta_{0} }[/math] is given by [math]\displaystyle{ \phi( \mathbf{x}_{0},0) }[/math], [math]\displaystyle{ \mathbf{x}_{0}\in\Delta_{0} }[/math], then by Floquet's theorem the potential satisfies

[math]\displaystyle{ \phi(\mathbf{x}_{m},0) = \phi(\mathbf{x}_{0},0) e^{im\sigma l}, }[/math]

and the displacement of the plate [math]\displaystyle{ \Delta_{m} }[/math] satisfies

[math]\displaystyle{ w(\mathbf{x}_{m}) = w(\mathbf{x}_{0}) e^{im\sigma l}, }[/math]

where [math]\displaystyle{ \mathbf{x}_{m} \in \Delta_{m} }[/math], [math]\displaystyle{ -\infty \lt m \lt \infty }[/math] and the phase difference is [math]\displaystyle{ \sigma = k\sin\theta }[/math] (see, for example, Linton 1998).

A standard approach to the solution of the equations of motion for the water is the Green Function Solution Method in which we transform the equations into a boundary integral equation using the Free-Surface Green Function. In doing so we obtain

[math]\displaystyle{ \phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0) +\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}} \left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi }[/math]

[math]\displaystyle{ G(\mathbf{x},\xi) }[/math] is the Free-Surface Green Function This can be written alternatively as

[math]\displaystyle{ \phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x}) +\int_{\Delta_{0}} \sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l} \phi(\xi) - G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi }[/math]

where the kernel [math]\displaystyle{ G_{\mathbf{P}} }[/math] (referred to as the periodic Green function) is given by

[math]\displaystyle{ G^{\mathbf{P}} (\mathbf{x};\xi) = \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l}. }[/math]

Accelerating the Convergence of the Periodic Green Function

The spatial representation of the periodic Green function given by equation is slowly convergent and in the far field the terms decay in magnitude like [math]\displaystyle{ O(n^{-1/2}) }[/math]. In this section we show how to accelerate the convergence. We begin with the asymptotic approximation of the Three-dimensional Free-Surface Green Function far from the source point,

[math]\displaystyle{ G(\mathbf{x},\xi) \sim -\frac{ik}{2} \,H_{0}( k |\mathbf{x}-\xi|), |\mathbf{x}-\xi| \to \infty }[/math]

Wehausen and Laitone 1960 where [math]\displaystyle{ H_0 \equiv H_{0}^{(1)} }[/math] is the Hankel function of the first kind of order zero Abramowitz and Stegun 1964. In Linton Linton 1998 various methods were described in which the convergence of the periodic Green functions was improved. One such method, which suits the particular problem being considered here, involves writing the periodic Green function as

[math]\displaystyle{ G_{\mathbf{P}} (\mathbf{x};\xi) = \sum_{m=-\infty}^{\infty} \left[ G\left(\mathbf{x};\xi)+(0,ml)\right) + \frac{ik}{2} H_{0} \Big(k\sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) e^{im\sigma l} \right] -\sum_{m=-\infty}^{\infty} \frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big) e^{im\sigma l} }[/math]

where [math]\displaystyle{ c }[/math] is a numerical smoothing parameter, introduced to avoid the singularity at [math]\displaystyle{ \mathbf{x} = \xi }[/math] in the Hankel function and

[math]\displaystyle{ X = x-\xi,\quad \mathrm{and} \quad Y_{m} = (y-\eta)-ml. }[/math]

Furthermore we use the fact that second slowly convergent sum can be transformed to

[math]\displaystyle{ -\sum_{m=-\infty}^{\infty} \frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big) e^{im\sigma l} -\frac{i}{l} \sum_{m=-\infty}^{\infty} \frac{e^{ik \mu_{m} |X+c| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}} }[/math]

Linton 1998 where [math]\displaystyle{ \sigma_{m} = \sigma + 2 m \pi/l }[/math] and

[math]\displaystyle{ \mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k} \right)^{2} \right]^{\frac{1}{2}}, }[/math]

where the positive real or positive imaginary part of the square root is taken. Combining these equations we obtain the accelerated version of the periodic Green function

[math]\displaystyle{ G_{\mathbf{P}} (\mathbf{x};\xi) = \sum_{m=-\infty}^{\infty} \left[ G\left(\mathbf{x};\xi+(0,ml)\right) + \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) \right] e^{im\sigma l} -\frac{i}{l}\sum_{m=-\infty}^{\infty} \frac{e^{ik \mu_{m}|X+cl| }e^{i\sigma_{m}Y_0}}{\mu_{m}}. }[/math]

The convergence of the two sums depends on the value of [math]\displaystyle{ c }[/math]. For small [math]\displaystyle{ c }[/math] the first sum converges rapidly while the second converges slowly. For large [math]\displaystyle{ c }[/math] the second sum converges rapidly while the first converges slowly. The smoothing parameter [math]\displaystyle{ c }[/math] must be carefully chosen to balance these two effects. Of course, the convergence also depends strongly on how close together the points [math]\displaystyle{ \mathbf{x} }[/math] and [math]\displaystyle{ \xi }[/math] are.

Note that some special combinations of wavelength [math]\displaystyle{ \lambda }[/math] and angle of incidence [math]\displaystyle{ \theta }[/math] cause the periodic Green function to diverge ( Scott 1998). This singularity is closely related to the diffracted waves and will be explained shortly.

