91影视

It has been asked to repeat problem 17.

Laplace鈥檚 equation in cylindrical coordinates is,

$$\begin{gathered} {\mathbf{鈭�}}^{\mathbf{2}}\mathit{u}\mathbf{=}\frac{\mathbf{1}}{\mathbf{r}}\frac{\mathbf{鈭�}}{\mathbf{鈭�}\mathbf{r}}\left(r\frac{鈭�u}{鈭�r}\right)\mathbf{+}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{2}}}\frac{{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{u}}{\mathbf{鈭�}{\mathbf{胃}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{u}}{\mathbf{鈭�}{\mathbf{z}}^{\mathbf{2}}} \\ \mathbf{=}\mathbf{0} \end{gathered}$$

And to separate the variable the solution assumed is of the form$\mathit{u}\mathbf{=}\mathit{R}\left(r\right)\mathit{螛}\left(胃\right)\mathit{Z}\left(z\right)$.

Take the$\left(x,y\right)$plane to be the plane of the circular support and take the origin at its center. Let$z\left(x,y,t\right)$be the displacement of the membrane from the$\left(x,y\right)$plane.

Then z satisfies the wave equation.

${鈭�}^{2}z=\frac{1}{v^2}\frac{{鈭�}^{2}z}{鈭�t^2}$

Put$z=F\left(x,y\right)T\left(t\right)$ into the previous differential equation a space differential equation is obtained and a time differential equation. Since the question is solving the problem for a circular membrane write${鈭�}^2$ in polar coordinates and then separate the propose a solution following the separation of variables for the space equation. So, the general solution is $z=\cos \left(n胃\right)\cos \left(kvt/a\right)$the circular membrane the frequencies are obtained as mentioned below.