Cylindrical Waves
Tags: #physics #cylindrical wavesEquation
$$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\ u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))$$Latex Code
\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\
u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))
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Introduction
Equation
Latex Code
\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\
u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))
Explanation
When the wave is cylindrical symmetry, the homogeneous wave equation becomes as above. This is a Bessel equation, with solutions that can be written as Hankel functions. For sufficient large values of r.
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