### 22 Wave PDE in 3D Spherical coordinates

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

problem number 150

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

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

Using the Physics convention for Spherical coordinates system.

Mathematica

$\text{DSolve}\left [u^{(0,0,0,2)}(r,\theta ,\phi ,t)=c^2 \left (\frac{\frac{u^{(0,2,0,0)}(r,\theta ,\phi ,t)}{r}+u^{(1,0,0,0)}(r,\theta ,\phi ,t)}{r}+u^{(2,0,0,0)}(r,\theta ,\phi ,t)+\frac{\csc (\theta ) \left (\sin (\theta ) u^{(1,0,0,0)}(r,\theta ,\phi ,t)+\frac{\cos (\theta ) u^{(0,1,0,0)}(r,\theta ,\phi ,t)}{r}+\frac{\csc (\theta ) u^{(0,0,2,0)}(r,\theta ,\phi ,t)}{r}\right )}{r}\right ),u(r,\theta ,\phi ,t),\{r,\theta ,\phi ,t\},\text{Assumptions}\to \{0<\theta <\pi \}\right ]$

Maple

$u \left ( r,\theta ,\phi ,t \right ) ={\frac{{{\rm e}^{1/2\, \left ( -\pi -2\,\phi \right ) \sqrt{{\it \_c}_{{3}}}-\sqrt{{\it \_c}_{{4}}}t}}\sqrt{2} \left ( \sin \left ( \theta \right ) \right ) ^{i\sqrt{{\it \_c}_{{3}}}} \left ({{\rm e}^{2\,\sqrt{{\it \_c}_{{4}}}t}}{\it \_C7}+{\it \_C8} \right ) \left ({{\rm e}^{2\,\sqrt{{\it \_c}_{{3}}}\phi }}{\it \_C5}+{\it \_C6} \right ) }{\sqrt{r}} \left ({\it \_C2}\,\BesselY \left ( 1/2\,\sqrt{{\frac{{c}^{2}+4\,{\it \_c}_{{1}}}{{c}^{2}}}},{\frac{\sqrt{-{\it \_c}_{{4}}}r}{c}} \right ) +{\it \_C1}\,\BesselJ \left ( 1/2\,\sqrt{{\frac{{c}^{2}+4\,{\it \_c}_{{1}}}{{c}^{2}}}},{\frac{\sqrt{-{\it \_c}_{{4}}}r}{c}} \right ) \right ) \left ( \cos \left ( \theta \right ){\mbox{_2F_1}(-1/4\,{\frac{-2\,\sqrt{-{\it \_c}_{{3}}}c+\sqrt{{c}^{2}+4\,{\it \_c}_{{1}}}-3\,c}{c}},1/4\,{\frac{2\,\sqrt{-{\it \_c}_{{3}}}c+\sqrt{{c}^{2}+4\,{\it \_c}_{{1}}}+3\,c}{c}};\,3/2;\,1/2\,\cos \left ( 2\,\theta \right ) +1/2)}{\it \_C3}+{\mbox{_2F_1}(1/4\,{\frac{2\,\sqrt{-{\it \_c}_{{3}}}c+\sqrt{{c}^{2}+4\,{\it \_c}_{{1}}}+c}{c}},-1/4\,{\frac{-2\,\sqrt{-{\it \_c}_{{3}}}c+\sqrt{{c}^{2}+4\,{\it \_c}_{{1}}}-c}{c}};\,1/2;\,1/2\,\cos \left ( 2\,\theta \right ) +1/2)}{\it \_C4} \right ) }$