Solution Found!
The wave equation in polar coordinates is urr + 1 r ur + 1 r2 u = 1 a2 utt . Show that
Chapter 11, Problem 4(choose chapter or problem)
The wave equation in polar coordinates is
\(u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}=\frac{1}{a^{2}} u_{t t}\).
Show that if \(u(r, \theta, t)=R(r) \Theta(\theta) T(t)\), and T satisfy the ordinary differential equations
\(\begin{aligned}
r^{2} R^{\prime \prime}+r R^{\prime}+\left(\lambda^{2} r^{2}-n^{2}\right) R & =0 \\
\Theta^{\prime \prime}+n^{2} \Theta & =0 \\
T^{\prime \prime}+\lambda^{2} a^{2} T & =0
\end{aligned}\)
Questions & Answers
QUESTION:
The wave equation in polar coordinates is
\(u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}=\frac{1}{a^{2}} u_{t t}\).
Show that if \(u(r, \theta, t)=R(r) \Theta(\theta) T(t)\), and T satisfy the ordinary differential equations
\(\begin{aligned}
r^{2} R^{\prime \prime}+r R^{\prime}+\left(\lambda^{2} r^{2}-n^{2}\right) R & =0 \\
\Theta^{\prime \prime}+n^{2} \Theta & =0 \\
T^{\prime \prime}+\lambda^{2} a^{2} T & =0
\end{aligned}\)
Step 1 of 5
Given:- The given wave equation in polar co-ordinates is \(u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}=\frac{1}{a^{2}} u_{t t} \ldots\)…. (i).