5.3.2 Cylindrical coordinates

5.3.2.1 [416] No I.C. no B.C.

5.3.2.1 [416] No I.C. no B.C.

problem number 416

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]];
 

Failed

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'));
 

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