Computation of Infinite Depth Green's Function - Revision history

Meylan at 10:34, 10 September 2009

2009-09-10T10:34:41Z

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			{{pages to be deleted}}

	The normal incidence Green's function for infinite depth has already been published [[Squire and Dixon 2000]], [[Dixon and Squire 2001]], the solution being expressed in terms of the auxilary sine and cosine integrals, or equivalently in terms of the exponential integral (see [[#Normal Incidence Simplification \|Appendix C.2.3]]). Similarly, the finite depth Green's function was derived by [[Evans and Porter 2003a]] and was also presented in [[Section 3.1]] of this thesis for both normal and oblique incidence. However, at present, the required derivatives <math>g^{(n)}(x)</math> of the infinite depth Green's function for oblique incidence may only be calculated exactly when <math>x=0^\pm</math>. Formulae for these quantities were published by [[Williams and Squire 2002]] and are also derived below in [[#Exact Calculation of Derivatives at the Origin \|Appendix C.2.2]]. Although the aforementioned derivatives may be calculated when <math>x\neq0</math> by Fast Fourier Transform, as shown in [[Section 3.2]], this is often not the most efficient approach, especially when the Green's function is only required to be evaluated at a few points. The following two sections outline two approaches that are more suitable for such a scenario.		The normal incidence Green's function for infinite depth has already been published [[Squire and Dixon 2000]], [[Dixon and Squire 2001]], the solution being expressed in terms of the auxilary sine and cosine integrals, or equivalently in terms of the exponential integral (see [[#Normal Incidence Simplification \|Appendix C.2.3]]). Similarly, the finite depth Green's function was derived by [[Evans and Porter 2003a]] and was also presented in [[Section 3.1]] of this thesis for both normal and oblique incidence. However, at present, the required derivatives <math>g^{(n)}(x)</math> of the infinite depth Green's function for oblique incidence may only be calculated exactly when <math>x=0^\pm</math>. Formulae for these quantities were published by [[Williams and Squire 2002]] and are also derived below in [[#Exact Calculation of Derivatives at the Origin \|Appendix C.2.2]]. Although the aforementioned derivatives may be calculated when <math>x\neq0</math> by Fast Fourier Transform, as shown in [[Section 3.2]], this is often not the most efficient approach, especially when the Green's function is only required to be evaluated at a few points. The following two sections outline two approaches that are more suitable for such a scenario.

Meylan: /* Normal Incidence Simplification */

2007-03-07T16:08:53Z

Normal Incidence Simplification

← Older revision		Revision as of 16:08, 7 March 2007
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	where <math>\epsilon_n'=\sgn\big(\mbox{e}[\gamma_n]\big)</math>. The cosine term needs to be added to allow for the difference between taking the Cauchy Principal value of the integral (3.4) and letting the poles <math>\pm\alpha_0</math> approach the real line from the upper or lower half plane as we did, while <math>g_\mbox{cos}</math> is the auxiliary cosine function [[Abramowitz and Stegun ~~1965~~]] which may be written as		where <math>\epsilon_n'=\sgn\big(\mbox{e}[\gamma_n]\big)</math>. The cosine term needs to be added to allow for the difference between taking the Cauchy Principal value of the integral (3.4) and letting the poles <math>\pm\alpha_0</math> approach the real line from the upper or lower half plane as we did, while <math>g_\mbox{cos}</math> is the auxiliary cosine function [[Abramowitz and Stegun 1964]] which may be written as

