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

(193)

where

$B_a=\frac{1-{\nu }_a^2}{E_a{t}_a}\frac{1}{{\mathrm{\Gamma }}^2}C$ (194)

(194)

Proof: substitution of Eq. (193) into Eq. (189) yields

$B_a\left(\frac{d^2{I}_1(\mathrm{\Gamma }r)}{d{r}^2}+\frac{1}{r}\frac{d{I}_1(\mathrm{\Gamma }r)}{dr}-\frac{I_1(\mathrm{\Gamma }r)}{r^2}\right)=\frac{1-{\nu }_a^2}{E_a}\frac{1}{t_a}C{I}_1(\mathrm{\Gamma }r)$ (195)

(195)

But $I_1(\mathrm{\Gamma }r)$ satisfies Eq. (180), i.e.,

$\frac{d^2{I}_1(\mathrm{\Gamma }r)}{d{r}^2}+\frac{1}{r}\frac{d{I}_1(\mathrm{\Gamma }r)}{dr}-\frac{I_1(\mathrm{\Gamma }r)}{r^2}={\mathrm{\Gamma }}^2{I}_1(\mathrm{\Gamma }r)$ (196)

(196)

Substitution of Eq. (196) into Eq. (195) gives Eq. (194). QED.

Addition of Eqs. (192) ...