The scattered waves (modes)

We begin with the accelerated periodic Green function, equation setting [math]\displaystyle{ c=0 }[/math] and considering the case when [math]\displaystyle{ X }[/math] is large (positive or negative). We also note that for [math]\displaystyle{ m }[/math] sufficiently small or large [math]\displaystyle{ i\mu_m }[/math] will be negative and the corresponding terms will decay. Therefore

[math]\displaystyle{ G_{\mathbf{P}} (\mathbf{x};\xi) \sim - \frac{i}{l} \sum_{m=-M}^{N} \frac{e^{ik\mu_{m}|X|}\, e^{i\sigma_{m}Y_0}} {\mu_{m}}, X \to \pm \infty }[/math]

where the integers [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] satisfy the following inequalities

[math]\displaystyle{ (M_N) \left. \begin{matrix} \sigma_{-M-1}\lt -k\lt \sigma_{-M},\\ \sigma_{N}\lt k\lt \sigma_{N+1}. \end{matrix} \right\} }[/math]

These equations can be written as

[math]\displaystyle{ \frac{l}{2\pi}\left(\sigma+k-2\pi \right) \lt M \lt \frac{l}{2\pi}\left( \sigma+k \right), }[/math]


[math]\displaystyle{ \frac{l}{2\pi}\left( k - \sigma \right) \gt N \gt \frac{l}{2\pi} \left( k-\sigma - 2\pi \right) }[/math]

Linton 1998. It is obvious that [math]\displaystyle{ G_{\mathbf{P}} }[/math] will diverge if [math]\displaystyle{ \sigma_m = \pm k }[/math]; these values correspond to cut-off frequencies which are an expected feature of periodic structures.

The diffracted waves

The diffracted waves are the plane waves which are observed as [math]\displaystyle{ x \to \pm \infty }[/math]. Their amplitude and form are obtained by substituting the limit of the periodic Green function as [math]\displaystyle{ x\to\pm\infty }[/math] into the boundary integral equation for the potential. This gives us

[math]\displaystyle{ \lim_{x\to\pm\infty} \phi^{s} ( \mathbf{x},0 ) = - \frac{i}{l} \sum_{m=-M}^{N} \int_{\Delta_0} \frac{e^{ik\mu_{m} |X| } e^{i\sigma_{m}Y_0}}{\mu_{m}} \left[ k\phi(\xi,0) - w(\xi) \right] d\xi, }[/math]

where [math]\displaystyle{ \phi^{s} = \phi-\phi^{\rm in} }[/math] is the scattered wave which is composed of a finite number of plane waves. For [math]\displaystyle{ x \to -\infty }[/math] the scattered wave is given by

[math]\displaystyle{ \lim_{x\to-\infty}\phi^{s} (\mathbf{x},0) = A_{m}^{-}\,e^{ik\mu_{m}x}e^{i\sigma_{m}y}, }[/math]

where the amplitudes [math]\displaystyle{ A_{m}^{-} }[/math] are

[math]\displaystyle{ A_{m}^{-} = -\frac{i}{\mu_{m}l} \int_{\Delta_0} e^{ik\mu_{m}\xi } e^{-i\sigma_{m}\eta} \left[ k\phi\left( \xi\right) - w (\xi) \right] d\xi. }[/math]

Likewise as [math]\displaystyle{ x \to \infty }[/math] the scattered wave is given by

[math]\displaystyle{ \lim_{x\to\infty}\phi^{s} (\mathbf{x},0) = A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y}, }[/math]

where [math]\displaystyle{ A_{m}^{+} }[/math] are

[math]\displaystyle{ A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0} e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta} \left[ k\phi (\xi,0) - w(\xi) \right] d\xi. }[/math]

The diffracted waves propagate at various angles with respect to the normal direction of the array. The angles of diffraction, [math]\displaystyle{ \psi_{m}^{\pm} }[/math], are given by

[math]\displaystyle{ \psi_{m}^{\pm} = \tan^{-1}\left( \frac{\sigma_{m}}{\pm k\mu_{m}}\right). (psi_m) }[/math]

Notice that for [math]\displaystyle{ m=0 }[/math] we have

[math]\displaystyle{ \psi_{0}^{\pm}=\pm\theta, (psi_0) }[/math]

where [math]\displaystyle{ \theta }[/math] is the incident angle. This is exactly as expected since we should always have a transmitted wave which travels in the same direction as the incident wave and a reflected wave which travels in the negative incident angle direction.

The fundamental reflected and transmitted waves

We need to be precise when we determine the wave of order zero at [math]\displaystyle{ x\to\infty }[/math] because we have to include the incident wave. There is always at least one set of propagating waves corresponding to [math]\displaystyle{ m=0 }[/math] which correspond to simple reflection and transmission. The coefficient, [math]\displaystyle{ R }[/math], for the fundamental reflected wave for the [math]\displaystyle{ m=0 }[/math] mode is given by

[math]\displaystyle{ R = A_{0}^{-} = -\frac{i}{\mu_{0}l}\int_{\Delta_0}e^{ik (\xi\cos\theta -\eta\sin\theta)}\left[ k\phi(\xi,0) - w(\xi)\right] d\xi. }[/math]

The coefficient, [math]\displaystyle{ T }[/math], for the fundamental transmitted wave for the [math]\displaystyle{ m=0 }[/math] mode is given by

[math]\displaystyle{ T = 1 + A_{0}^{+} = 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta +\eta\sin\theta)}\left[ k\phi(\xi,0) - w(\xi) \right] d\xi. }[/math]

Conservation of energy

The diffracted wave, taking into account the correction for [math]\displaystyle{ T }[/math], must satisfy the energy flux equation. This simply says that the energy of the incoming wave must be equal to the energy of the outgoing waves. This gives us

[math]\displaystyle{ \cos\theta = \left( |R|^2+|T|^2 \right) \cos\theta + \sum_{m=-M,\,m \neq 0}^{N} \left( |A_{m}^{-}|^2 \cos\psi_{m}^{-} +|A_{m}^{+}|^2 \cos\psi_{m}^{+} \right). }[/math]

The energy balance equation can be used as an accuracy check on the numerical results.