Cylindrical Waves Equation Formula Latex Code and Summary

Cylindrical Waves

Tags: #physics #cylindrical waves

Equation

$$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\ u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))$$

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

$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\ u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))$

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))

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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2023-01-28 00:00
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