├── INFO.tex ├── LICENSE ├── README.md ├── examples ├── BigCylinderDiffraction.nb ├── ImpulseWave.nb ├── RandomScatterers.nb ├── ScatteringCoefficients.nb ├── Source.nb └── TwoBodyScattering.nb ├── media ├── 45-Wave_10-Scatterers.GIF ├── 45-Wave_12-Scatterers-Neuman.GIF ├── BigCylinderDiffraction.jpg ├── BugCylinderDiffraction.pdf ├── TwoBodyFrequency.jpg ├── TwoBodyFrequency.pdf ├── TwoBodyScattering.gif └── TwoWallsBodyScattering.gif ├── src └── MultipleScattering2D.wl └── theory ├── DiscreteFourier.aux ├── DiscreteFourier.fdb_latexmk ├── DiscreteFourier.fls ├── DiscreteFourier.log ├── DiscreteFourier.nb ├── DiscreteFourier.out ├── DiscreteFourier.pdf ├── DiscreteFourier.synctex.gz ├── DiscreteFourier.tex ├── DiscreteFourierOffset.nb ├── ScatteringOperator.aux ├── ScatteringOperator.fdb_latexmk ├── ScatteringOperator.fls ├── ScatteringOperator.log ├── ScatteringOperator.out ├── ScatteringOperator.pdf ├── ScatteringOperator.synctex.gz └── ScatteringOperator.tex /INFO.tex: -------------------------------------------------------------------------------- 1 | A package to calculate multiple scattering according to the 2D wave equation. All scatterers at present need to be cylinders or the same size, but can be placed anywhere. The incident wave can be anything, but the default for the package is the 2D green's function. There are functions to calculate the total wave in frequency and time, together with examples on how to plot them. 2 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | MIT License 2 | 3 | Copyright (c) 2017 Art Gower 4 | 5 | Permission is hereby granted, free of charge, to any person obtaining a copy 6 | of this software and associated documentation files (the "Software"), to deal 7 | in the Software without restriction, including without limitation the rights 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 | copies of the Software, and to permit persons to whom the Software is 10 | furnished to do so, subject to the following conditions: 11 | 12 | The above copyright notice and this permission notice shall be included in all 13 | copies or substantial portions of the Software. 14 | 15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 | SOFTWARE. 22 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | ![compatibility](https://img.shields.io/badge/Mathematica-8.x_9.x_10.x_11.x-brightgreen.svg) 2 | 3 | ## MultipleScattering2D.wl - exact multiple scattering in 2D for Mathematica 4 | 5 | A package to calculate multiple scattering according to the 2D wave equation. The best place to get started is the notebook [TwoBodyScattering.nb](examples/TwoBodyScattering.nb). The example [ImpulseWave.nb](examples/ImpulseWave.nb) uses randomly place scatterers, similar to the gifs below, whereas [Source.nb](examples/Source.nb) shows how to change source function. 6 | 7 | At present all scatterers need to be cylinders of the same size, but can be placed anywhere. The incident wave can be anything, but the default for the package is the 2D green's function. There are functions to calculate the total wave in frequency and time, together with examples on how to plot them. For details on the maths see [Martin (1995)](https://pdfs.semanticscholar.org/8bd3/38ec62affc5c89592a9d6d13f1ee6a7d7e53.pdf). 8 | 9 | ## Examples 10 | 11 | Scattering of a plane wave from two cylinders for one frequency generated by [TwoBodyScattering.nb](examples/TwoBodyScattering.nb) 12 | 13 | ``` 14 | Get[NotebookDirectory[] <> "/src/MultipleScattering2D.wl"] 15 | 16 | (*radius of the scatterers*) 17 | radius = 0.1; 18 | (*max number of hankel functions per scatterer*) 19 | N0 = 2; 20 | (*angular frequency*) 21 | ws = {6.}; 22 | 23 | options = { 24 | "SourceWave" -> (Exp[I #2 #1[[1]]] &),(*chose a plane wave*) 25 | "BoundaryCondition" -> "Dirchlett"(*"BoundaryCondition"->"Neumann"*) 26 | }; 27 | 28 | (*Position of the scatterers*) 29 | Xs = {{-.3, .4}, {0.5, .4}}; 30 | 31 | (*reciever mesh*) 32 | rngX = Range[-8 radius, 8 radius, radius/4]; 33 | rngY = Range[0., 8 radius, radius/4]; 34 | listeners = ListenersOutsideScatterers[radius, Xs, rngX, rngY]; 35 | 36 | (*Calculate response at every mesh point *) 37 | responses = FrequencyFromScatterers[listeners, Xs, radius, N0, ws]; 38 | 39 | (*plot the absolute value of the result*) 40 | data = Flatten@{listeners[[#]], Abs@responses[[#, 1, 2]]} & /@ 41 | Range[Length@responses]; 42 | p1 = ListDensityPlot[data, InterpolationOrder -> 1, PlotLegends -> Automatic ]; 43 | p2 = DrawScatterers[Xs, radius]; 44 | Show[p1, p2, AspectRatio -> Automatic] 45 | 46 | ``` 47 | 48 | ![TwoBodyScattering](media/TwoBodyFrequency.jpg) 49 | 50 | Diffraction of a point source (2D Green's function), from one big cylinder for one frequency generated by [BigCylinderDiffraction.nb](examples/BigCylinderDiffraction.nb) 51 | 52 | ![OneBigCylinder](media/BigCylinderDiffraction.jpg) 53 | 54 | 55 | By lining up cylinders above and below the source, we effectively create two walls: 56 | 57 | ![TwoWalls](media/TwoWallsBodyScattering.gif) 58 | 59 | 60 | ## Randomly placed cyclinders 61 | The package's focus is on scattering from random media, for example see the notebook [ImpulseWave.nb](examples/ImpulseWave.nb) 62 | #### Dirichlet boundary conditions 63 | 64 | ![dirichlet](media/45-Wave_10-Scatterers.GIF) 65 | 66 | #### Neumann boundary conditions 67 | 68 | ![neumann](media/45-Wave_12-Scatterers-Neuman.GIF) 69 | -------------------------------------------------------------------------------- /examples/RandomScatterers.nb: -------------------------------------------------------------------------------- 1 | (* Content-type: application/vnd.wolfram.mathematica *) 2 | 3 | (*** Wolfram Notebook File ***) 4 | (* http://www.wolfram.com/nb *) 5 | 6 | (* CreatedBy='Mathematica 10.4' *) 7 | 8 | (*CacheID: 234*) 9 | (* Internal cache information: 10 | NotebookFileLineBreakTest 11 | NotebookFileLineBreakTest 12 | NotebookDataPosition[ 158, 7] 13 | NotebookDataLength[ 10684, 280] 14 | NotebookOptionsPosition[ 10048, 252] 15 | NotebookOutlinePosition[ 10416, 268] 16 | CellTagsIndexPosition[ 10373, 265] 17 | WindowFrame->Normal*) 18 | 19 | (* 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old commands 15 | %\usepackage{epsfig} 16 | % The amssymb package provides various useful mathematical symbols 17 | \usepackage{amssymb} 18 | \usepackage{amsmath} 19 | %\usepackage{amsthm} 20 | \usepackage{mathdots} 21 | %\usepackage{mathaccents} 22 | \evensidemargin=0cm \oddsidemargin=0cm \setlength{\textwidth}{16cm} 23 | \usepackage{setspace} 24 | \usepackage{stmaryrd} 25 | \usepackage{theorem} 26 | \usepackage{hyperref} 27 | 28 | \newtheorem{theorem}{Theorem} 29 | \newtheorem{corollary}{Corollary} 30 | \newtheorem{lema}{Lema} 31 | \theorembodyfont{\normalfont} 32 | \newtheorem{definition}{Definition} 33 | \newtheorem{example}{Example} 34 | \newtheorem{solution}{Solution} 35 | \def \proof{\noindent {\bf \emph{Proof:}} } 36 | 37 | \newcommand{\be}{\begin{equation}} 38 | \newcommand{\en}{\end{equation}} 39 | \def\bga#1\ega{\begin{gather}#1\end{gather}} % suggested in technote.tex 40 | \def\bgas#1\egas{\begin{gather*}#1\end{gather*}} 41 | 42 | \def\bal#1\eal{\begin{align}#1\end{align}} % suggested in technote.tex 43 | \def\bals#1\eals{\begin{align*}#1\end{align*}} 44 | 45 | \newcommand{\initial}[1]{{#1}_\circ} 46 | \renewcommand{\thefootnote}{\fnsymbol{footnote}} 47 | 48 | \DeclareMathOperator{\sign}{sgn} 49 | \DeclareMathOperator{\divergence}{div} 50 | \DeclareMathOperator{\tr}{tr} 51 | \DeclareMathOperator{\Ord}{\mathcal{O}} 52 | \DeclareMathOperator{\GRAD}{grad} 53 | \DeclareMathOperator{\DIV}{DIV} 54 | 55 | \newcommand{\filler}{\hspace*{\fill}} 56 | \newcommand{\ii}{\textrm{i}} 57 | \newcommand{\ee}{\textrm{e}} 58 | \renewcommand{\vec}[1]{\boldsymbol{#1}} 59 | \def \bm#1{\mbox{\boldmath{$#1$}}} % this is used to write boldface Greek 60 | 61 | \graphicspath{{../Images/}} 62 | \graphicspath{{Media/}} 63 | 64 | \doublespacing 65 | 66 | \setlength{\topmargin}{0cm} \addtolength{\textheight}{2cm} 67 | 68 | \begin{document} 69 | 70 | \title{\sc{ Discrete Fourier Transforms}} 71 | \author{ 72 | Artur L Gower\footnotemark[2] \\[12pt] 73 | %$^a$ School of Mathematics, Statistics and Applied Mathematics,\\ 74 | %National University of Ireland Galway;\\ 75 | } 76 | \date{\today} 77 | \maketitle 78 | 79 | \footnotetext[2]{School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK.} 80 | 81 | \begin{abstract} 82 | Here we give details on Green's functions and fourier transforms used in the in the Mathematica package \href{https://github.com/arturgower/MultipleScattering-Mathematica}{MultipleScattering2D}. 83 | One unusually feature in the package is that we use do a discrete Fourier transforms without including the zero frequency $\omega =0$. The reason for avoiding $\omega =0$ is for functions, like Hankel functions, which have a singularity at $\omega =0$. The notebook \url{DiscreteFourierOffset.nb}, in the same directory, shows how to implement these methods and compares them with known analytic results. 84 | \end{abstract} 85 | 86 | \noindent 87 | {\textit{Keywords:} Discrete Fourier, Discrete offset Fourier, Greens functions} 88 | 89 | 90 | %%%%%%%%%%%% 91 | 92 | \section{Incident wave} 93 | 94 | We look to solve the 3D wave equation 95 | \be 96 | \mathcal L \{\varphi \}(\vec x,t) = \frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}(\vec x,t) - \nabla^2 \varphi(\vec x,t) = \frac{1}{c^2}B(\vec x,t), 97 | \label{eqn:ScalarWaveEquation} 98 | \en 99 | with the conditions 100 | \be 101 | \varphi(\vec x,0^{-}) =0 , \quad \dot \varphi( \vec x,0^{-}) =0 \quad \text{and} \quad \lim_{\|\vec x\| \to 0} \varphi(\vec x,t) =0, 102 | \en 103 | where $B$ is the body force\footnote{This $B$ is technically only a body force for an elastic SH-wave. The interpretation of $B$ depends on the physical interpretation of $\varphi$.}. To solve this we use the Delta Dirac $\delta(\vec x) = \delta(x)\delta(y)\delta(z) = \delta (r)/(4 \pi r^2)$ if $r$ is the radius of a spherical coordinate system, and first solve the wave equation in spherical coordinates 104 | \be 105 | \frac{1}{c^2}\frac{\partial^2 g}{\partial t^2} - \frac{1}{r^2} \frac{\partial}{\partial r} \left (r^2 \frac{\partial g}{\partial r} \right ) = \frac{\delta(r)}{4 \pi r^2} \delta (t) 106 | \en 107 | with $g(r,t) =0$ for $t<0$. This is not trivially solved, as when differentiating $r^{-1}$ a distribution appears on the origin. The solution can be found in p. 92 Achenbach (1973)\footnote{There he changes to spherical coordinates, substitutes $\varphi(r,t) = \Phi(r,t)/r$, and with witchcraft solves the resulting scalar wave equation, picking only the outgoing wave} . 108 | \be 109 | g(r, t) = \frac{1}{4 \pi r} \delta(t - r/c). 110 | \label{eqn:ScalarGreen3D} 111 | \en 112 | %Note replacing $\delta$ for some function $f$ above would give the solution for when $B(x,t) = c^2 f(t) \delta( \vec x)$. 113 | If we let $B(\vec x,t) = \delta (\vec x) b(t)$, then the solution to Eq.~\eqref{eqn:ScalarWaveEquation} without a scatterer, i.e. for the incident wave, becomes 114 | \begin{multline} 115 | \varphi^I(\vec x,t) = \int g(\vec x - \vec \xi,t -\tau)\frac{1}{c^2} B(\vec \xi, \tau) d \vec \xi d\tau = 116 | \\ 117 | \int g(\vec x - \vec \xi,t -\tau)\frac{1}{c^2} \delta(\vec \xi) b(\tau) d \vec \xi d\tau = \int \delta(t- \tau - r/c) \frac{b(\tau)}{4 c^2 \pi r} d\tau = \frac{b(t- r/c)}{4 c^2 \pi r}. 118 | \label{eqn:ScalarPointImpact} 119 | % \int \frac{1}{4 \pi} \frac{1}{c^2} B\left ( \sqrt{r^2 + r\xi^2 \mp 2 r r_\xi \cos\theta_\xi \sin\phi_\xi} , t -r_\xi /c \right) r_\xi \sin \phi_\xi d \phi_\xi d \theta_\xi d r_\xi , 120 | \end{multline} 121 | which is the solution to Eq.~\eqref{eqn:ScalarWaveEquation} because 122 | \[ 123 | \mathcal L \{\varphi^I \}(\vec x,t) = \int \mathcal L \{ g \}(\vec x -\vec \xi,t -\tau)\frac{1}{c^2}B(\vec \xi, \tau) d \xi d \tau = \int \delta(\vec x -\vec \xi,t -\tau) \frac{1}{c^2} B(\vec \xi, \tau) d \xi d \tau = \frac{1}{c^2}B(\vec x,t). 124 | \] 125 | Let us adopt the Fourier transform convention: 126 | \[ 127 | f(t) = \frac{1}{2 \pi} \int_{-\infty}^\infty \hat f(\omega) \ee^{- \ii \omega t} d \omega \;\; \text{and} \;\; \hat f(\omega) = \int_{-\infty}^\infty f(t) \ee^{\ii \omega t} d t. 128 | \] 129 | 130 | If we took one frequency $g(r,t) = \hat g(r,\omega) \ee^{-\ii \omega t}$ and solved for $\hat g$ with $B = c^2 \ee^{- \ii \omega t} \delta(\vec x)$, one frequency of $c^2 \delta(\vec x) \delta(t)$, the solution using only outgoing waves would be 131 | \be 132 | \hat g(r, \omega) = \frac{\ee^{\ii k r}}{4 \pi r}, 133 | \en 134 | with $k = \omega /c$, which after a Fourier transform would give the causal 3D Greens function~\eqref{eqn:ScalarGreen3D} as expected. 135 | 136 | For the frequency decomposition of the 2D Greens function $\hat g_2$, we imagine that all functions will be independent of the $z$ coordinate. So to use $\hat g$ in a convolution we need to first integrate over $z$: 137 | \be 138 | \hat g_2 = \int_{-\infty}^\infty \hat g(r, \omega) dz = \int_{-\infty}^\infty \frac{\ee^{\ii k \sqrt{ r^2 + z^2} }}{4 \pi \sqrt{ r^2 + z^2}} dz = \frac{\ii}{4}H_0^{(1)}(k r), 139 | \en 140 | where $r^2 = x^2 + y^2$ and $\hat g_2$ is an outgoing wave solution to 141 | \be 142 | k^2 \hat g_2 + \nabla^2_2 \hat g_2 = - \delta(x)\delta(y), 143 | \en 144 | where $\hat B(\vec x , \omega) = c^2 \delta(\vec x)$ and $\nabla_2$ is the gradient in $x$ and $y$. Note that Graf's and Gegenbauer's addition formulas are very useful to rewrite any bessel or hankel function. 145 | 146 | To calculate $g_2$ we can take the 3D Greens~\eqref{eqn:ScalarGreen3D} substitute $r \to \sqrt{r^2 + z^2}$ and integrate in $z$ to get 147 | \be 148 | g_2 = \int_{-\infty}^{\infty} \frac{\delta(c t - \sqrt{r^2 + z^2})}{4 \pi \sqrt{r^2 + z^2}} c dz = \frac{c}{2 \pi} \frac{H_s(t -|r|/c)}{\sqrt{c^2 t^2 - r^2}}, 149 | \en 150 | where $H_s$ is the Heavside step function, so that $H_s(t-|r|/c)$ is zero for $r > ct$. 151 | 152 | We can now calculate the 2D incident wave for $B(\vec x,t) = \delta(\vec x) b (t)$ by using the procedure~\eqref{eqn:ScalarPointImpact} for $g_2$ to find 153 | \be 154 | \varphi^I(\vec x,t) = 155 | \int_{-\infty}^{\infty} g_2(r ,t -\tau) \frac{b(\tau)}{c^2} d\tau = \int_{|r|/c}^{\text{max} \{t, \frac{|r|}{c}\} } \frac{1}{2 \pi c} \frac{b(t-\tau)}{\sqrt{c^2 \tau^2 - r^2}} d\tau, 156 | %\int_{-\infty}^{t-r/c} \frac{1}{2 \pi c} \frac{b(\tau) }{\sqrt{c^2 (t- \tau)^2 - r^2}} d\tau, 157 | \label{eqn:ScalarPointImpact2D} 158 | \en 159 | where we changed variables $\tau \leftarrow t -\tau$ so that we can differentiate the above expression in $t$ more easily (specially numerically) and assumed that $b(-t) =0$ for $t>0$. 