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]];
\[\left \{\left \{u(r,\phi ,z,t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-\sqrt {c_9} \phi -z \sqrt {c_{10}}-t \sqrt {c_{11}}} \left (\operatorname {BesselJ}\left (\sqrt {-c_9},\frac {r \sqrt {c^2 c_{10}-c_{11}}}{\sqrt {c^2}}\right ) c_1+\operatorname {BesselY}\left (\sqrt {-c_9},\frac {r \sqrt {c^2 c_{10}-c_{11}}}{\sqrt {c^2}}\right ) c_2\right ) \left (e^{2 \phi \sqrt {c_9}} c_3+c_4\right ) \left (e^{2 z \sqrt {c_{10}}} c_5+c_6\right ) \left (e^{2 t \sqrt {c_{11}}} c_7+c_8\right ) & c\neq 0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
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'));
\[u \left (r , \phi , z , t\right ) = \left (c_{3} c_{5} c_{7} {\mathrm e}^{2 \sqrt {\textit {\_c}_{3}}\, z +2 \sqrt {\textit {\_c}_{4}}\, t +2 \sqrt {\textit {\_c}_{2}}\, \phi }+c_{3} c_{5} c_{8} {\mathrm e}^{2 \sqrt {\textit {\_c}_{3}}\, z +2 \sqrt {\textit {\_c}_{2}}\, \phi }+c_{3} c_{6} c_{7} {\mathrm e}^{2 \sqrt {\textit {\_c}_{4}}\, t +2 \sqrt {\textit {\_c}_{2}}\, \phi }+c_{4} c_{5} c_{7} {\mathrm e}^{2 \sqrt {\textit {\_c}_{3}}\, z +2 \sqrt {\textit {\_c}_{4}}\, t}+c_{3} c_{6} c_{8} {\mathrm e}^{2 \sqrt {\textit {\_c}_{2}}\, \phi }+c_{4} \left (c_{8} c_{5} {\mathrm e}^{2 \sqrt {\textit {\_c}_{3}}\, z}+c_{6} \left (c_{7} {\mathrm e}^{2 \sqrt {\textit {\_c}_{4}}\, t}+c_{8} \right )\right )\right ) \left (c_{1} \operatorname {BesselJ}\left (\sqrt {-\textit {\_c}_{2}}, \frac {\sqrt {\textit {\_c}_{3} c^{2}-\textit {\_c}_{4}}\, r}{c}\right )+c_{2} \operatorname {BesselY}\left (\sqrt {-\textit {\_c}_{2}}, \frac {\sqrt {\textit {\_c}_{3} c^{2}-\textit {\_c}_{4}}\, r}{c}\right )\right ) {\mathrm e}^{-\sqrt {\textit {\_c}_{3}}\, z -\sqrt {\textit {\_c}_{4}}\, t -\sqrt {\textit {\_c}_{2}}\, \phi }\]