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

(100)

Proof of first line of Eq. (99):

(101)

A similar proof applies to the second line of Eq. (99). It can be easily verified that orthogonality properties similar to Eq. (99) also apply to the interval $[0,2\pi ]$, i.e.,

$\begin{array}{l}{\int }_0^{2\pi }\sin p\varphi \sin q\varphi d\varphi =\pi {\delta }_{pq}\hfill \\ {\int }_0^{2\pi }\cos p\varphi \cos q\varphi d\varphi =\pi {\delta }_{pq}\hfill \end{array}$ (102)

(102)

For vibrations and waves analysis, one can apply a similar argument to harmonic ...