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

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image (428)
image
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 Φimage in spherical coordinates, i.e.,

2Φ=1r2r(r2Φr)+1r2sinθθ(sinθΦθ)+1r2sin2θ2Φϕ2(Laplacianinsphericalcoord.) (429)

image (429)

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

2Φ=1r2r(r2Φr)(sphericalsymmetry ...

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