Transient response of a cylindrical cavity in three-dimensional elastic space under impact loading: an analytical solution - Nature
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Appendix$$ E_{1} = \left( {\lambda k_{1}^{2} - \lambda \gamma^{2} } \right)K_{n} (k_{1} r) + \left( {2\mu k_{1}^{2} + \frac{{2\mu n^{2} }}{{r^{2} }} - \frac{2\mu n}{{r^{2} }}} \right)K_{n} (k_{1} r) + \frac{{2\mu k_{1} }}{r}K_{n + 1} (k_{1} r) $$$$ E_{2} = \left( {\frac{ - 2\mu n}{{r^{2} }} + \frac{{2\mu n^{2} }}{{r^{2} }}} \right)K_{n} (k_{2} r) - \frac{{2\mu nk_{2} }}{r}K_{n + 1} (k_{2} r) $$$$ E_{3} = \left( {2\mu \lambda k_{2}^{2} + \frac{{2\mu n^{2} \lambda }}{{r^{2} }} - \frac{2\mu n\lambda }{{r^{2} }}} \right)K_{n} (k_{2} r) + \frac{{2\mu \lambda k_{2} }}{r}K_{n + 1} (k_{2} r) $$$$ E_{4} = \left( { - \lambda k_{1}^{2} + \lambda \gamma^{2} } \right)K_{n} (k_{1} r) - \left( {2\mu k_{1}^{2} + \frac{{2\mu n^{2} }}{{r^{2} }} - \frac{2\mu n}{{r^{2} }}} \right)K_{n} (k_{1} r) - \frac{{2\mu k_{1} }}{r}K_{n + 1} (k_{1} r) $$$$ E_{5} = \left( {\frac{2\mu n}{{r^{2} }} - \frac{{2\mu n^{2} }}{{r^{2} }}} \right)K_{n} (k_{2} r) + \frac{{2\mu nk_{2} }}{r}K_{n + 1} (k_{2} r) $$$$ E_{6} = \left( { - 2\mu \lambda k_{2}^{2} - \frac{{2\mu n^{2} \lambda }}{{r^{2} }} + \frac{2\mu n\lambda }{{r^{2} }}} \right)K_{n} (k_{2} r) - \frac{{2\mu \lambda k_{2} }}{r}K_{n + 1} (k_{2} r) $$$$ E_{7} = \left(
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