ROUGH DRAFT authorea.com/6690

# Introduction

The purpose of the lab was to find the normal modes of rectangular and cylindrical geometries and using that data of these frequencies obtain the speed of sound. Before the lab was started the first 33 modes for the rectangular box were found theoretically up to the $$(4,0,0)$$ mode to compare to our results and the first 17 modes of the cylindrical up to the $$(4,2,0)$$. The physics used to to find the resonant frequencies of the box come from seperation of variables of the pressure wave equation and the frequencies for the cylindrical geometry were found seperating variables again but in cylindrical coordinates to the pressure wave equation and examining the zeroes of the deriviatives of the bessel function. For practial purposes, c was taken to be $$345 \frac{m}{s}$$ to approximate resonant frequencies.

# Theory

## 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 $f=\frac{c}{2}\sqrt{\left(\frac{n_x}{L_x}\right)^2+\left(\frac{n_y}{L_y}\right)^2+\left(\frac{n_z}{L_z}\right)^2}$

## 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 are nodes. Now the resonant frequencies are given by $f_{m,n,n_z}=\frac{c}{2}\sqrt{\left(\frac{j'_{mn}}{\pi R}\right)^2+\left(\frac{n_z}{L_z}\right)^2}$ as before c is assumed to be $$345 \frac{m}{s}$$.R is the radius if the cylinder, $$L_z$$ is the height of the cylinder, and n is the mode number.