Commençons par calculer le laplacien de s :
![{\displaystyle {\begin{aligned}r^{2}{\frac {\partial s}{\partial r}}&=r^{2}{\frac {\partial }{\partial r}}\left({\frac {1}{r}}\Psi (r,t)\right)\\&=r^{2}{\frac {\partial ({\frac {1}{r}})}{\partial r}}\Psi (r,t)+r{\frac {\partial \Psi }{\partial r}}\\&=-\Psi (r,t)+r{\frac {\partial \Psi }{\partial r}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81fa2dd4b766df575a9d132d5fb022cc0e69a34d)
![{\displaystyle {\begin{aligned}{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial s}{\partial r}}\right)&={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(-\Psi (r,t)+r{\frac {\partial \Psi }{\partial r}}\right)\\&=-{\frac {1}{r^{2}}}{\frac {\partial \Psi }{\partial r}}+{\frac {1}{r^{2}}}\left({\frac {\partial \Psi }{\partial r}}+r{\frac {\partial ^{2}\Psi }{\partial r^{2}}}\right)\\&={\frac {1}{r}}{\frac {\partial ^{2}\Psi }{\partial r^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6f2ba01f5b77cae2badc63317de5e19f03127c)
Les autres termes du laplacien sont nuls, donc
.
De plus, l'équation de propagation donne :
![{\displaystyle {\begin{aligned}\Delta s&={\frac {1}{c^{2}}}{\frac {\partial ^{2}s}{\partial t^{2}}}\\&={\frac {1}{r}}{\frac {1}{c^{2}}}{\frac {\partial ^{2}\Psi }{\partial t^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/851c7a502c199862b91ee210125b588b7f0b07d7)
On arrive finalement à l'équation de propagation suivante :
.
On sait alors que
Les solutions à l'équation de propagation sous les conditions de l'énoncé sont alors de la forme
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