(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.,

$\begin{array}{l}{\int}_{0}^{2\pi}\mathrm{sin}p\varphi \mathrm{sin}q\varphi d\varphi =\pi {\delta}_{pq}\hfill \\ {\int}_{0}^{2\pi}\mathrm{cos}p\varphi \mathrm{cos}q\varphi d\varphi =\pi {\delta}_{pq}\hfill \end{array}$ (102)

(102)

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

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