Wave equation#
The standard form of the wave equation in spherical coordinates is given by
(1)#\[\begin{equation}
\frac{\partial^2 (rP)}{\partial t^2}=c^2\frac{\partial^2(rP)}{\partial^2r}
\end{equation}\]
the solution of which is
(2)#\[\begin{equation}
rP=f(ct\pm r)
\end{equation}\]
where \(f(ct\pm r)\) is any function that depends on \((ct\pm r)\)
The general solution includes both outgoing (minus sign) and incoming waves( plus sign)
For periodic sound waves one can write
(3)#\[\begin{equation}
f(ct\pm r) = A(ct\pm r)\cos\Biggl(\frac{2\pi}{\lambda}(ct\pm r)\Biggr)
= A(ct\pm r)\cos(\omega t\pm kr)
\end{equation}\]