Structural Health Monitoring with Piezoelectric Wafer Active Sensors, 2nd Edition

(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.,

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

(429)

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

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