	<center><math>		<center><math>

Meylan at 16:07, 7 March 2007

2007-03-07T16:07:09Z

← Older revision		Revision as of 16:07, 7 March 2007
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	The normal incidence Green's function for infinite depth has already been published [[Squire and Dixon 2000]], [[Dixon and Squire 2001]], the solution being expressed in terms of the auxilary sine and cosine integrals, or equivalently in terms of the exponential integral (see [[#Normal Incidence Simplification \|Appendix C.2.3]]). Similarly, the finite depth Green's function was derived by [[Evans and Porter 2003a]] and was also presented in [[Section 3.1]] of this thesis for both normal and oblique incidence. However, at present, the required derivatives <math>g^{(n)}(x)</math> of the infinite depth Green's function for oblique incidence may only be calculated exactly when <math>x=0^\pm</math>. Formulae for these quantities were published by [[Williams and Squire 2002]] and are also derived below in [[#Exact Calculation of Derivatives at the Origin \|Appendix C.2.2]]. Although the aforementioned derivatives may be calculated when <math>x\neq0</math> by Fast Fourier Transform, as shown in [[Section 3.2]], this is often not the most efficient approach, especially when the Green's function is only required to be evaluated at a few points. The following two sections outline two approaches that are more suitable for such a scenario.		The normal incidence Green's function for infinite depth has already been published [[Squire and Dixon 2000]], [[Dixon and Squire 2001]], the solution being expressed in terms of the auxilary sine and cosine integrals, or equivalently in terms of the exponential integral (see [[#Normal Incidence Simplification \|Appendix C.2.3]]). Similarly, the finite depth Green's function was derived by [[Evans and Porter 2003a]] and was also presented in [[Section 3.1]] of this thesis for both normal and oblique incidence. However, at present, the required derivatives <math>g^{(n)}(x)</math> of the infinite depth Green's function for oblique incidence may only be calculated exactly when <math>x=0^\pm</math>. Formulae for these quantities were published by [[Williams and Squire 2002]] and are also derived below in [[#Exact Calculation of Derivatives at the Origin \|Appendix C.2.2]]. Although the aforementioned derivatives may be calculated when <math>x\neq0</math> by Fast Fourier Transform, as shown in [[Section 3.2]], this is often not the most efficient approach, especially when the Green's function is only required to be evaluated at a few points. The following two sections outline two approaches that are more suitable for such a scenario.

	The first uses a partial fractions decomposition to reduce them (the required derivatives of <math>g</math>, or more specifically of <math>g_1</math>) into the sum of five simpler transforms which each satisfy linear inhomogeneous second order ordinary differential equations (ODEs). The inhomogeneous term is the zeroth order modified Bessel function of the second kind, <math>K_0</math>, which has a standard series representation ([[Abramowitz and Stegun ~~1965~~]], p375). Using a method that is similar to both the Method of Undetermined Coefficients ([[Boyce and DiPrima 1997]], p164 ff.) and the Method of Frobenius ([[Boyce and DiPrima 1997]], p262 ff.), we search for solutions that have similar expansions to that of <math>K_0</math>, and obtain simple recurrence relations for the series coefficients. The series obtained are convergent for all values of <math>x</math>, with less terms needed in the expansion when <math>x</math> is smaller.		The first uses a partial fractions decomposition to reduce them (the required derivatives of <math>g</math>, or more specifically of <math>g_1</math>) into the sum of five simpler transforms which each satisfy linear inhomogeneous second order ordinary differential equations (ODEs). The inhomogeneous term is the zeroth order modified Bessel function of the second kind, <math>K_0</math>, which has a standard series representation ([[Abramowitz and Stegun 1964]], p375). Using a method that is similar to both the Method of Undetermined Coefficients ([[Boyce and DiPrima 1997]], p164 ff.) and the Method of Frobenius ([[Boyce and DiPrima 1997]], p262 ff.), we search for solutions that have similar expansions to that of <math>K_0</math>, and obtain simple recurrence relations for the series coefficients. The series obtained are convergent for all values of <math>x</math>, with less terms needed in the expansion when <math>x</math> is smaller.

	A second approach, which uses a contour integral to transform the inversion integral into one that may be calculated more easily by numerical quadrature, is more efficient for larger values of <math>x</math>. It is described below in [[#Contour Integration \|Section C.2]], but first we will introduce the series expansion method.		A second approach, which uses a contour integral to transform the inversion integral into one that may be calculated more easily by numerical quadrature, is more efficient for larger values of <math>x</math>. It is described below in [[#Contour Integration \|Section C.2]], but first we will introduce the series expansion method.

