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'));
\[u \left ( r,\phi ,z,t \right ) ={\frac { \left ( \left ( {{\rm e}^{\sqrt {{\it \_c}_{{2}}}\phi }} \right ) ^{2}{\it \_C3}+{\it \_C4} \right ) \left ( \left ( {{\rm e}^{\sqrt {{\it \_c}_{{4}}}t}} \right ) ^{2}{\it \_C7}+{\it \_C8} \right ) \left ( \left ( {{\rm e}^{\sqrt {{\it \_c}_{{3}}}z}} \right ) ^{2}{\it \_C5}+{\it \_C6} \right ) }{{{\rm e}^{\sqrt {{\it \_c}_{{2}}}\phi }}{{\rm e}^{\sqrt {{\it \_c}_{{3}}}z}}{{\rm e}^{\sqrt {{\it \_c}_{{4}}}t}}} \left ( {\it \_C1}\,\BesselJ \left ( \sqrt {-{\it \_c}_{{2}}},{\frac {\sqrt {{\it \_c}_{{3}}{c}^{2}-{\it \_c}_{{4}}}r}{c}} \right ) +{\it \_C2}\,\BesselY \left ( \sqrt {-{\it \_c}_{{2}}},{\frac {\sqrt {{\it \_c}_{{3}}{c}^{2}-{\it \_c}_{{4}}}r}{c}} \right ) \right ) }\]