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:
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)}
