The total energy \(\mathcal{E}\) is useful in characterizing bandwidth: \begin{align}\label{eq:E} \mathcal{E} &= \tfrac{1}{2} \rho c^2 \int_{0}^{\ell} \left(\frac{\partial u}{\partial x}\right)^2 dx + \tfrac{1}{2} EI \int_{0}^{\ell} \left(\frac{\partial^2 u}{\partial x^2}\right)^2 dx + \tfrac{1}{2} \rho\int_{0}^{\ell} \left(\frac{\partial u}{\partial t}\right)^2 dx \\ &= \frac{\rho c^2 u_0^2\pi^2}{4\ell} \sum_{n=1}^{\infty} [a_n^2 n^2 (1 + B n^2) + b_n^2 n^2 (1 + Bn^2 -\Gamma_n^2/n^2)] \end{align}
Normalizing \(\mathcal{E}\) to \((\rho c^2 u_0^2\pi^2/4\ell)\sum_{n=1}^{\infty} (a_n^2 + b_n^2)\) yields \begin{align}\label{eq:beta} \boxed{\beta_\mathrm{rms} = \left\{\frac{\sum_{n=1}^{\infty} [a_n^2 n^2 (1 + B n^2) + b_n^2 n^2 (1 + Bn^2 -\Gamma_n^2/n^2)]}{\sum_{n=1}^{\infty} (a_n^2 + b_n^2)}\right\}^{1/2}} \end{align}