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\section{Theory}  \subsection{Rectangular Geometry} 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 $\delta p=\delta p_0e^{i\omega t}\cos kx$ and since its a max at the wall then its derivative is zero there. Again enforcing boundary conditions yields $k_i=\frac{n_i\pi}{L_i}$, where i is used for a general component. Recall that $\lambda _i=\frac{2\pi}{k_i}$, now the resonant frequencies are given by $f_{ni}=\frac{c}{\lambda i}$. This generalizes to the 3D rectangular box used in the experiment with the same boundary conditions just in the x,y,z direction and this yields the resonant frequencies \[f=\frac{c}{2}\sqrt{\left(\frac{n_x}{L_x}\right)^2+\left(\frac{n_y}{L_y}\right)^2+\left(\frac{n_z}{L_z}\right)^2}\]  \subsection{Cylindrical Geometry}  This is solved in a similar way as the rectangular geometry but instead we have $\nabla^2$ in cylindrical coordinates. Now $\delta p=\delta p_0J_m\left(k_rr\right)\cos(m\theta)\cos\left(k_zz\right)e^{i\omega t}$. m is an integer due to the periodicity of $\delta p$.Here look at the points where the bessul function has zeros in slope, thats where the change in pressure will be zero and hence a node. That is when $\frac{\partial J_m\left(k_rr\right)}{\partial r}=0$. Define $j'_{mn}$ as the points where there arethe  nodes. Now the resonant frequencies are given by \[f_{m,n,n_z}=\frac{c}{2}\sqrt{\left(\frac{j'_{mn}}{\pi R}\right)^2+\left(\frac{h_z}{L_z}\right)^2}\]