#### 6.3.2 Cylindrical coordinates

6.3.2.1 [417] No I.C. no B.C.

##### 6.3.2.1 [417] No I.C. no B.C.

problem number 417

Solve for $$u(r,\phi ,z,t)$$ the wave PDE in 3D $u_{tt} = c^2 \nabla ^2 u$
$\left \{\left \{u(r,\phi ,z,t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-\sqrt {c_9} \phi -z \sqrt {c_{10}}-t \sqrt {c_{11}}} \left (J_{\sqrt {-c_9}}\left (\frac {r \sqrt {c^2 c_{10}-c_{11}}}{\sqrt {c^2}}\right ) c_1+Y_{\sqrt {-c_9}}\left (\frac {r \sqrt {c^2 c_{10}-c_{11}}}{\sqrt {c^2}}\right ) c_2\right ) \left (e^{2 \phi \sqrt {c_9}} c_3+c_4\right ) \left (e^{2 z \sqrt {c_{10}}} c_5+c_6\right ) \left (e^{2 t \sqrt {c_{11}}} c_7+c_8\right ) & c\neq 0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}$
$u \left (r , \phi , z , t\right ) = \left (c_{3} {\mathrm e}^{2 \phi \sqrt {\textit {\_c}_{2}}}+c_{4}\right ) \left (c_{1} \BesselJ \left (\sqrt {-\textit {\_c}_{2}}, \frac {\sqrt {\textit {\_c}_{3} c^{2}-\textit {\_c}_{4}}\, r}{c}\right )+c_{2} \BesselY \left (\sqrt {-\textit {\_c}_{2}}, \frac {\sqrt {\textit {\_c}_{3} c^{2}-\textit {\_c}_{4}}\, r}{c}\right )\right ) \left (c_{7} {\mathrm e}^{2 t \sqrt {\textit {\_c}_{4}}}+c_{8}\right ) \left (c_{5} {\mathrm e}^{2 z \sqrt {\textit {\_c}_{3}}}+c_{6}\right ) {\mathrm e}^{-\phi \sqrt {\textit {\_c}_{2}}} {\mathrm e}^{-z \sqrt {\textit {\_c}_{3}}} {\mathrm e}^{-t \sqrt {\textit {\_c}_{4}}}$