\documentclass[a4	paper]{article}

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{setspace}
\usepackage{graphicx}
\usepackage{epstopdf}

\usepackage{letltxmacro}
\makeatletter
\let\oldr@@t\r@@t
\def\r@@t#1#2{%
\setbox0=\hbox{$\oldr@@t#1{#2\,}$}\dimen0=\ht0
\advance\dimen0-0.2\ht0
\setbox2=\hbox{\vrule height\ht0 depth -\dimen0}%
{\box0\lower0.4pt\box2}}
\LetLtxMacro{\oldsqrt}{\sqrt}
\renewcommand*{\sqrt}[2][\ ]{\oldsqrt[#1]{#2}}
\makeatother

\oddsidemargin=20pt
\topmargin=-1cm
\textheight=650pt
\textwidth=414pt
\footskip=30pt

\begin{document}

\title{Sound Lens}
\author{David Harrison}
\date{March 2012}
\maketitle

\section{Introduction}
\normalsize
~
~
In the 19th century, a device was invented for the purpose of focusing sound waves. It consisted of a balloon, the same shape as a converging lens, filled with a gas, such that the velocity of sound in the balloon was different from the velocity in air.

~

Find the focal length of such a lens, having radii of curvature of 2m, filled with N$_{2}$O, assuming the gases to be ideal and the temperature inside the balloon to be the same as the surrounding air. Neglect any effect from the balloon material.

~

The effective molecular mass of air can be taken as 29 and the relative molecular mass of N$_{2}$O is 44. The ratio of principal specific heats, $\gamma$, is 1.40 for diatomic and 1.33 for triatomic gases.

~

\section{Speed of Sound}
~
~
Since the temperature is not specified, it is not possible to determine the speed of sound in either of the gases. However, a ratio of the two can be determined.

~

The formula for the speed of sound in a gas is given by:

\begin{align}
	v &= \sqrt{\dfrac{\gamma RT}{M}}
\nonumber
\intertext{\indent Since air is almost completely composed of diatomic gases, $\gamma_{air}$ can be taken as 1.40, and $\gamma_{nitrous}$ as 1.33.}
	v_{air} &= \sqrt{\dfrac{1.40 * RT}{29}}
\nonumber\\
	v_{nitrous} &= \sqrt{\dfrac{1.33 * RT}{44}}
\nonumber\\
	\dfrac{v_{air}}{v_{nitrous}} &= \sqrt{\dfrac{1.40 * RT * 44}{1.33 * RT * 29}}
\nonumber\\
	\dfrac{v_{air}}{v_{nitrous}} &= \sqrt{\dfrac{140 * 44}{133 * 29}}
\nonumber\\
	\dfrac{v_{air}}{v_{nitrous}} &= \sqrt{\dfrac{6160}{3857}}
\end{align}

\section{Lens Makers Equation}
~
~
The following formula is the lensmakers equation, where $n_{1}$ is the refractive index of the material outside the lens, and $n_{2}$ is the refractive index of the material inside the lens:
\begin{align*}
	\dfrac{1}{f} &= \left( \dfrac{n_{2}}{n_{1}} - 1\right) \left( \dfrac{1}{R_{1}} - \dfrac{1}{R_{2}} \right)
\intertext{\indent Now recall that for light, the refractive index, n, is given by:}
	n &= \dfrac{c}{v}
\intertext{\indent Where c is the speed of light in a vacuum and v is the speed of light through some medium. However, sound waves cannot travel in a vacuum, so a similar definition does not make much sense for sound, but thanks to the the division of one refractive index by another in the lens makers equation, the speed of light terms cancel away and leave a simple ratio of velocities:}
	\dfrac{n_{2}}{n_{1}} &= \dfrac{c * v_{1}}{c * v_{2}}\\
	\dfrac{n_{2}}{n_{1}} &= \dfrac{v_{1}}{v_{2}}\\
	\dfrac{n_{nitrous}}{n_{air}} &= \dfrac{v_{air}}{v_{nitrous}}
\intertext{\indent Recall equation 1 from section 2, where this is exactly what was calculated:}
	\dfrac{v_{air}}{v_{nitrous}} &= \sqrt{\dfrac{6160}{3857}}
\intertext{\indent Using this, the focal length can be calculated:}
	\dfrac{1}{f} &= \left( \sqrt{\dfrac{6160}{3857}} - 1\right) \left( \dfrac{1}{R_{1}} - \dfrac{1}{R_{2}} \right)\\
	\dfrac{1}{f} &= \left( \sqrt{\dfrac{6160}{3857}} - 1\right) \left( \dfrac{1}{2} - \dfrac{1}{-2} \right)\\
	\dfrac{1}{f} &= \left( \sqrt{\dfrac{6160}{3857}} - 1\right) \left( \dfrac{1}{2} + \dfrac{1}{2} \right)\\
	\dfrac{1}{f} &= \left( \sqrt{1.5971} - 1\right) * 1\\
	\dfrac{1}{f} &= 1.26 - 1\\
	\dfrac{1}{f} &= 0.26\\
	f &= 3.8\ m
\end{align*}

Therefore the focal length of this particular sound lens is 3.8 metres.

\end{document}