160 | Alternatively, Eq.~\eqref{eqn:ScalarPointImpact2D} can also be written in terms of the Fourier transforms $\hat g_2(r, \omega)$ and $\hat b(\omega)$ as 161 | \be 162 | \varphi^I(\vec x,t) = 163 | \frac{1}{c^2} \int_{-\infty}^{\infty} g_2(r ,\tau) b(t -\tau) d\tau = \frac{1}{2 \pi c^2} \int_{-\infty}^{\infty} \ee^{-\ii \omega t} \hat g_2(r ,\omega) \hat b(\omega) d \omega. 164 | \label{eqn:ConvolutionWithGreen2D} 165 | \en 166 | 167 | Assuming that $b(t) = 0$ for $t \not \in [0, T]$, then we turn into a numerical method by approximating $b(t)$ with its truncated Fourier series. 168 | 169 | \be 170 | \varphi^I(\vec x,t) \approx \frac{1}{T c^2} \sum_{n = - N}^N \ee^{-\ii \frac { 2 \pi n }{T} t} \hat g_2(r ,2 \pi n /T) \hat b_N^n . 171 | \label{eqn:ConvolutionWithGreen2DN} 172 | \en 173 | One issue is that $\hat g_2$ has a singularity at $\omega =0$. More generally, every Hankel function of the first type has a singularity at $\omega =0$, which we will deal with carefully in the next section. 174 | 175 | \subsection{The offset Discrete Fourier Transform} 176 | Approximating an impulse $b(t)$ with its truncated Fourier series $b_N(t)$ means that 177 | \be 178 | b_N(t) = \frac{1}{T} \sum_{n = -N}^{ N} \hat b_N^n \ee^{- \ii t \frac{2 \pi n}{T}}, \;\; \text{where} \;\; \hat b_N^n := \hat b_N\left (\frac{2 \pi n}{T}\right ) = \int_0^T b(t) \ee^{\ii \frac{2 \pi n t}{T}} dt, 179 | \en 180 | %which can be used to approximate the Fourier transformed $\hat b(\omega)$ for $\omega \in [- 2 \pi \frac{N}{T}, 2 \pi \frac{N}{T}]$ using 181 | %\be 182 | %c_n = \int_{0}^T b_N(t) \ee^{\ii t \frac{2 \pi n}{T} } dt = \hat b_N(2 \pi n/T) , 183 | %\en 184 | %for $n= - N, -N+1, \ldots, N$. 185 | Alternatively we can fix $\omega_n = n \delta \omega$, so that $T = 2\pi/\delta \omega$ and $N= \Omega/\delta \omega$. 186 | 187 | We can turn this into the discrete Fourier transform by using only the points 188 | %$b_N(m T /(2N +1))$ with $m=0, 1, \ldots 2N$, from which we get 189 | \be 190 | b_N^m := b_N \left (\frac{m T}{2N +1}\right )= \frac{1}{T} \sum_{n = -N}^{ N} \hat b^n_N \ee^{-2 \pi \ii \frac{ m n}{2N +1}}, 191 | %= \frac{1}{T} \sum_{n = 0}^{ 2 N} \hat b^{n- N}_N \ee^{-2 \pi \ii \frac{ m n}{2 N}} \ee^{2 \pi \ii m \frac{N}{2 N +1}}, 192 | \en 193 | for $m= 0, 1, \ldots, 2 N$. We can now apply some linear algebra to extract the coefficients $T^{-1}\hat b^{n}_N$ of the vectors 194 | \be 195 | (\vec v_n)^m:= \ee^{-2 \pi \ii \frac{ m n }{2 N + 1}} \;\; \text{ for } n=0, 1, \ldots 2 N, 196 | %\ee^{2 \pi \ii m \frac{N}{2 N +1}} \;\; \text{ for } n=0, 1, \ldots 2 N, 197 | \en 198 | where $\vec v_n \cdot \bar{\vec v}_j = (2 N +1) \delta_{nj}$, to reach that 199 | \be 200 | \hat b^{n}_N = \frac{T}{(2 N +1)} \sum_{m=0}^{2 N} b_N^m \ee^{2 \pi \ii \frac{ m n }{2 N + 1}}, 201 | \en 202 | which is the definition of the discrete Fourier transform. 203 | 204 | For wave problems there is often a singularity at $\omega =0$, a frequency which we used above. The easiest way to deal with this is just to take $\hat b^0_N =0$ which results in the wrong constant being added to the whole time signal. This wrong constant can be corrected by attempting to make the signal casual, which works well if we know the time signal is a pulse. Failing that, the whole frequency range can be offset by constant. 205 | 206 | Suppose we wish to approximate $b(t)$ by 207 | \be 208 | b(t) \approx \frac{1}{T}\sum_{n=-N}^N \hat \beta^n_N \ee^{-\ii t( 2 \pi n/T + \delta)}, 209 | \en 210 | then we see that by multiplying by $\ee^{\ii t( 2 \pi m/T + \delta)}$, for $m= -N, \ldots, N$, on both sides\footnote{Check the proof of converges of the Fourier series and see if it can be adapted to this case.} and integrating we get 211 | \be 212 | \hat \beta^n_N = \int_0^T b(t) \ee^{\ii t( 2 \pi n/T + \delta)} dt = \hat b\left (\frac{ 2 \pi n}{T} + \delta \right), 213 | \en 214 | which looks like the Discrete Fourier transform of $\beta(t) = b(t)\ee^{\ii t \delta}$. In fact substituting $b(t)$ for $\beta(t)$ we get 215 | \be 216 | \beta(t) \approx \frac{1}{T}\sum_{n=-N}^N \hat \beta^n_N \ee^{-\ii 2 \pi n t /T }, 217 | \en 218 | from which we know that by taking $\beta_N^m = \beta(m T /(2 N +1))$ we have that 219 | \be 220 | \beta_N^m = \frac{1}{T}\sum_{n=-N}^N \hat \beta^n_N \ee^{-2 \pi \ii \frac{n m}{2 N +1} } \quad \text{and} \quad 221 | \hat \beta_N^n = \frac{T}{2N +1}\sum_{m=0}^{2N} \beta^m_N \ee^{2 \pi \ii \frac{n m}{2 N +1} }. 222 | \en 223 | The above translated back to $b(t)$ gives us 224 | \bga 225 | b\left (\frac{ m T} {2 N +1} \right) \approx \ee^{- \ii \delta \frac{ m T} {2 N +1}} \frac{1}{T}\sum_{n=-N}^N \hat b\left (\frac{ 2 \pi n}{T} + \delta \right) \ee^{-2 \pi \ii \frac{n m}{2 N +1} } \quad \text{and} 226 | \label{eqn:InverseOffsetDFT} \\ 227 | \hat b\left (\frac{ 2 \pi n}{T} + \delta \right) \approx \frac{T}{2N +1}\sum_{m=0}^{2N} \ee^{ \ii \delta \frac{ m T} {2 N +1}} b\left (\frac{ m T} {2 N +1} \right) \ee^{2 \pi \ii \frac{n m}{2 N +1} }, 228 | \label{eqn:OffsetDFT} 229 | \ega 230 | where $m =0, \ldots, 2N$ for $t\in [0,T]$, and $n =-N, \ldots, N$ for $\omega \in [ \delta - 2 \pi N /T , \delta + 2 \pi N /T ]$. In our methods the more important parameter is the maximum frequency $\Omega \approx 2 \pi N /T$, followed by $N$, so that $T = 2 \pi N/ \Omega$, which should be bigger than the period of $b(t)$. 231 | 232 | Typically the discrete Fourier, and its inverse, are implemented so that both $n$ and $m$ run from $0$ to $2N$. That is the input to a DFT is $(\hat \beta_N^0, \hat \beta_N^1, \ldots, \hat \beta_N^{2 N})$. So we must adjust Eq.~\eqref{eqn:InverseOffsetDFT} so that $n$ runs from $0$ to $2N$. Turning to Eq.\eqref{eqn:OffsetDFT} we see that 233 | \[ 234 | \hat b\left ( -\frac{ 2 \pi n}{T} + \delta \right) = \hat b_{-n} = \hat b_{-n + 2N +1}, 235 | \] 236 | which for $n = 1, 2, \ldots, N$ gives $\hat b_{-1} = \hat b_{2N}$, $ \hat b_{-2} = \hat b_{2N -1}$, \ldots, $ \hat b_{-N} = \hat b_{N +1}$ . We use this to rewrite Eq.~\eqref{eqn:InverseOffsetDFT} as 237 | \bga 238 | b_m \approx \ee^{- \ii \delta \frac{ m T} {2 N +1}} \frac{1}{T}\sum_{n=0}^{2N} \hat b_n \ee^{-2 \pi \ii \frac{n m}{2 N +1} }. 239 | \label{eqn:InverseOffsetDFT} 240 | \ega 241 | To be clear, the input vector for DFT would typically be 242 | \[ 243 | \hat{\vec b} = (\hat b_0, \hat b_1, \ldots, \hat b_N, \hat b_{-N}, \ldots, \hat b_{-1}), 244 | \] 245 | calculated from $\vec \omega = \delta + (1,2\pi/T, \ldots, 2N \pi/T,- N \pi/T, \ldots,- \pi/T )$. Below are some examples. 246 | 247 | The default Mathematica DFT calculates 248 | \bga 249 | \hat {\vec f_n} = \mathrm{Fourier}[\vec f]_n = \frac{1}{\sqrt{2N+1}} \sum_{m=0}^{2 N} f_m \ee^{2 \pi \ii \frac{n m}{2N+1}}, 250 | \\ 251 | {\vec f_m} = \mathrm{InverseFourier}[\hat{\vec f}]_m = \frac{1}{\sqrt{2N+1}} \sum_{n=0}^{2 N} \hat f_n \ee^{-2 \pi \ii \frac{n m}{2N+1}}. 252 | \ega 253 | So taking $\hat f_n = \hat b_n \sqrt{2N +1}/T$ leads to $b_m = \ee^{-\ii \delta \frac{m T}{2N+1}} f_m$. 254 | 255 | Julia's fft reverse the role of the forward and back DFT. For Julia's taking $\hat f_n = \hat b_{-n} (2N +1)/T$ leads to $b_m = \ee^{\ii \delta \frac{m T}{2N+1}} f_m$. 