Meylan: /* Ordinary Differential Equations */

2007-03-07T16:06:17Z

Ordinary Differential Equations

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	\end{matrix}</math></center>		\end{matrix}</math></center>

	where <math>\gamma_e</math> is the Euler constant, <math>\chi_m=\sum_{r=1}^m1/r</math>, and <math>I_0</math> is the modified Bessel function of the first kind (also given by [[Abramowitz and Stegun ~~1965~~]]):		where <math>\gamma_e</math> is the Euler constant, <math>\chi_m=\sum_{r=1}^m1/r</math>, and <math>I_0</math> is the modified Bessel function of the first kind (also given by [[Abramowitz and Stegun 1964]]):

	<center><math>\begin{matrix}		<center><math>\begin{matrix}

Meylan: /* Ordinary Differential Equations */

2007-03-07T16:06:00Z

Ordinary Differential Equations

← Older revision		Revision as of 16:06, 7 March 2007
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	where <math>K_0</math> is a zeroth order modified Bessel function of the second type ([[Abramowitz and Stegun ~~1965~~]]). For <math>\|=Arg= (w)\|<\pi</math>, it has the expansion		where <math>K_0</math> is a zeroth order modified Bessel function of the second type ([[Abramowitz and Stegun 1964]]). For <math>\|=Arg= (w)\|<\pi</math>, it has the expansion

	<center><math>\begin{matrix}		<center><math>\begin{matrix}

Meylan: /* Normal Incidence Simplification */

2007-03-07T16:05:10Z

Normal Incidence Simplification

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	\end{matrix}</math></center>		\end{matrix}</math></center>

	where <math>E_1</math> is the well-known exponential integral function ([[Abramowitz and Stegun ~~1965~~]], p228ff.) defined as follows		where <math>E_1</math> is the well-known exponential integral function ([[Abramowitz and Stegun 1964]], p228ff.) defined as follows

	<center><math>		<center><math>

Syan077: /* Normal Incidence Simplification */

2006-12-25T09:05:09Z

Normal Incidence Simplification

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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	2\gamma_n^2g_{3n}(x)&=\int_l^\infty\left(\frac2k-\frac1{k+\mbox{i}\gamma_n}-\frac1{k-\mbox{i}\gamma_n}\right)\mbox{e}^{-\kappa x}\mbox{d}\kappa\\		&2\gamma_n^2g_{3n}(x)=\int_l^\infty\left(\frac2k-\frac1{k+\mbox{i}\gamma_n}-\frac1{k-\mbox{i}\gamma_n}\right)\mbox{e}^{-\kappa x}\mbox{d}\kappa\\
	&=2K_0(lx)-\int_l^\infty\left(\frac1{k+\mbox{i}\gamma_n}+\frac1{k-\mbox{i}\gamma_n}\right)\mbox{e}^{-\kappa x}\mbox{d}\kappa. &\qquad (C.28)		&=2K_0(lx)-\int_l^\infty\left(\frac1{k+\mbox{i}\gamma_n}+\frac1{k-\mbox{i}\gamma_n}\right)\mbox{e}^{-\kappa x}\mbox{d}\kappa. &\qquad (C.28)
	\end{matrix}</math></center>		\end{matrix}</math></center>
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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	g_3^{(2n)}(x)&=-\frac1\pi\sum_{m=-2}^2A_m'(\mbox{i}\alpha_m)^{2n}g_{3m}(x)\\		&g_3^{(2n)}(x)=-\frac1\pi\sum_{m=-2}^2A_m'(\mbox{i}\alpha_m)^{2n}g_{3m}(x)\\
	&=\frac1{2\pi}\sum_{m=-2}^2A_m'(\mbox{i}\alpha_m)^{2n}\int_l^\infty\left(\frac1{k+\mbox{i}\gamma_n}+\frac1{k-\mbox{i}\gamma_n}\right)\mbox{e}^{-\kappa x}\mbox{d}\kappa. &\qquad (C.29)		&=\frac1{2\pi}\sum_{m=-2}^2A_m'(\mbox{i}\alpha_m)^{2n}\int_l^\infty\left(\frac1{k+\mbox{i}\gamma_n}+\frac1{k-\mbox{i}\gamma_n}\right)\mbox{e}^{-\kappa x}\mbox{d}\kappa. &\qquad (C.29)
	\end{matrix}</math></center>		\end{matrix}</math></center>
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	<center><math>		<center><math>
	E_1(w)=\int_w^\infty \frac{\mbox{e}^{-t}}t \mbox{d} t		E_1(w)=\int_w^\infty\frac{\mbox{e}^{-t}}t\mbox{d}t=-\gamma-\log w - \sum_{n=1}^\infty \frac{(-w)^n}{n\,n!} \quad \mbox{for} \ \mid \mbox{Arg} (w) \mid < \pi. \qquad (C.31)
	=-\gamma-\log w-\sum_{n=1}^\infty\frac{(-w)^n}{n\,n!}
	\quad &\mbox{for} \ \|\~~text~~{Arg= (w)\|<\pi }. \qquad (C.31)
	</math></center>		</math></center>

