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pkirwin 2021-03-01 15:43:09 -07:00
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\usepackage{fancyhdr}
\usepackage{titlesec}
\usepackage{lipsum}
\usepackage{enumitem}
\usepackage{cancel}
\titleformat{\subsection}[runin]{\normalfont \large \bfseries}
{\thesubsection}{1em}{}
@ -171,29 +171,30 @@ legend({'Numerical','Analytical'},'Location','northwest');
The analytical solution is
\begin{equation}
\Phi(x) = A sin \left( \frac{n \pi}{L} x \right)
\phi(x) = A sin \left( \frac{n \pi}{L} x \right)
\label{eq:analytical}
\end{equation}
In order to normalise this equation it must conform to the following:
\begin{equation}
\int_{0}^{L} \left| \Phi(x) \right| ^2 dx = 1
\int_{0}^{L} \left| \phi(x) \right| ^2 dx = 1.
\label{eq:normalise}
\end{equation}
Using the following identity:
We use the following identity:
\begin{equation}
foo
\int\sin^2{(\alpha x)}\:dx = \frac{1}{2} x -\frac{1}{4\alpha}\sin{(2\alpha x)}.
\label{eq:integral_ident}
\end{equation}
Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate to the above identity.
Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate (\ref{eq:analytical})
to the above identity.
Evaluating the integral gives us the following relationship
\begin{equation*}
\frac{1}{A^2} = \frac{1}{2} L - \frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right) - \frac{1}{2} \cdot 0 + \frac{L}{4n \pi} sin(0)
\frac{1}{A^2} = \frac{1}{2} L - \cancel{\frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right)} - \cancel{\frac{1}{2} \cdot 0} + \cancel{\frac{L}{4n \pi} sin(0)}
\end{equation*}
From this, we find:
@ -224,7 +225,15 @@ legend({'Numerical','Analytical'},'Location','northwest');
\item %1c
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
\item %1ci
\lipsum[1]
From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that
$\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities
$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and
$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where
$\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \ell}{L}$.
Fro
\item %1cii
\lipsum[1]
\item %1ciii