256 | 257 | Note that when we cut out the discontinuity from $\hat b$, the Discrete Fourier Transform will approximate $\hat b$ for some bell shaped curve (and likewise for the time curve). If $\delta$ is to small in comparison to $\delta \omega$ this bell shape will be much too big and rounded to approximate the sharp step from the hankel function. For example, numerically we find that for $\delta = 0.01 \delta \omega$ are already significantly off, whereas for $\delta =0.1 \delta \omega$ they results are always decent (within $1\%$ error when reconstructing the time signal). 258 | 259 | 260 | Convolution formulas such as~\eqref{eqn:ConvolutionWithGreen2D} can also be written in terms of the Offset Discrete Fourier Transform. Let 261 | \be 262 | h(t) = \frac{1}{c^2} \int_{-\infty}^\infty \ee^{-\ii \omega t} \hat g(\omega) \hat b(\omega) d \omega. 263 | \en 264 | Assuming $\Omega$ an $N$ is given, we can represent the above by the inverse of the Offset Discrete Fourier Transform by using Eq.~\eqref{eqn:InverseOffsetDFT} to get 265 | %\[ 266 | %h\left ( \frac{ m T} {2 N +1} \right) \approx \ee^{- \ii \delta \frac{ m T} {2 N +1}} \frac{1}{T c^2} \sum_{n = -N}^N \ee^{- 2 \pi \ii \frac{ n m } {2 N +1} } \hat g(2 \pi n /T +\delta) \hat b(2 \pi n /T +\delta) . 267 | %\] 268 | \be 269 | h\left ( \frac{m \pi}{\Omega} \frac{2 N } {2 N +1} \right) \approx \ee^{- \ii \delta \frac{m \pi}{\Omega} \frac{2 N} {2 N +1}} \frac{\Omega}{2 \pi N c^2} \sum_{n = -N}^N \ee^{- 2 \pi \ii \frac{ n m } {2 N +1} } \hat g\left (\Omega \frac{n}{N} +\delta \right) \hat b\left ( \Omega \frac{n}{N} +\delta \right) . 270 | \en 271 | \subsection{Scattered wave} 272 | 273 | The Fourier transform of the outgoing waves from a cylinder can be anything in the form 274 | \be 275 | \hat \varphi^S(r, \omega) = \sum_{n= -\infty}^{ \infty} a_n H^{(1)}_n( \omega r/c) \ee^{\ii n \theta} \approx \sum_{n= -Na}^{ Na} a_n H^{(1)}_n( \omega r/c) \ee^{\ii n \theta} 276 | \label{eqn:ScatteredForm} 277 | \en 278 | where we call $Na$ the number of angular modes. To choose $Na=6$ for the package add \texttt{"NAngularModes"->6} to the function call. 279 | 280 | To calculate the scattered wave in time $\varphi^S$, we do the same operation on $\hat \varphi_S$ that we did on $\hat g_2$ in Eq.\eqref{eqn:ConvolutionWithGreen2DN} 281 | \begin{multline} 282 | \varphi^S(r, t) \approx \frac{1}{2 \pi c^2} \int_{- 2 \pi N/T}^{ 2 \pi N/T} \ee^{-\ii \omega t} \hat \varphi^S(r ,\omega) \hat b_N(\omega) d \omega 283 | \\ 284 | = \frac{1}{2\pi c} \sum_{n= -\infty}^{ \infty} \int_{- 2 \pi N/T}^{ 2 \pi N/T} \ee^{ -\ii c k t + \ii n \theta } a_n(k) H^{(1)}_n( k r) \hat b_N(c k) d k. 285 | \label{eqn:ScatteredConvolutionN} 286 | \end{multline} 287 | The coefficients $a_n$ will be determined by the boundary conditions. 288 | 289 | 290 | \subsubsection{Boundary condition} 291 | 292 | To compare the incident and scattered wave we use Graf's addition formula~\footnote{see \url{http://dlmf.nist.gov/10.23\#E7}} applied to 293 | \be 294 | H^{(1),(2)}_0 (k \|\vec r- \vec r_I \|) = \sum_{n=-\infty}^{\infty} \ee^{\ii n (\theta -\theta_1)} J_n (k r) H^{(1),(2)}_n(k r_I) \quad \text{if} \;\; r< r_I, 295 | \en 296 | %~\eqref{eqn:ScatteredWave} we rewrite\footnote{See obscure reference on p.159 of Pao and Mow, and use $J_{n}= (-1)^n J_{-n}$ together with $H_{n}= (-1)^n H_{-n}$.} 297 | %\be 298 | %\hat g_2 (r,r_I) = (1 + \delta_m^0)\frac{\ii}{8} \sum_{m=-\infty}^\infty \ee^{\ii m(\theta-\theta_1)} 299 | %\begin{cases} 300 | %J_m(k r_I) H_m^{(1)}(k r),& r > r_I,\\ 301 | %J_m(k r) H_m^{(1)}(k r_I),& r < r_I, 302 | %\end{cases} 303 | %\en 304 | where the body force originates at $\vec r_I = r_I (\cos \theta_I,\sin \theta_I)$ and $\vec r = r (\cos \theta,\sin \theta)$. 305 | %and $\delta_m^0 =1$ only if $m=0$. 306 | If $\hat \varphi^I = \hat g_2( \|\vec r- \vec r_I\|)$ and $\hat \varphi^S = a_{ n} H^{(1)}_n (k r) \ee^{\ii n \theta}$ then the boundary condition $\hat \varphi_I + \hat \varphi_S =0$ on $r= r_S< r_I$ implies 307 | %\varphi^S= $\sum_{n=-\infty}^\infty a_{ n} H^{(1)}_n (k r) \ee^{-\ii k c t} \ee^{\ii n \theta}$ 308 | \bga 309 | \frac{\ii}{4} \ee^{- \ii n \theta_I} J_n (k r_S) H^{(1)}_n(k r_I) + a_n H_n^{(1)}(k r_S) =0 \implies 310 | \\ 311 | a_n =-\frac{\ii}{4} \ee^{- \ii n \theta_I} \frac{J_n (k r_S)}{H_n^{(1)}(k r_S)} H^{(1)}_n(k r_I). 312 | \label{eqn:BCDirichlet} 313 | \ega 314 | while the boundary condition $\partial \hat \varphi^I /\partial r + \partial \hat \varphi^S/ \partial r =0$ on $r= r_S < r_I$ implies that 315 | \bga 316 | % a_n k H^{'(1)}_n (k r_S) + (1+\delta_n^0) \frac{i \ee^{\ii n \theta_I}}{8} k J_n'(k r_S) H_n^{(1)}(k r_I) =0 317 | %\implies 318 | %\notag \\ 319 | a_n =- \frac{i}{4} \ee^{- \ii n \theta_I} \frac{ J_n'(k r_S)}{ H^{'(1)}_n (k r_S)} H_n^{(1)}(k r_I). 320 | \label{eqn:BCNeumann} 321 | \ega 322 | 323 | \subsection{Wave frequency to time response} 324 | 325 | This module is used for more complicated scattering, such as MST. Any outgoing wave from a scatterer can be expanded as 326 | \[ 327 | \hat \psi^{S}(\vec r, k) \approx \sum_{n=-Na}^{Na} a_{n} H_n (k r) \ee^{\ii n \theta}, 328 | \] 329 | for a cylindrical coordinate system with origin at the scatterer, where $H_n:= H_n^{(1)}$ a Hankel function of the first kind. To recover the wave in time we need to approximate the inverse Fourier transform: 330 | \[ 331 | \psi^{S}(\vec r ,t ) = \frac{c}{2 \pi}\int_{-\infty}^\infty \hat \psi^{S}(\vec r ,k ) \ee^{-\ii c k t} d k \approx \frac{c}{2 \pi} \sum_{n=-Na}^{Na}\ee^{\ii n \theta} \int_{- 2 \pi N /T}^{2 \pi N /T} a_{n} H_n (k r) \ee^{-\ii c k t} d k, 332 | \] 333 | where $T$ is the period of the incident wave and $N$ is the number of frequency modes. 334 | 335 | Assuming that the scatterers are small compared with the wavelength $k r_S << 1$, where $r_S$ is the radius of the scatterer, then the most general form for outgoing waves from the $j$-th scatterer is 336 | \bga 337 | \psi^{S}(\vec r) = a H_0 (k r) + (c \cos \theta + s \sin \theta ) H_1 (k r) , 338 | \ega 339 | which means that $a_0 = a$, $a_1 = (s + c)/4$ and $a_{-1} = (s-c)/4$. 340 | 341 | 342 | 343 | 344 | 345 | 346 | 347 | \end{document} 348 | -------------------------------------------------------------------------------- /theory/ScatteringOperator.