### 23 Wave PDE in 3D Cylindrical coordinates

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#### 23.1 No initial and no boundary conditions given

problem number 151

Solve for $$u(r,\phi ,z,t)$$ the wave PDE in 3D
$u_{tt} = c^2 \nabla ^2 u$
$\text{DSolve}\left [u^{(0,0,0,2)}(r,\phi ,z,t)=c^2 \left (u^{(0,0,2,0)}(r,\phi ,z,t)+\frac{\frac{u^{(0,2,0,0)}(r,\phi ,z,t)}{r}+u^{(1,0,0,0)}(r,\phi ,z,t)}{r}+u^{(2,0,0,0)}(r,\phi ,z,t)\right ),u(r,\phi ,z,t),\{r,\phi ,z,t\}\right ]$
$u \left ( r,\phi ,z,t \right ) = \left ({\it \_C7}\,{\it \_C5}\,{\it \_C3}\,{{\rm e}^{2\,\sqrt{{\it \_c}_{{2}}}\phi +2\,\sqrt{{\it \_c}_{{3}}}z+2\,\sqrt{{\it \_c}_{{4}}}t}}+{\it \_C8}\,{\it \_C3}\,{\it \_C5}\,{{\rm e}^{2\,\sqrt{{\it \_c}_{{2}}}\phi +2\,\sqrt{{\it \_c}_{{3}}}z}}+{\it \_C6}\,{\it \_C7}\,{\it \_C3}\,{{\rm e}^{2\,\sqrt{{\it \_c}_{{2}}}\phi +2\,\sqrt{{\it \_c}_{{4}}}t}}+{\it \_C7}\,{\it \_C4}\,{\it \_C5}\,{{\rm e}^{2\,\sqrt{{\it \_c}_{{3}}}z+2\,\sqrt{{\it \_c}_{{4}}}t}}+{\it \_C6}\,{\it \_C8}\,{\it \_C3}\,{{\rm e}^{2\,\sqrt{{\it \_c}_{{2}}}\phi }}+{\it \_C4}\, \left ({\it \_C8}\,{\it \_C5}\,{{\rm e}^{2\,\sqrt{{\it \_c}_{{3}}}z}}+{\it \_C6}\, \left ({{\rm e}^{2\,\sqrt{{\it \_c}_{{4}}}t}}{\it \_C7}+{\it \_C8} \right ) \right ) \right ){{\rm e}^{-\sqrt{{\it \_c}_{{2}}}\phi -\sqrt{{\it \_c}_{{3}}}z-\sqrt{{\it \_c}_{{4}}}t}} \left ({\it \_C1}\,\BesselJ \left ( \sqrt{-{\it \_c}_{{2}}},{\frac{\sqrt{{\it \_c}_{{3}}{c}^{2}-{\it \_c}_{{4}}}r}{c}} \right ) +{\it \_C2}\,\BesselY \left ( \sqrt{-{\it \_c}_{{2}}},{\frac{\sqrt{{\it \_c}_{{3}}{c}^{2}-{\it \_c}_{{4}}}r}{c}} \right ) \right )$