Added Jan 10, 2019.
Solve for \(u(r,\phi ,z,t)\) the wave PDE in 3D \[ u_{tt} = c^2 \nabla ^2 u \]
Mathematica ✓
ClearAll["Global`*"]; lap = Laplacian[u[r, phi, z, t], {r, phi, z}, "Cylindrical"]; pde = D[u[r, phi, z, t], {t, 2}] == c^2*lap; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[r, phi, z, t], {r, phi, z, t}], 60*10]];
\[\left \{\left \{u(r,\phi ,z,t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-\sqrt {c_9} \phi -z \sqrt {c_{10}}-t \sqrt {c_{11}}} \left (J_{\sqrt {-c_9}}\left (\frac {r \sqrt {c^2 c_{10}-c_{11}}}{\sqrt {c^2}}\right ) c_1+Y_{\sqrt {-c_9}}\left (\frac {r \sqrt {c^2 c_{10}-c_{11}}}{\sqrt {c^2}}\right ) c_2\right ) \left (e^{2 \phi \sqrt {c_9}} c_3+c_4\right ) \left (e^{2 z \sqrt {c_{10}}} c_5+c_6\right ) \left (e^{2 t \sqrt {c_{11}}} c_7+c_8\right ) & c\neq 0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✓
restart; lap :=VectorCalculus:-Laplacian( u(r,phi,z,t), 'cylindrical'[r,phi,z] ); pde := diff(u(r,phi,z,t),t$2)= c^2* lap; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(r,phi,z,t),'build')),output='realtime'));
\[u \left (r , \phi , z , t\right ) = \left (c_{3} {\mathrm e}^{2 \phi \sqrt {\mathit {\_c}_{2}}}+c_{4}\right ) \left (c_{1} \BesselJ \left (\sqrt {-\mathit {\_c}_{2}}, \frac {\sqrt {\mathit {\_c}_{3} c^{2}-\mathit {\_c}_{4}}\, r}{c}\right )+c_{2} \BesselY \left (\sqrt {-\mathit {\_c}_{2}}, \frac {\sqrt {\mathit {\_c}_{3} c^{2}-\mathit {\_c}_{4}}\, r}{c}\right )\right ) \left (c_{7} {\mathrm e}^{2 t \sqrt {\mathit {\_c}_{4}}}+c_{8}\right ) \left (c_{5} {\mathrm e}^{2 z \sqrt {\mathit {\_c}_{3}}}+c_{6}\right ) {\mathrm e}^{-\phi \sqrt {\mathit {\_c}_{2}}} {\mathrm e}^{-z \sqrt {\mathit {\_c}_{3}}} {\mathrm e}^{-t \sqrt {\mathit {\_c}_{4}}}\]