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldcontentsline\contentsline 6 | \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} 7 | \global\let\oldnewlabel\newlabel 8 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 9 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 10 | \AtEndDocument{\ifx\hyper@anchor\@undefined 11 | \let\contentsline\oldcontentsline 12 | \let\newlabel\oldnewlabel 13 | \fi} 14 | \fi} 15 | \global\let\hyper@last\relax 16 | \gdef\HyperFirstAtBeginDocument#1{#1} 17 | \providecommand\HyField@AuxAddToFields[1]{} 18 | \providecommand\HyField@AuxAddToCoFields[2]{} 19 | \citation{martin_2006} 20 | \@writefile{toc}{\contentsline {section}{\numberline {1}The General Setup}{1}{section.1}} 21 | \newlabel{eqn:ExcitedWave}{{2}{2}{The General Setup}{equation.1.2}{}} 22 | \newlabel{eqn:LinearScattering}{{3}{2}{The General Setup}{equation.1.3}{}} 23 | \newlabel{eqn:TwoScatterers}{{6}{2}{The General Setup}{equation.1.6}{}} 24 | \@writefile{toc}{\contentsline {section}{\numberline {2}Cylindrical Scatterers}{2}{section.2}} 25 | \newlabel{eqn:OutingSj}{{7}{2}{Cylindrical Scatterers}{equation.2.7}{}} 26 | \newlabel{eqn:Graf}{{8}{3}{Cylindrical Scatterers}{equation.2.8}{}} 27 | \newlabel{eqn:TjOpNeumann}{{11}{3}{Cylindrical Scatterers}{equation.2.11}{}} 28 | \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Boundary conditions and point scatterers}{4}{subsection.2.1}} 29 | \newlabel{sec:BoundaryConditions}{{2.1}{4}{Boundary conditions and point scatterers}{subsection.2.1}{}} 30 | \newlabel{eqn:aDirichlet}{{19}{4}{Boundary conditions and point scatterers}{equation.2.19}{}} 31 | \newlabel{eqn:aNeumann}{{21}{5}{Boundary conditions and point scatterers}{equation.2.21}{}} 32 | \newlabel{eqn:PsiTaylor}{{22}{5}{Boundary conditions and point scatterers}{equation.2.22}{}} 33 | \newlabel{eqn:PsiTaylor}{{23}{5}{Boundary conditions and point scatterers}{equation.2.23}{}} 34 | \newlabel{eqn:DPsiTaylor}{{25}{5}{Boundary conditions and point scatterers}{equation.2.25}{}} 35 | \newlabel{eqn:ScatteringOpDirichlet}{{26}{6}{Boundary conditions and point scatterers}{equation.2.26}{}} 36 | \newlabel{eqn:ScatteringOpNeumann}{{27}{6}{Boundary conditions and point scatterers}{equation.2.27}{}} 37 | \newlabel{eqn:DirchlettHankelExpansion}{{28}{6}{Boundary conditions and point scatterers}{equation.2.28}{}} 38 | \newlabel{eqn:NeumannHankelExpansion}{{29}{6}{Boundary conditions and point scatterers}{equation.2.29}{}} 39 | \bibcite{martin_2006}{1} 40 | \newlabel{eqn:PsiETerm}{{33}{7}{Boundary conditions and point scatterers}{equation.2.33}{}} 41 | -------------------------------------------------------------------------------- /theory/ScatteringOperator.fdb_latexmk: 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\setlength{\topmargin}{1cm} 14 | % \addtolength{\textheight}{0cm} 15 | 16 | \usepackage{graphicx} 17 | \usepackage{amssymb,amsmath,mathtools} 18 | 19 | \usepackage[mathscr]{eucal} %just for the font \mathscr 20 | \usepackage{mathdots} 21 | \usepackage{setspace} 22 | % \usepackage{stmaryrd} 23 | \usepackage{hyperref} 24 | 25 | 26 | \newcommand{\be}{\begin{equation}} 27 | \newcommand{\en}{\end{equation}} 28 | \def\bga#1\ega{\begin{gather}#1\end{gather}} % suggested in technote.tex 29 | 30 | \def\bal#1\eal{\begin{align}#1\end{align}} % suggested in technote.tex 31 | \def\bals#1\eals{\begin{align*}#1\end{align*}} 32 | 33 | \renewcommand{\thefootnote}{\fnsymbol{footnote}} 34 | 35 | \DeclareMathOperator{\sign}{sgn} 36 | \DeclareMathOperator{\Ord}{\mathcal{O}} 37 | 38 | \newcommand{\filler}{\hspace*{\fill}} 39 | \newcommand{\T}{\mathscr T} 40 | \newcommand{\ii}{\textrm{i}} 41 | \newcommand{\ee}{\textrm{e}} 42 | 43 | \renewcommand{\vec}[1]{\boldsymbol{#1}} 44 | \def \bm#1{\mbox{\boldmath{$#1$}}} % this is used to write boldface Greek 45 | 46 | % \doublespacing 47 | 48 | \begin{document} 49 | 50 | \title{Calculating scattering coefficients from a cylinder} 51 | \author{ 52 | Artur L. Gower$^{1}$ \\[12pt] 53 | $^1$ School of Mathematics, University of Manchester \\ 54 | Oxford Road, Manchester M13 9PL, UK 55 | } 56 | \date{\today} 57 | \maketitle 58 | 59 | \begin{abstract} 60 | Here we derive some of the theory used in the Mathematica package \href{https://github.com/arturgower/MultipleScattering-Mathematica}{MultipleScattering2D}. Specifically the function \texttt{ScatteringCoefficientOperator}, which calculates the scattering coefficients for any source/exciting function, is given by combing equations~(\ref{eqn:ScatteringOpDirichlet},\ref{eqn:PsiTaylor},\ref{eqn:aDirichlet}) for Dirichlet boundary conditions. Similar equations are given for Neumann too. 61 | \end{abstract} 62 | 63 | \noindent 64 | {\textit{Keywords:} Multi-pole method, multiple scattering, scattering operator } 65 | 66 | 67 | %%%%%%%%%%%% 68 | 69 | \section{The General Setup} 70 | For a review on multiple scattering from a finite number of obstacles see \cite{martin_2006}. 71 | 72 | Consider a homogeneous isotropic medium that can propagate wave according to the scalar Helmholtz equation 73 | \be 74 | (\Delta + k^2)\psi(\vec r) =0, 75 | \en 76 | where $k$ is real for acoustics and possibly imaginary for fluids. Let there be $N$ scatterers, with the centre located at $\vec r_1$, $\vec r_1$, ..., $\vec r_N$. If we excite the scatterers by sending an incident wave $\psi^I(\vec r)$, we can then write the total wave field $\psi(\vec r| \vec r_1, \ldots, \vec r_N) = \psi^I(\vec r) + \sum_{j=1}^N \psi^S(\vec r; \vec r_j| \vec r_1, \ldots, \vec r_N)$, where $\psi^S_j(\vec r) := \psi^S(\vec r; \vec r_j| \vec r_1, \ldots, \vec r_N)$ is the outward going wave\footnote{We are only concerned with what the scatterer is emitting, and not waves that might bounce around inside.} being emitted from the boundary of the $j$-th scatterer. The term $\vec r_1, \ldots, \vec r_N$ indicates the dependence of $\psi$ and $\psi^S_j$'s on the position of all the scatterers. Each scattered wave $\psi^S_j$ is excited by 77 | \be 78 | \psi^E_j(\vec r) = \psi^I(\vec r) + \sum_{i \not = j} \psi^S_i(\vec r), 79 | \label{eqn:ExcitedWave} 80 | \en 81 | which we can relate to $\psi^S_j$ through the boundary condition on the $j$-th scatterer to give 82 | \be 83 | \psi^S_j(\vec r) = \T_j \left \{ \psi^E_j \right \}(\vec r), 84 | \label{eqn:LinearScattering} 85 | \en 86 | where $\T_j\{\circ\}(\vec r)$ is the linear \emph{scattering operator} which acts on the whole function $\circ$. Dirchlett and Neumann boundary condtions are examples of when $\T_j$ is a linear operator. By expanding $\psi^S_j$ as in Eq.\eqref{eqn:LinearScattering}, then substituting $\psi^E_j$ from Eq.\eqref{eqn:ExcitedWave} and then repeating the process by expanding $\psi^S_i$ with eq.\eqref{eqn:LinearScattering} again we obtain 87 | \be 88 | \psi^S_j(\vec r) = \T_j \{ \psi^I \}(\vec r) + \sum_{i \not = j} \T_j\circ \T_i \left \{ \psi^I \right \}(\vec r) + \sum_{i \not = j} \sum_{n \not = i} \T_j \circ \T_i \circ \T_n \left \{ \psi^I\right \}(\vec r) + \ldots. 89 | \en 90 | For two scatterers this becomes, 91 | \be 92 | \psi^S_1(\vec r) = \T_1 \{ \psi^I \}(\vec r) + \T_1\circ \T_2 \left \{ \psi^I \right \}(\vec r) + \T_1 \circ \T_2 \circ \T_1 \left \{ \psi^I\right \}(\vec r) + \ldots. 93 | \en 94 | If on the other hand we stopped expanding the series at 95 | \bga 96 | \psi^S_1(\vec r) = \T_1 \left \{ \psi^I \right \}(\vec r) + \T_1 \left \{ \psi^S_2 \right \}(\vec r), \notag \\ 97 | \psi^S_2(\vec r) = \T_2 \left \{ \psi^I \right \}(\vec r) + \T_2 \left \{ \psi^S_1 \right \}(\vec r). 98 | \label{eqn:TwoScatterers} 99 | \ega 100 | If we could expand each $\psi^S_j$ into a general outgoing wave 101 | \[ 102 | \psi^S_j(\vec r) = \sum_{n=-\infty}^\infty s_{jn} \mathcal C_n ( r) \mathcal A_n(\theta, \phi), 103 | \] 104 | then we could solve for the coefficients $s_{jn}$ by substituting into Eqs.\eqref{eqn:TwoScatterers}. 105 | 106 | \section{Cylindrical Scatterers} 107 | 108 | The outgoing wave from a scatterer at $\vec r_j$ be expanded as 109 | \be 110 | \psi^{S_j}(\vec r) = \sum_{n=-\infty}^\infty a_{jn} H_n (k \|\vec r - \vec r_j \|) \ee^{\ii n \alpha_j}, 111 | \label{eqn:OutingSj} 112 | \en 113 | where $H_n:= H_n^{(1)}$ is a Hankel function of the first kind and $\alpha_j$ is the angle between $\vec r - \vec r_j$ and the $x$--axis. 114 | 115 | We can expand the outgoing waves~\eqref{eqn:OutingSj} by using Graf's addition theorem\footnote{I double checked this with both \url{http://dlmf.nist.gov/10.23} and \url{http://www.wikiwaves.org/Graf's_Addition_Theorem}.