Appendix B The partial differential equation
\begin{equation*}
\frac{\partial^{2}u}{\partial t^{2}} =v^{2}\left(\frac{\partial^{2}u}{\partial x^{2}} +\frac{\partial^{2}u}{\partial y^{2}}\right)
\end{equation*}
can be converted into the polar coordinates by the parameters
\begin{equation*}
x=r\cos\theta \quad \text{and}\quad y=r\sin\theta
\end{equation*}
so that
\begin{equation*}
r^{2}=x^{2}+y^{2}
\end{equation*}
and
\begin{equation*}
\tan\theta =\frac{y}{x},
\end{equation*}
then,
\begin{equation*}
\frac{\partial r}{\partial x}=\frac{x}{r}, \frac{\partial r}{\partial y}=\frac{y}{r}, \frac{\partial \theta}{\partial x}=-\frac{y}{r^{2}}, \frac{\partial \theta}{\partial y}=\frac{x}{r^{2}}
\end{equation*}
\begin{equation*}
\therefore \frac{\partial r}{\partial x} =\cos\theta, \frac{\partial r}{\partial y} =\sin\theta ,\frac{\partial \theta}{\partial x} =-\frac{\sin\theta}{r}, \frac{\partial \theta}{\partial y} =\frac{\cos\theta}{r}.
\end{equation*}
Thus,
\begin{equation*}
\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r}\frac{\partial r}{\partial x}+ \frac{\partial u}{\partial \theta}\frac{\partial \theta}{\partial x} = \cos\theta\frac{\partial u}{\partial r}-\frac{\sin\theta}{r}\frac{\partial u}{\partial \theta}
\end{equation*}
i.e.,
\begin{equation}
\frac{\partial }{\partial x}\equiv \left(\cos\theta\frac{\partial }{\partial r}-\frac{\sin\theta}{r}\frac{\partial }{\partial \theta}\right) \tag{B.0.1}
\end{equation}
similarly,
\begin{equation}
\frac{\partial }{\partial y}\equiv \left(\sin\theta\frac{\partial }{\partial r}+\frac{\cos\theta}{r}\frac{\partial }{\partial \theta}\right) \tag{B.0.2}
\end{equation}
also,
\begin{equation*}
\frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial }{\partial x}\left(\frac{\partial u}{\partial x}\right) = \left(\cos\theta\frac{\partial }{\partial r}-\frac{\sin\theta}{r}\frac{\partial }{\partial \theta}\right)\left(\cos\theta\frac{\partial u}{\partial r}-\frac{\sin\theta}{r}\frac{\partial u}{\partial \theta}\right)
\end{equation*}
\begin{equation*}
= \cos^{2}\theta \frac{\partial^{2} u}{\partial r^{2}}-\frac{\cos\theta\cdot \sin\theta}{r}\frac{\partial^{2} u}{\partial r \partial\theta}+\frac{\sin\theta\cdot \cos\theta}{r^{2}}\frac{\partial u}{\partial \theta}
\end{equation*}
\begin{equation*}
-\frac{\cos\theta\cdot \sin\theta}{r}\frac{\partial^{2} u}{\partial r \partial\theta}+\frac{\sin^{2}\theta}{r}\frac{\partial u}{\partial r}+ \frac{\sin^{2}\theta}{r^{2}}\frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\sin\theta\cdot \cos\theta}{r^{2}}\frac{\partial u}{\partial \theta}
\end{equation*}
and
\begin{equation*}
\frac{\partial^{2} u}{\partial y^{2}} =\left(\sin\theta\frac{\partial }{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)\left(\sin\theta\frac{\partial u}{\partial r}+\frac{\cos\theta}{r}\frac{\partial u}{\partial \theta}\right)
\end{equation*}
\begin{equation*}
= \sin^{2}\theta \frac{\partial^{2} u}{\partial r^{2}} + \frac{\sin\theta\cdot \cos\theta}{r}\frac{\partial^{2} u}{\partial r \partial\theta}-\frac{\sin\theta\cdot \cos\theta}{r^{2}}\frac{\partial u}{\partial\theta} + \frac{\cos\theta\cdot \sin\theta}{r}\frac{\partial^{2} u}{\partial r \partial\theta}
\end{equation*}
\begin{equation*}
+\frac{\cos^{2}\theta}{r}\frac{\partial u}{\partial r}+\frac{\cos^{2}\theta}{r^{2}}\frac{\partial^{2} u}{\partial \theta^{2}}-\frac{\cos\theta\cdot \sin\theta}{r^{2}}\frac{\partial u}{\partial \theta}
\end{equation*}
Hence,
\begin{equation*}
\frac{\partial^{2} u}{\partial t^{2}} = v^{2}\left(\frac{\partial^{2} u}{\partial x^{2}} +\frac{\partial^{2} u}{\partial y^{2}}\right) = v^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2} u}{\partial \theta^{2}}\right)
\end{equation*}