	Note that, since no imaginary solutions exist for the infinite depth dispersion relation, <math>\big\|=Arg= [\pm \mbox{i}\gamma_nx]\big\|<\pi</math>, so the above formulae are valid. Also note that the <math>g_3^{2n}</math> will be everywhere bounded, despite the logarithmic singularities in <math>E_1</math>. This also follows from the rules (C.26), allowing us to deduce that taking the limit in (C.30) as <math>x \to 0^+</math> gives		Note that, since no imaginary solutions exist for the infinite depth dispersion relation, <math>\big\| \mbox{Arg} [\pm \mbox{i}\gamma_nx]\big\|<\pi</math>, so the above formulae are valid. Also note that the <math>g_3^{2n}</math> will be everywhere bounded, despite the logarithmic singularities in <math>E_1</math>. This also follows from the rules (C.26), allowing us to deduce that taking the limit in (C.30) as <math>x \to 0^+</math> gives

	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	g_3^{(2n)}(0^+)&=-\frac1{2\pi}\sum_{m=-2}^2A_m'(\mbox{i}\gamma_m)^{2n}		&g_3^{(2n)}(0^+)=-\frac1{2\pi}\sum_{m=-2}^2A_m'(\mbox{i}\gamma_m)^{2n}
	\big(\log(\mbox{i}\gamma_m)+\log(-\mbox{i}\gamma_m)\big)\\		\big(\log(\mbox{i}\gamma_m)+\log(-\mbox{i}\gamma_m)\big)\\
	&=-\frac1\pi\sum_{m=-2}^2A_m'(\mbox{i}\gamma_m)^{2n}\log\gamma_m		&=-\frac1\pi\sum_{m=-2}^2A_m'(\mbox{i}\gamma_m)^{2n}\log\gamma_m
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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	g(x-\xi)&=\frac1{a^2}\partial^2_{z'\zeta'}G_\mbox{SD}(x'-\xi',0,0)+\mbox{i} A_0\cos\gamma_0x\\		&g(x-\xi)=\frac1{a^2}\partial^2_{z'\zeta'}G_\mbox{SD}(x'-\xi',0,0)+\mbox{i} A_0\cos\gamma_0x\\
	&=\mbox{i}\sum_{m=-2}^0A'_m\epsilon_me^{\mbox{i}\epsilon_m\gamma_mx}		&=\mbox{i}\sum_{m=-2}^0A'_m\epsilon_me^{\mbox{i}\epsilon_m\gamma_mx}
	+\frac1\pi\sum_{m=-2}^2A_m'g_\mbox{cos}(\epsilon_m'\gamma_mx), &\qquad (C.36)		+\frac1\pi\sum_{m=-2}^2A_m'g_\mbox{cos}(\epsilon_m'\gamma_mx), &\qquad (C.36)

Syan077: /* Exact Calculation of Derivatives at the Origin */

2006-12-25T08:57:48Z

Exact Calculation of Derivatives at the Origin

← Older revision		Revision as of 08:57, 25 December 2006
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	\end{matrix}</math></center>		\end{matrix}</math></center>

	where <math>\Theta_n=(\gamma_n+\alpha_n)/(\gamma_n-\alpha_n)</math>. The latter step is justifed since if <math>\im[\alpha_n]>0</math> the argument of the first logarithm will be in the first quadrant, while the argument of the second logarithm will be in the second quadrant. This means that we can now write		where <math>\Theta_n=(\gamma_n+\alpha_n)/(\gamma_n-\alpha_n)</math>. The latter step is justifed since if <math>\mathfrak{im}[\alpha_n]>0</math> the argument of the first logarithm will be in the first quadrant, while the argument of the second logarithm will be in the second quadrant. This means that we can now write

	<center><math>		<center><math>
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	</math></center>		</math></center>