}: 116 | %\be 117 | %\mathscr C_\nu (k \| \vec r - \vec r_1\|) \ee^{\pm \ii \nu (\alpha_1 -\theta_{12})} = 118 | %\sum_{m=-\infty}^\infty \mathscr C_{\nu +m} (k \| \vec r_1 - \vec r_2 \|) J_m(k \|\vec r- \vec r_2 \|) \ee^{ \pm \ii m (\pi +\theta_{ij} - \alpha_2)}, 119 | %\label{eqn:Graf} 120 | %\en 121 | %I calculate this $\pm$ above thinking that the angle $\alpha_{12}$ between to vectors $\vec r_1$ and $\vec r_2$ satisfies $\alpha_{12} = \alpha_{21}$, but that is not true. In fact $\alpha_{12} = 2 \pi - \alpha_{21}$. 122 | \be 123 | \mathscr C_\nu (k \| \vec r - \vec r_1\|) \ee^{ \ii \nu (\alpha_1 -\theta_{12})} = 124 | \sum_{m=-\infty}^\infty \mathscr C_{\nu +m} (k \| \vec r_1 - \vec r_2 \|) J_m(k \|\vec r- \vec r_2 \|) \ee^{ \ii m (\pi +\theta_{12} - \alpha_2)}, 125 | \label{eqn:Graf} 126 | \en 127 | provided $\|\vec r- \vec r_2 \| < \|\vec r_1- \vec r_2 \|$, where $\theta_{12}$ and $\alpha_2$ are respectively the angular cylindrical coordinate of $\vec r_1 - \vec r_2$ and $\vec r -\vec r_2$. 128 | %\bga 129 | %\cos \chi = \frac{(\vec r_i-\vec r_j)\cdot \vec r_i}{\|\vec r_i-\vec r_j \| r_i} = \frac{r_i}{\|\vec r_i -\vec r_j\|} - \cos \theta_{ij} \frac{r_j}{\|\vec r_i -\vec r_j\|}, 130 | %\\ 131 | %\sin \chi = \frac{\|(\vec r_i-\vec r_j)\times \vec r_i \|}{\|\vec r_i-\vec r_j \| r_i} = \sin \theta_{ij} \frac{r_j}{\|\vec r_i -\vec r_j\|} \implies 132 | %\\ 133 | %\ee^{ \ii \nu \chi} = \left ( \frac{r_i - r_j \ee^{- \ii \theta_{ij} }}{\|\vec r_i -\vec r_j\|} \right )^\nu. 134 | %\ega 135 | In the addition theorem $\mathscr C_\nu$ can be substituted with any of the Bessel functions $J_\nu,Y_\nu,H_\nu^{(1)}$ and $H_\nu^{(2)}$, or linear combinations of them. 136 | 137 | Using Eq.\eqref{eqn:Graf} we write $\psi^{S_j}$ in terms of an origin centered at the $i$-th scatterer by substituting 138 | \[ 139 | H_n (k \| \vec r - \vec r_j\|) \ee^{ \ii n \alpha_j} = 140 | \sum_{m=-\infty}^\infty H_{n -m} (k \| \vec r_j - \vec r_i \|) J_m(k \|\vec r- \vec r_i \|) \ee^{ \ii m \alpha_i +\ii (n - m) \theta_{ij}}, 141 | \] 142 | into the $\psi^{S_j}$ to arrive at 143 | \bga 144 | \psi^{S_j}(\vec r) = \sum_{n=-\infty}^\infty \sum_{m=-\infty}^\infty a_{jn} H_{n -m} (k \| \vec r_j - \vec r_i \|) J_m(k \|\vec r- \vec r_i \|) \ee^{ \ii m \alpha_i +\ii (n - m) \theta_{ij}}. 145 | %\\ 146 | %\psi^{S_j}(\vec r) = \sum_{n=-\infty}^\infty \sum_{m=-\infty}^\infty a_{jn} H_{n +m} (k r_j) J_m(k r) \ee^{\ii ( n-m) \theta + \ii ( m \theta_j - n \chi )}. 147 | %\\= \sum_{n=-\infty}^\infty \sum_{p=-\infty}^\infty a_{jn} H_{2 n -p} (k r_j) J_{n-p}(k r) \ee^{\ii p \theta + \ii ( (n-p) \theta_j - n \chi )}, 148 | \ega 149 | %where to get the last equation we renamed $m = n- p$. 150 | 151 | In Section~\ref{sec:BoundaryConditions} we derive some explicit forms for the scattering operator $\T$, given by 152 | Eqs.\eqref{eqn:ScatteringOpDirichlet} and \eqref{eqn:ScatteringOpNeumann}. Assuming that the scatterers are small compared with the wavelength $k r_S << 1$, where $r_S$ is the radius of the scatterer, then the most general form for outgoing waves from the $j$-th scatterer is 153 | \bga 154 | \psi^{S_j}(\vec r) = a_{j} H_0 (k \|\vec r - \vec r_j \|) + (c_{j} \cos (\vec r - \vec r_j) + s_{j} \sin (\vec r - \vec r_j) ) H_1 (k \|\vec r - \vec r_j \|) , 155 | \ega 156 | with $c_{j}=s_{j} =0$ for Dirichlet boundary conditions, where $\cos(\vec r - \vec r_j) = \cos \vartheta$ and $\vartheta$ is the angle between $\vec r - \vec r_j$ and the $x$-axis. 157 | In which case, using the results in Section~\ref{sec:BoundaryConditions}, the scattering operator for Neumann boundary conditions can be written as 158 | \begin{multline} 159 | \T_j : \psi^E(r, \theta) \to \frac{ \ii \pi r_S^2}{4} (\psi^E_{\vec r_j,xx} + \psi^E_{\vec r_j,yy}) H_0(k \| \vec r - \vec r_j \|) 160 | \\ 161 | + \frac{i \pi r^{2}_S}{2} k( \psi^E_{\vec r_j,x} \cos ( \vec r - \vec r_j ) + \psi^E_{\vec r_j,y} \sin ( \vec r - \vec r_j )) H_1(k \| \vec r - \vec r_j \|), 162 | \label{eqn:TjOpNeumann} 163 | \end{multline} 164 | accurate up to order $r^{3}_S$ in the scatterer radius $r_S$, where the subscript $\vec r_j$ on $\psi^E_{\vec r_j}$ means that $\psi^E$ and its derivatives are evaluated at $\vec r = \vec r_j$ after differentiation. This implies that to solve Eqns.~\eqref{eqn:TwoScatterers} for every $\vec r$ we need to equate the coefficients of $H_0(k \| \vec r - \vec r_j \|)$, $\cos ( \vec r - \vec r_j ) H_1(k \| \vec r - \vec r_j \|)$ and $\sin ( \vec r - \vec r_j ) H_1(k \| \vec r - \vec r_j \|)$ each to be zero, from which we get 165 | \bga 166 | a_{i} = \frac{ \ii \pi r_S^2}{4} (\psi^I_{\vec r_i,xx} + \psi^I_{\vec r_i,yy}+ \psi^{S_j}_{\vec r_i,xx} + \psi^{S_j}_{\vec r_i,yy}), 167 | \\ 168 | c_i = \frac{i \pi r^{2}_S}{2} k( \psi^{I}_{\vec r_i,x}+ \psi^{S_j}_{\vec r_i,x}), \;\; s_i = \frac{i \pi r^{2}_S}{2} k( \psi^{I}_{\vec r_i,y}+ \psi^{S_j}_{\vec r_i,y}). 169 | \ega 170 | To simplify we choose, without loss of generality, $x_2 = x_1$ and $- y_1 = y_2 = Y/2$ which we use to reduce Eqns.~\eqref{eqn:TwoScatterers} to 171 | \bga 172 | \frac{k^2}{2} (H_0 - H_2) a_1 + \frac{k^2}{4} (3 H_1 - H_3)s_1 - \frac{4 \ii }{\pi r_S^2} a_{2} = \psi^I_{\vec r_2,xx} + \psi^I_{\vec r_2,yy}, 173 | \\ 174 | \frac{H_1}{Y}c_1 +\frac{2 i }{\pi k r_S^2}c_2 =-\psi^I_{\vec r_2, x}, \;\; 175 | k H_1 a_1 -\frac{2 i}{\pi k r_S^2}s_2-\frac{k }{2} (H_0-H_2) s_1 =\psi^I_{\vec r_2, y} 176 | \ega 177 | where the Hankel functions $H_0$, $H_1$, $H_2$ and $H_3$ are evaluated at $k Y$, with $Y = \| \vec r_1 - \vec r_2 \|$, and to reach the above equations we have used some recurrence relations to calculate the derivatives of $H_0$ and $H_1$. 178 | 179 | %We need 180 | %\bga 181 | %\psi^S_{0,xx} + \psi^S_{0,yy} = a_{2} \left (\frac{k}{r_j}H_0'(k r_j) +k^2 H_0''( k r_j)\right ) + (c_{2} \cos (\vec r - \vec r_2) 182 | %\\ 183 | %+ s_{2} \sin (\vec r - \vec r_2) ) \left( \frac{k}{r_j}H_1'(k r_j) +k^2 H_1''( k r_j) \right) 184 | %\ega 185 | 186 | 187 | 188 | Apply $\T_1$ on $\psi^S_2$ with the centre of scatterer 1 as the origin we get 189 | \be 190 | \T_1 \psi^S_2(\vec r) = a_{2} H_0 (k \|\vec r - \vec r_2 \|) + (c_{2} \cos (\vec r - \vec r_2) + s_{2} \sin (\vec r - \vec r_2) ) H_1 (k \|\vec r - \vec r_2 \|) , 191 | \en 192 | 193 | 194 | 195 | The following formulas will be useful, for $f(\|\vec r - \vec r_j\|)$ we have 196 | \bga 197 | f_{,x} (k r_j) = \frac {k x_j}{r_j} f'( k r_j) , \; \; 198 | f_{,y} (k r_j) = \frac {k y_j}{r_j} f' ( k r_j), \\ 199 | f_{,yy} (k r_j) + f_{,xx} (k r_j) = \frac{k}{r_j}f'(k r_j) +k^2 f''( k r_j). 200 | \ega 201 | % and 202 | % \bga 203 | % \partial_x \cos(\vec r - \vec r_j) |_{\vec r =0} = \frac{y_j^2}{r_j^{3}}, \; \; \; \partial_x \sin(\vec r - \vec r_j) |_{\vec r=0} = -\frac{x_j y_j}{r_j^3}, \\ 204 | % \partial_y \cos(\vec r - \vec r_j) |_{\vec r =0} = - \frac{x_j y_j}{r_j^{3}}, \; \; \; \partial_y \sin(\vec r - \vec r_j) |_{\vec r =0} = \frac{x_j^2}{r_j^3}, \\ 205 | % \left ( \partial_{xx}+ \partial_{yy} \right )\cos(\vec r - \vec r_j) |_{\vec r =0} = \frac{x_j}{r_j^3}, \;\; \left ( \partial_{xx}+ \partial_{yy} \right )\sin(\vec r - \vec r_j) |_{\vec r =0} = \frac{y_j}{r_j^3}. 