arma-thesis

git clone https://git.igankevich.com/arma-thesis.git
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commit c01ff6a26b1531a9be5432d2a5db5ac39a5f601b
parent 0b364de7556734a488c3a31f88d7533d83ad911f
Author: Ivan Gankevich <igankevich@ya.ru>
Date:   Sun,  9 Apr 2017 16:13:52 +0300

Add explanation why there is no such formula for 3-d case.

Diffstat:
arma-thesis-ru.org | 5+++--
arma-thesis.org | 23++++++++++++-----------
preamble.tex | 1+
3 files changed, 16 insertions(+), 13 deletions(-)

diff --git a/arma-thesis-ru.org b/arma-thesis-ru.org @@ -1319,7 +1319,7 @@ eqref:eq-solution-2d-full до подхода было бы предпочтительнее. Выполняя замену, применяя преобразование Фурье к обеим частям равенства и -подставляя результат в eqref:eq-guessed-sol-3d, . получаем выражение для +подставляя результат в eqref:eq-guessed-sol-3d, получаем выражение для \(\phi\): \begin{equation*} \phi(x,y,z,t) = \InverseFourierY{ @@ -1664,7 +1664,8 @@ arma.plot_ramp_up_interval(label="Интервал разгона") также включается в выражение для коэффициентов \(E(u)\)). Численные эксперименты показывают, что нормировка хоть и позволяет получить адекватное поле скоростей, оно мало отличается от выражений из линейной теории волн, в которых члены с -\(\zeta\) опускаются. +\(\zeta\) опускаются. Как следствие, формула для трехмерного случая не +выводилась. #+name: tab-delta-functions #+caption: Формулы для вычисления \(\Fun{z}\) и \(\FunSecond{z}\) из [[#sec:pressure-2d]], использующие нормировку для исключения неоднозначности определения дельта функции комплексного аргумента. diff --git a/arma-thesis.org b/arma-thesis.org @@ -1874,17 +1874,18 @@ two-dimensional pressure determination problem there are functions \(\FunSecond{z}=\InverseFourierY{\Sinh{2\pi{u}{z}}}{x}\) which has multiple analytic representations and are difficult to compute. Each function is a Fourier transform of linear combination of exponents which reduces to poorly -defined Dirac delta function of a complex argument (see table\nbsp{}[[tab-delta-functions]]). -The usual way of handling this type of functions is to write them as -multiplication of Dirac delta functions of real and imaginary part, however, -this approach does not work here, because applying inverse Fourier transform to -this representation does not produce exponent, which severely warp resulting -velocity field. In order to get unique analytic definition normalisation factor -\(1/\Sinh{2\pi{u}{h}}\) (which is also included in formula for \(E(u)\)) may be -used. Despite the fact that normalisation allows to obtain adequate velocity -potential field, numerical experiments show that there is little difference -between this field and the one produced by formulae from linear wave theory, in -which terms with \(\zeta\) are omitted. +defined Dirac delta function of a complex argument (see +table\nbsp{}[[tab-delta-functions]]). The usual way of handling this type of +functions is to write them as multiplication of Dirac delta functions of real +and imaginary part, however, this approach does not work here, because applying +inverse Fourier transform to this representation does not produce exponent, +which severely warp resulting velocity field. In order to get unique analytic +definition normalisation factor \(1/\Sinh{2\pi{u}{h}}\) (which is also included +in formula for \(E(u)\)) may be used. Despite the fact that normalisation allows +to obtain adequate velocity potential field, numerical experiments show that +there is little difference between this field and the one produced by formulae +from linear wave theory, in which terms with \(\zeta\) are omitted. As a result, +the formula for three-dimensional case was not derived. #+name: tab-delta-functions #+caption: Formulae for computing \(\Fun{z}\) and \(\FunSecond{z}\) from [[#sec:pressure-2d]], that use normalisation to eliminate uncertainty from definition of Dirac delta function of complex argument. diff --git a/preamble.tex b/preamble.tex @@ -47,6 +47,7 @@ \newcommand{\Fun}[1]{\mathcal{D}_1\left(x,#1\right)} \newcommand{\FunSecond}[1]{\mathcal{D}_2\left(x,#1\right)} \newcommand{\FunThird}[1]{\mathcal{D}_2\left(x,#1\right)} +\newcommand{\FunThreeD}[1]{\mathcal{D}_3\left(x,y,#1\right)} \newcommand{\Sinh}[1]{\cosh\left(#1\right)} \newcommand{\SinhX}[1]{\sinh\left(#1\right)}