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No credit card required (428) Figure 5.25 (a) spherical waves in 3-D; (b) cylindrical waves in 3-D; (c) circular waves in 2-D.

Recall the Laplacian of $\mathrm{\Phi }$ in spherical coordinates, i.e.,

${\nabla }^{2}\mathrm{\Phi }=\frac{1}{{r}^{2}}\frac{\partial }{\partial r}\left({r}^{2}\frac{\partial \mathrm{\Phi }}{\partial r}\right)+\frac{1}{{r}^{2}\mathrm{sin}\theta }\frac{\partial }{\partial \theta }\left(\mathrm{sin}\theta \frac{\partial \mathrm{\Phi }}{\partial \theta }\right)+\frac{1}{{r}^{2}{\mathrm{sin}}^{2}\theta }\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\varphi }^{2}}\left(\mathrm{Laplacian}\mathrm{in}\mathrm{spherical}\mathrm{coord}.\right)$ (429) (429)

For spherical symmetry, Eq. (429) depends on $r$ only, i.e.,

${\nabla }^{2}\mathrm{\Phi }=\frac{1}{{r}^{2}}\frac{\partial }{\partial r}\left({r}^{2}\frac{\partial \mathrm{\Phi }}{\partial r}\right)\left(\mathrm{spherical}\mathrm{symmetry ...}$

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