	Now, for small <math>l</math>, <math>\alpha_n \sim \epsilon_n\gamma_n\big(1-l^2/2\gamma_n+O(l^4)\big)</math>, where <math>\epsilon_n=\mbox{sgn}(\im[\gamma_n])</math>, so at first glance it seems like the above expression will become singular as <math>l \to 0</math>. However, this is not the case. If we define <math>A'_n=A_n\alpha_n/\gamma_n=-\gamma_n^2/(4\gamma_n^5+1)</math>, <math>A_n\sim\epsilon_nA_n'+O(l^2)</math> and it is easily shown that for <math>n=0,1,2,3</math> they satisfy the rules		Now, for small <math>l</math>, <math>\alpha_n\sim\epsilon_n\gamma_n\big(1-l^2/2\gamma_n+ O(l^4)\big)</math>, where <math>\epsilon_n=\mbox{sgn}(\mathfrak{im}[\gamma_n])</math>, so at first glance it seems like the above expression will become singular as <math>l \to 0</math>. However, this is not the case. If we define <math>A'_n=A_n\alpha_n/\gamma_n=-\gamma_n^2/ (4\gamma_n^5+1)</math>, <math>A_n\sim\epsilon_nA_n'+O(l^2)</math> and it is easily shown that for <math>n=0,1,2,3</math> they satisfy the rules

	<center><math>		<center><math>
	\sum_{m=-2}^{2}A_m'\gamma_m^{2n}=0 \quad &\mbox{for} n=0,\ldots,3, \qquad (C.26)		\sum_{m=-2}^{2}A_m'\gamma_m^{2n}=0 \quad \mbox{for} \ n=0,\ldots,3, \qquad (C.26)
	</math></center>		</math></center>

Syan077: /* Contour Integration */

2006-12-25T08:51:59Z

Contour Integration

Show changes

Syan077: /* Trial Series Solution */

2006-12-25T08:18:23Z

Trial Series Solution

← Older revision		Revision as of 08:18, 25 December 2006
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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	J''(w)+\alpha^2J(w)=K_0(w)\quad=for ~~<math>~~w>0~~</math>=~~ , &\qquad (C.8a)\\		J''(w)+\alpha^2J(w)=K_0(w) \quad \mbox{for} \ w>0, &\qquad (C.8a)\\
	J(0^+)=J'(0^+)=0. &\qquad (C.8b)		J(0^+)=J'(0^+)=0. &\qquad (C.8b)
	\end{matrix}</math></center>		\end{matrix}</math></center>
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	<center><math>		<center><math>
	J(w)=\sum_{m=0}^\infty\big(b_m+c_m\log w\big)w^{2m+2}, &\qquad (C.9)		J(w) = \sum_{m=0}^\infty \big( b_m + c_m \log w \big) w^{2m+2}, \qquad (C.9)
	</math></center>		</math></center>

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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	J''(w)+\alpha^2J&=2b_0+3c_0+2c_0\log w\\		&J''(w)+\alpha^2J=2b_0+3c_0+2c_0\log w\\
	&\quad+\sum_{m=1}^\infty\big( (2m+2)(2m+1)b_{m}+\alpha^2b_{m-1}+(4m+3)c_m\big)w^{2m}\\		&\quad+\sum_{m=1}^\infty\big( (2m+2)(2m+1)b_{m}+\alpha^2b_{m-1}+(4m+3)c_m\big)w^{2m}\\
	&\quad+\sum_{m=1}^\infty\big((2m+2)(2m+1)c_m+\alpha^2c_{m-1}\big)w^{2m}\log w. &\qquad (C.11)		&\quad+\sum_{m=1}^\infty\big((2m+2)(2m+1)c_m+\alpha^2c_{m-1}\big)w^{2m}\log w. &\qquad (C.11)
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	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	(2m+2)(2m+1)b_{m}+\alpha^2b_{m-1}+(4m+3)c_{m}=a_{m}d_{m}		(2m+2)(2m+1)b_{m}+\alpha^2b_{m-1}+(4m+3)c_{m}=a_{m}d_{m}
	\quad&=for ~~<math>~~m\geq1~~</math>=~~ , &\qquad (C.12a)\\		\quad &\mbox{for} \ m\geq1, &\qquad (C.12a)\\
	(2m+2)(2m+1)c_m+\alpha^2c_{m-1}=-a_m \quad & \mbox{for} m\geq1, &\qquad (C.12b)		(2m+2)(2m+1)c_m+\alpha^2c_{m-1}=-a_m \quad & \mbox{for} \ m\geq1, &\qquad (C.12b)
	\end{matrix}</math></center>		\end{matrix}</math></center>