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Chris Spencer edited Theory .tex
about 10 years ago
Commit id: 78eea6f39bb9cb0ca4a3ab2d01b2fac0bf86661f
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\section{Theory}
The wave equation is given as \[\nabla^2p-\frac{1}{c^2}\frac{\partial^2\rho}{\partial^2t}=0\] where $p$ is the pressure, $\rho$ is the density, and $c=\sqrt{\frac{\gamma RT}{M}}$. $\gamma$ is the ratio of specific heat at constant pressure and volume $=\frac{C_p}{C_v}$, R is the gas constant, and M is the molecular weight of air. In this experiment it is assumed that air is an ideal gas. In a 1D box, boundary conditions dictate that at x=0 and x=L the velocity of the wave will be zero. The velocity,u, is connected to pressue by the euler equation
\[\frac{\partial{u}}{\partial{t}}=-\frac{1}{\rho}\frac{{\partial{\sigma{p}}{\partial{x}}}\] \[\frac{\partial{u}}{\partial{t}}=-\frac{1}{\rho}\frac{{\partial{\sigma{p}}{\partial x}\]