206 | % \ega 207 | 208 | 209 | \subsection{Boundary conditions and point scatterers} 210 | \label{sec:BoundaryConditions} 211 | Here we develop expressions for the scattering operator $\T$ and establish how point scatterers behave for different boundary conditions. Given an incident wave $\psi^E$ and outgoing cylindrical wave $\psi^S = a_n H_n(k r) \ee^{\ii n \theta}$, together $\psi^E + \psi^S$ must satisfy the boundary condition on the cylinder with radius $r = r_S$. To simplify we choose the origin of our coordinate system to be the centre of the scatterer and will use $\bar r := r k$. 212 | 213 | For a Dirichlet boundary conditions we have 214 | %\bga 215 | %\psi^E(r_S, \theta) + \psi^S(r_S, \theta) =0 \;\; \text{ for } \; 0\leq \theta < 2 \pi \implies 216 | %a_n = - H_n(r_S)^{-1} \mathcal C_n(r_S). 217 | %\ega 218 | \bga 219 | \psi^E(\bar r_S, \theta) + \psi^S(\bar r_S,\theta) =0 \;\; \text{ for } \; 0\leq \theta < 2 \pi \implies 220 | a_n = - \frac{H_n(\bar r_S)^{-1}}{2\pi}\int_{0}^{2 \pi}\psi^E(\bar r_S, \theta)\ee^{-\ii n \theta} d \theta, 221 | \label{eqn:aDirichlet} 222 | \ega 223 | noting that $\ee^{\ii n \theta}$ forms a complete basis for functions on $\theta \in [0, 2\pi]$. 224 | 225 | For a Neumann boundary conditions we have 226 | \bga 227 | \frac{\partial \psi^E}{\partial \bar r}(\bar r_S, \theta) + \frac{\partial \psi^S}{\partial \bar r}(\bar r_S, \theta) =0 \;\; \text{ for } \; 0\leq \theta < 2 \pi \implies 228 | \\ 229 | a_n = - \frac{\partial_{\bar r} H_n( \bar r_S)^{-1}}{2\pi}\int_{0}^{2 \pi}\partial_{\bar r} \psi^E(\bar r_S, \theta)\ee^{-\ii n \theta} d \theta, 230 | \label{eqn:aNeumann} 231 | \ega 232 | 233 | Our main interest is to model point scatterers for which $\bar r_S \to 0$. In this limit the incident wave should converge in an open ball to its Taylor series at $r_S$ (as the wave field should be perfectly smooth), so we can write 234 | \begin{multline} 235 | \psi^E(\bar r_S, \vartheta) = \sum_{k= 0}^\infty \sum_{m= 0}^p \frac{\partial^p \psi^E_0}{\partial_{x}^m \partial_{y}^{p-m}} \frac{ r_S^p (\cos \vartheta)^m (\sin \vartheta)^{p-m} }{m! (p-m)!} 236 | \\ 237 | = \sum_{p= 0}^\infty \sum_{m= 0}^p \frac{\partial^p \psi^E_0}{\partial_{x}^m \partial_{ y}^{p-m}} \frac{ r_S^p }{m! (p-m)!} \sum_{m_1= 0}^m \sum_{p_1= 0}^{p-m} \frac{\ee^{\ii (p-2 m_1-2 p_1)\vartheta} }{(-1)^{p_1} 2^p \ii^{p-m} } \binom{m}{m_1} \binom{p-m}{p_1}, 238 | %Formula checked from scratch in Mathematica! 239 | \label{eqn:PsiTaylor} 240 | \end{multline} 241 | where the subscript $0$ on $\psi^E_0$ and its derivatives means that the expression was evaluated at $x=y=0$ after differentiation. When substituting this expression into the integral in Eq.\eqref{eqn:aDirichlet}, only terms with $p-2 m_1-2 p_1 =n$ will have a non-zero contribution, because every other term after multiplying with $\ee^{-\ii n \vartheta}$ and integrating in $\vartheta$ over $0$ to $2 \pi$ will be zero. Likewise, only terms with $p-2 m_1-2 p_1 =n$ will have a non-zero contribution to Eq.\eqref{eqn:aNeumann}. 242 | So letting $p_1 = j-n/2- m_1$ and $p = 2 j$, so that $p-2 m_1-2 p_1 =n$, and integrating over $\vartheta$ we conclude that 243 | %So letting $p_1 = (|n|- n)/2 +j- m_1$ and $p = |n|+ 2 j$, so that $p-2 m_1-2 k_1 =n$, and integrating over $\vartheta$ we conclude that 244 | %\[ 245 | %\sum_{j= 0}^\infty \sum_{m= 0}^{|n|+ 2 j} \frac{\partial^{ |n|+ 2 j} \psi^E_0}{\partial_{\bar x}^m \partial_{\bar y}^{{ |n|+ 2 j}-m}} \frac{\bar r_S^{ |n|+ 2 j} }{m! (k-m)!} \sum_{m_1= 0}^m \frac{(-1)^{(|n|- n)/2 +j- m_1} }{ 2^{ |n|+ 2 j} \ii^{ |n|+ 2 j -m} } \binom{m}{m_1} \binom{ |n|+ 2 j-m}{(|n|- n)/2 +j- m_1}, 246 | %\] 247 | %So letting $k_1 = (k- n)/2- m_1$ and $k = n+ 2 j$ in Eq.~\eqref{eqn:PsiTaylor} and integrating over $\vartheta$ we can conclude 248 | %\begin{multline} 249 | %p_n= \frac{1}{2 \pi}\int_{0}^{2 \pi} \psi^E(\bar r_s, \vartheta)\ee^{- \ii n \vartheta} d \vartheta = 250 | %\sum_{j= \text{max} \{0, - \ceil{n/2} \} }^\infty \left ( \frac{\bar r_S}{2} \right )^{2 j+n} \sum_{m= 0}^{n+ 2 j} \frac{\partial^{2 j+n} \psi^E_0}{\partial_{\bar x}^m \partial_{\bar y}^{{ 2 j+n} -m}} \frac{C^m_{j, 2 j +n} \ii^{m{-2 j - n}} }{m! ({2 j +n}-m)!} , 251 | %%Formula checked from scratch in Mathematica for the first two C's just below! 252 | %\label{eqn:PsiTaylor} 253 | %\end{multline} 254 | %where 255 | %\be 256 | %C^m_{j, 2 j +n} = \sum_{m_1= \text{max} \{0, -n -j\} }^{\text{min}\{m, j\} } (-1)^{j-m_1} \binom{m}{m_1} \binom{{2 j +n} -m}{j-m_1} 257 | %\en 258 | %\be 259 | %C^m_{j, p} = \sum_{m_1= \text{max} \{0, j-p\} }^{\text{min}\{m, j\}} (-1)^{j-m_1} \binom{m}{m_1} \binom{p -m}{j-m_1}. 260 | %\en 261 | %Note that because $k = n+ 2 j$, $n$ can be negative and the minimal value for $k$ is $0$, then $2 j \geq -n $. Also, as $k_1$ was substituted for $j-m_1$, then $-n -j \leq m_1 \leq j$, which combined with the restriction $0 \leq m_1 \leq m$, becomes 262 | %$ 263 | %\text{max } \{0, -n -j\} \leq m_1 \leq \text{ min } \{m, j\}. 264 | %$ 265 | %\begin{multline} 266 | %p_n= \frac{1}{2 \pi}\int_{0}^{2 \pi} \psi^E(\bar r_s, \vartheta)\ee^{- \ii n \vartheta} d \vartheta = \sum_{j= 0}^\infty \left (\frac{\bar r_S}{2}\right)^{ |n|+ 2 j} \sum_{m= 0}^{|n|+ 2 j} \frac{\partial^{ |n|+ 2 j} \psi^E_0}{\partial_{\bar x}^m \partial_{\bar y}^{{ |n|+ 2 j}-m}} \frac{\ii^{ m-|n|- 2 j } C^m_{n,j} }{m! (|n|+ 2 j -m)!} , 267 | %%Formula checked from scratch in Mathematica for the C's just below! 268 | %\label{eqn:PsiTaylor} 269 | %\end{multline} 270 | %where 271 | %\be 272 | %C^m_{n,j} = \sum_{m_1= m_1^\text{min}}^{m_1^\text{max}} (-1)^{\frac{|n|- n}{2} +j- m_1} \binom{m}{m_1} \binom{ |n|+ 2 j-m}{(|n|- n)/2 +j- m_1}, 273 | %\en 274 | \be 275 | \mathcal P_n\{\psi^E \}= \frac{1}{2 \pi}\int_{0}^{2 \pi} \psi^E(\bar r_s, \vartheta)\ee^{- \ii n \vartheta} d \vartheta = \sum_{j= |n|/2}^\infty \left (\frac{r_S}{2}\right)^{ 2 j} \sum_{m= 0}^{2 j} \frac{\partial^{ 2 j} \psi^E_0}{\partial_{x}^m \partial_{y}^{{ 2 j}-m}} \frac{\ii^{ m-2 j} C^m_{n, j } }{m! (2 j -m)!} , 276 | %Formula checked from scratch in Mathematica for the C's just below! 277 | \label{eqn:PsiTaylor} 278 | \en 279 | where 280 | \be 281 | C^m_{n,j} = \sum_{m_1= \text{max } \{0, m -j -n/2 \}}^{\text{ min } \{m, j -n/2\}} (-1)^{-n/2 +j- m_1} \binom{m}{m_1} \binom{ 2 j -m}{ j -n/2- m_1}, 282 | \en 283 | %where as $p_1$ was substituted for $(|n|- n)/2 +j- m_1$, then from $0\leq p_1 \leq p-m $ we conclude that $m_1 \leq (|n|- n)/2 +j $ and $-(|n|+ n)/2 -j +m \leq m_1$, which combined with the restriction $0 \leq m_1 \leq m$ implies that $m_1^\text{min} = \text{max } \{0, -(|n|+ n)/2 -j +m\}$ and $m_1^\text{max } = \text{ min } \{m, (|n|- n)/2 +j\}$. 284 | where $j$ (same as all the dummy indexes) increases in increments of 1, and as $p_1$ was substituted for $j -n/2- m_1$, then from $0\leq p_1 \leq p-m $ we conclude that $m_1 \leq j -n/2 $ and $-j -n/2 +m \leq m_1$, which combined with the restriction $0 \leq m_1 \leq m$ implies that $ \text{max } \{0, m -j -n/2 \}\leq m_1 \leq \text{ min } \{m, j -n/2\}$. 285 | %For only $j= |n|/2$ the above can be reduced to 286 | %\bga 287 | %p_n= \frac{1}{2 \pi}\int_{0}^{2 \pi} \psi^E(\bar r_s, \vartheta)\ee^{- \ii n \vartheta} d \vartheta = \sum_{m= 0}^{|n|} \frac{\partial^{|n|} \psi^E_0}{\partial_{\bar x}^m \partial_{\bar y}^{{|n|}-m}} \frac{(\ii\sign n)^{m-{|n|}} }{m! ({|n|}-m)!