Wave PDE in 3D Cylindrical coordinates

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26.1 No initial and no boundary conditions given

Mathematica ✗

ClearAll[u, t, r, z, phi]; 
 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]];

\[ \text {Failed} \]

Maple ✓

 
u:='u';t:='t'; phi:='phi';r:='r';z:='z'; 
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')); 
sol:=simplify(sol);

\[ u \left ( r,\phi ,z,t \right ) ={{\rm e}^{-\sqrt {{\it \_c}_{{2}}}\phi -\sqrt {{\it \_c}_{{3}}}z-\sqrt {{\it \_c}_{{4}}}t}} \left ( {\it \_C1}\,\BesselJ \left ( \sqrt {-{\it \_c}_{{2}}},{\frac {\sqrt {{\it \_c}_{{3}}{c}^{2}-{\it \_c}_{{4}}}r}{c}} \right ) +{\it \_C2}\,\BesselY \left ( \sqrt {-{\it \_c}_{{2}}},{\frac {\sqrt {{\it \_c}_{{3}}{c}^{2}-{\it \_c}_{{4}}}r}{c}} \right ) \right ) \left ( {\it \_C7}\,{\it \_C5}\,{\it \_C3}\,{{\rm e}^{2\,\sqrt {{\it \_c}_{{2}}}\phi +2\,\sqrt {{\it \_c}_{{3}}}z+2\,\sqrt {{\it \_c}_{{4}}}t}}+{\it \_C8}\,{\it \_C3}\,{\it \_C5}\,{{\rm e}^{2\,\sqrt {{\it \_c}_{{2}}}\phi +2\,\sqrt {{\it \_c}_{{3}}}z}}+{\it \_C6}\,{\it \_C7}\,{\it \_C3}\,{{\rm e}^{2\,\sqrt {{\it \_c}_{{2}}}\phi +2\,\sqrt {{\it \_c}_{{4}}}t}}+{\it \_C7}\,{\it \_C4}\,{\it \_C5}\,{{\rm e}^{2\,\sqrt {{\it \_c}_{{3}}}z+2\,\sqrt {{\it \_c}_{{4}}}t}}+{\it \_C6}\,{\it \_C8}\,{\it \_C3}\,{{\rm e}^{2\,\sqrt {{\it \_c}_{{2}}}\phi }}+ \left ( {\it \_C8}\,{\it \_C5}\,{{\rm e}^{2\,\sqrt {{\it \_c}_{{3}}}z}}+{\it \_C6}\, \left ( {{\rm e}^{2\,\sqrt {{\it \_c}_{{4}}}t}}{\it \_C7}+{\it \_C8} \right ) \right ) {\it \_C4} \right ) \]

26 Wave PDE in 3D Cylindrical coordinates

26.1 No initial and no boundary conditions given