26 Wave PDE in 3D Cylindrical coordinates

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26.1 No initial and no boundary conditions given

problem number 167

Added Jan 10, 2019.

Solve for $u(r,\varphi ,z,t)$ the wave PDE in 3D

$$u_{tt}=c^2{\mathrm{\nabla }}^{2}u$$

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);
 

$$u(r,\varphi ,z,t)={\mathrm{e}}^{-\sqrt{{\mathit{\_}\mathit{c}}_2}\varphi -\sqrt{{\mathit{\_}\mathit{c}}_3}z-\sqrt{{\mathit{\_}\mathit{c}}_4}t}(\mathit{\_}\mathit{C}\mathit{1}\mathrm{BesselJ}(\sqrt{-{\mathit{\_}\mathit{c}}_2},\frac{\sqrt{{\mathit{\_}\mathit{c}}_3{c}^2-{\mathit{\_}\mathit{c}}_4}r}{c})+\mathit{\_}\mathit{C}\mathit{2}\mathrm{BesselY}(\sqrt{-{\mathit{\_}\mathit{c}}_2},\frac{\sqrt{{\mathit{\_}\mathit{c}}_3{c}^2-{\mathit{\_}\mathit{c}}_4}r}{c}))(\mathit{\_}\mathit{C}\mathit{7}\mathit{\_}\mathit{C}\mathit{5}\mathit{\_}\mathit{C}\mathit{3}{\mathrm{e}}^{2\sqrt{{\mathit{\_}\mathit{c}}_2}\varphi +2\sqrt{{\mathit{\_}\mathit{c}}_3}z+2\sqrt{{\mathit{\_}\mathit{c}}_4}t}+\mathit{\_}\mathit{C}\mathit{8}\mathit{\_}\mathit{C}\mathit{3}\mathit{\_}\mathit{C}\mathit{5}{\mathrm{e}}^{2\sqrt{{\mathit{\_}\mathit{c}}_2}\varphi +2\sqrt{{\mathit{\_}\mathit{c}}_3}z}+\mathit{\_}\mathit{C}\mathit{6}\mathit{\_}\mathit{C}\mathit{7}\mathit{\_}\mathit{C}\mathit{3}{\mathrm{e}}^{2\sqrt{{\mathit{\_}\mathit{c}}_2}\varphi +2\sqrt{{\mathit{\_}\mathit{c}}_4}t}+\mathit{\_}\mathit{C}\mathit{7}\mathit{\_}\mathit{C}\mathit{4}\mathit{\_}\mathit{C}\mathit{5}{\mathrm{e}}^{2\sqrt{{\mathit{\_}\mathit{c}}_3}z+2\sqrt{{\mathit{\_}\mathit{c}}_4}t}+\mathit{\_}\mathit{C}\mathit{6}\mathit{\_}\mathit{C}\mathit{8}\mathit{\_}\mathit{C}\mathit{3}{\mathrm{e}}^{2\sqrt{{\mathit{\_}\mathit{c}}_2}\varphi }+(\mathit{\_}\mathit{C}\mathit{8}\mathit{\_}\mathit{C}\mathit{5}{\mathrm{e}}^{2\sqrt{{\mathit{\_}\mathit{c}}_3}z}+\mathit{\_}\mathit{C}\mathit{6}({\mathrm{e}}^{2\sqrt{{\mathit{\_}\mathit{c}}_4}t}\mathit{\_}\mathit{C}\mathit{7}+\mathit{\_}\mathit{C}\mathit{8}))\mathit{\_}\mathit{C}\mathit{4})$$