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Chris Spencer edited Theory .tex
about 10 years ago
Commit id: db792bd8c27c15f4122daf7192c14e866b302a53
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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 \[\frac{\partial u}{\partial t}=-\frac{1}{\rho}\frac{\partial \delta p}{\partial x}\] where $\delta p$ is the pressure oscillation. Enforcing boundary conditions gives that the pressure oscillation is a maximum at a wall. The resulting pressure is $\delta p=\delta p_0e^{i\omega
t}\coskx$ t}\cos kx$