5.3.2 Cylindrical coordinates

5.3.2.1 [415] No I.C. no B.C.

problem number 415

Added Jan 10, 2019.

Solve for $u(r,\varphi ,z,t)$ the wave PDE in 3D $$u_{tt}=c^2{\mathrm{\nabla }}^{2}u$$

Figure 5.110: PDE specification

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]];
 

$$\left\{\{u(r,\varphi ,z,t)\to \begin{array}{cc}\{& \begin{array}{cc}{e}^{-\sqrt{c_9}\varphi -z\sqrt{c_{10}}-t\sqrt{c_{11}}}(J_{\sqrt{-c_9}}\left(\frac{r\sqrt{c^2{c}_{10}-c_{11}}}{\sqrt{c^2}}\right)c_1+Y_{\sqrt{-c_9}}\left(\frac{r\sqrt{c^2{c}_{10}-c_{11}}}{\sqrt{c^2}}\right)c_2)(e^{2\varphi \sqrt{c_9}}{c}_3+c_4)(e^{2z\sqrt{c_{10}}}{c}_5+c_6)(e^{2t\sqrt{c_{11}}}{c}_7+c_8)& c\ne 0\\ \text{Indeterminate}& \text{True}\end{array}\end{array}\}\right\}$$

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'));
 

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