}\frac{\bar r_S^{|n|}}{2^{|n|}} + \mathcal O(\bar r_S^{|n|+2}), 288 | %%\\ 289 | %%dp_n= \frac{1}{2 \pi}\int_{0}^{2 \pi} \partial_{\bar r}\psi^E(\bar r_s, \vartheta)\ee^{- \ii n \vartheta} d \vartheta = \sum_{m= 0}^n \frac{\partial^n \psi^E_0}{\partial_{\bar x}^m \partial_{\bar y}^{n-m}} \frac{\ii^{m-n} }{m! (n-m)!}\frac{\bar r_S^n}{2 ^n}, 290 | %\ega 291 | 292 | For the Neumann boundary condition we need 293 | \be 294 | \frac{1}{2 \pi}\int_{0}^{2 \pi} \partial_{\bar r}\psi^E(\bar r_s, \vartheta)\ee^{- \ii n \vartheta} d \vartheta = k^{-1} \partial_{r_S}\mathcal P_n\{\psi^E \}. 295 | \label{eqn:DPsiTaylor} 296 | \en 297 | 298 | The terms~\eqref{eqn:PsiTaylor} suggest that the series of the scattering operators converge absolutely if $r_S \leq 2$, assuming the derivatives of $\psi^E$ 299 | %, $H_n^{(1)}(\bar r)/ H_n^{(1)}(\bar r_S)$ and $H_n^{(1)}(\bar r)/ \partial_{\bar r} H_n^{(1)}(\bar r_S)$ 300 | are uniformly bounded for every $n$ and $\bar r> \bar r_S$ and $k> \delta>0$ (as $w^E$ will often have a singularity at $k=0$). So for $r_S \leq 2$ we can truncate the series~\eqref{eqn:PsiTaylor} at $j = N_j/2$ for every $n$, which we denote by $\mathcal P_n^{N_j} := \mathcal P_n + \mathcal O(r_s^{N_j +2}) $. 301 | 302 | Eq.~\eqref{eqn:PsiTaylor} together with Eq.~\eqref{eqn:aDirichlet} imply that the scattering operator becomes 303 | \be 304 | \T : \psi^E( \bar r, \theta) \to \psi^S(\bar r, \theta) = - \sum_{n = -\infty}^\infty\frac{H_n(\bar r)}{ H_n(\bar r_S)} \mathcal P_n\{ \psi^E \} \ee^{\ii n \theta} , 305 | \label{eqn:ScatteringOpDirichlet} 306 | \en 307 | which is indeed linear in $\psi^E$. Similarly for the Neumann boundary condition the scattering operator becomes 308 | %\be 309 | %\T : \mathcal C_n(r) \ee^{\ii n \theta} \to - \frac{ H_n(r)}{H_n(r_S)} \mathcal C_n(r_S) \ee^{\ii n \theta} \implies \T\{ \} 310 | %\en 311 | \be 312 | \T : \psi^E(\bar r, \theta) \to \psi^S(\bar r, \theta) = - \sum_{n = -\infty}^\infty \frac{H_n(\bar r)}{\partial_{\bar r} H_n( \bar r_S)} k^{-1} \partial_{r_S} \mathcal P_n\{ \psi^E \} \ee^{\ii n \theta}, 313 | \label{eqn:ScatteringOpNeumann} 314 | \en 315 | which is also linear in $\psi^E$. 316 | 317 | To examine the limit where the particles radius $r_S$ is small in comparison to the wavelength $k r_S \to 0$, we first note that 318 | \be 319 | \frac{H_n(\bar r)}{H_n( \bar r_S)} = H_n(\bar r) \left\{ 320 | \begin{array}{ll} 321 | \frac{\ii \pi (k r_S)^n }{2^{n} (n-1)!} + \mathcal O( k r_S)^{n +2} , \quad \quad \quad \; \;\; n > 0, \\ 322 | \frac{\pi}{ \pi +2 \ii (\gamma_0 +\log(k r_S/2))} + \mathcal O( k r_S)^{2}, \quad n =0, 323 | \end{array} 324 | \right. 325 | \label{eqn:DirchlettHankelExpansion} 326 | \en 327 | and 328 | \be 329 | \frac{H_n(\bar r)}{\partial_{\bar r_S} H_n( \bar r_S)} = H_n(\bar r) \left\{ 330 | \begin{array}{ll} 331 | - \frac{\pi \ii (k r_S)^{n +1} }{2^{n} n!} + \mathcal O( k r_S)^{n +3} , \quad n > 0, \\ 332 | - \frac{\pi \ii }{2} k r_S + \mathcal O( k r_S)^{3}, \quad \quad \quad \; \, n =0, 333 | \end{array} 334 | \right. 335 | \label{eqn:NeumannHankelExpansion} 336 | \en 337 | here $\gamma_0 = 0.5772$, the term $H_n(\bar r)$ has not been asymptotically expanded as $\bar r$ can be of any size, and note that for $n <0$ we have $H_n(\bar r_S) = (-1)^n H_{|n|}(\bar r_S)$. So, for example, if we want to asymptotically expand the scattered wave $w^S$ up too $\mathcal O (\bar r_S^2)$ for the Dirchlett boundary condition, we need to expand 338 | %$\mathcal P_{0} \{ \psi^E\}, \, \mathcal P_{-1} \{ \psi^E\}$ and $ \mathcal P_{1} \{ \psi^E\}$ up too $\mathcal O(\bar r_S^2)$, $\mathcal O(\bar r_S^1)$ and $\mathcal O(\bar r_S^1)$, respectively, for a Dirchlett boundary condition 339 | \bal 340 | \psi^S(\bar r, \theta) = & - \frac{H_1(\bar r)}{ H_1( \bar r_S)} \mathcal P_{-1}\{ \psi^E \} \ee^{- \ii \theta} - \frac{H_0(\bar r)}{ H_0( \bar r_S)}\mathcal P_0\{ \psi^E \} - \frac{H_1(\bar r)}{ H_1( \bar r_S)} \mathcal P_1\{ \psi^E \} \ee^{\ii \theta} + \mathcal O (\bar r_S^2) 341 | \\ 342 | = & - H_0(k r) \frac{\pi \psi^E_0 }{2 \ii \log (\bar r_S /2 )+\pi +2 \ii \gamma_0 } + \mathcal O (\bar r_S^2), 343 | \eal 344 | while to expand $w^S$ up too $\mathcal O (\bar r_S^3)$ for the Neumann boundary condition, we need 345 | \bal 346 | \psi^S(\bar r, \theta) = &- \frac{k^{-1} H_1(\bar r)}{\partial_{\bar r} H_1( \bar r_S)} \partial_{r_S} \mathcal P_1\{ \psi^E \} \ee^{\ii \theta} - \frac{k^{-1} H_0(\bar r)}{\partial_{\bar r} H_0( \bar r_S)} \partial_{r_S} \mathcal P_{0}\{ \psi^E \} 347 | \notag \\ 348 | & - \frac{k^{-1} H_1(\bar r)}{\partial_{\bar r} H_1( \bar r_S)} \partial_{r_S} \mathcal P_{-1}\{ \psi^E \} \ee^{-\ii \theta} + \mathcal O (\bar r_S^3) 349 | \notag \\ 350 | = & \frac{i \pi r^{2}_S}{2} k( \partial_x \psi^E_{0} \cos \theta + \partial_y \psi^E_{0} \sin \theta) H_1(k r ) 351 | \notag \\ 352 | & +\frac{ \ii \pi r_S^2}{4} (\partial_x^2 \psi^E_{0} + \partial_y^2 \psi^E_{0}) H_0 (k r) +\mathcal O(\bar r_S^3). 353 | \eal 354 | 355 | 356 | If we are interested in lower frequencies for $k$, then we must be careful with the dependence that $\psi^E$ has on the wavenumber $k$. In general $\psi^E$ will be a sum of terms of the form 357 | \be 358 | \psi^E_q= H_q (k \|\vec r-\vec r_E\|) \ee^{\ii q \arctan(\vec r - \vec r_E) }, 359 | \label{eqn:PsiETerm} 360 | \en 361 | where $\arctan(x,y) = \arctan (y/x)$, $\vec r_E$ is a constant vector, $c_q$ is determined by boundary conditions and we assume they are uniformly bounded for every $k$. If we are to approximate $\mathcal P_n \{\psi^E_q\}$($k \partial_{r_S}\mathcal P_n \{\psi^E_q\}$) up too $\mathcal O ( r_s^{N_j +2})$($\mathcal O ( r_s^{N_j +1})$) by truncating at $j = N_j/2$, then we should investigate the term left out $j= N_j/2 +1$. Using~\eqref{eqn:PsiETerm} and expanding the term $j= N_j/2+1$ in $\mathcal P_n\{\psi^E_q\}$ in a series of powers of $k$, the lowest order term will be 362 | \be 363 | \mathcal P_n\{\psi^E_q\} -\mathcal P_n^{N_j}\{\psi^E_q\} = k^{-|q|} r_S^{N_j + 2 } \mathcal O(1) + \mathcal O(r_S^{N_j+2} k^{2 -|q|}), 364 | \en 365 | which is multiplied with Eq.~\eqref{eqn:DirchlettHankelExpansion} 366 | %\be 367 | %\frac{H_n(\bar r)}{H_n( \bar r_S)} = \left(\frac{r_S}{r}\right)^{|n|}+ \mathcal O (r_S^{|n|}) \mathcal O (k^2) + \mathcal O (r_S^{|n|+1}) , 368 | %\en 369 | to get the scattering operator. 370 | For the Neumann boundary condition, 371 | \be 372 | k^{-1} \left (\partial_{r_S} \mathcal P_n\{\psi^E_q\} - \partial_{r_S} \mathcal P_n^{N_j}\{\psi^E_q\} \right) = k^{-q-1} r_S^{N_j + 1 } \mathcal O(1) + \mathcal O(r_S^{N_j+1})\mathcal O( k^{1 - q}), 373 | \en 374 | which is multiplied with Eq.~\eqref{eqn:NeumannHankelExpansion}. 375 | %\be 376 | %\frac{H_n(\bar r)}{\partial_{\bar r} H_n( \bar r_S)} = -\frac{k r_S }{|n|} \left(\frac{r_S}{r}\right)^{|n|}+ \mathcal O (r_S^{|n|}) \mathcal O (k^3). 377 | %\en 378 | 379 | \begin{thebibliography}{9} 380 | 381 | \bibitem{martin_2006} 382 | Martin, Paul A. Multiple scattering: interaction of time-harmonic waves with N obstacles. Vol. 107. Cambridge University Press, 2006. 383 | 384 | \end{thebibliography} 385 | 386 | 387 | \end{document} 388 | --------------------------------------------------------------------------------