Introduce dimensionless displacement, position, time, loss, and attack parameters, respectively given by \begin{align}\label{eq:dim} U &\equiv \frac{u}{u_0}\,,\quad X \equiv \frac{x}{\ell}\,,\quad \Omega \equiv \frac{\pi ct}{\ell} \,,\quad \Gamma_n \equiv \frac{\gamma_n \ell}{\pi c}\,,\quad \Theta \equiv \frac{\pi c\tau}{\ell} \end{align}
Then the expansion coefficients become \begin{align}\label{eq:coeff:dim} a_n &= 2 e^{-\Delta^2} \Delta \! \int_{0}^{1} \!\! f(X) \sin(n\pi X) \,dX\, ,\\ b_n &= 2 e^{-\Delta^2} \frac{1 - 2\Delta^2}{\Theta n'}\int_{0}^{1} \!\! f(X) \sin(n\pi X) \,dX %I_n = \int_{0}^{1} \!\! f(X) \sin(n\pi X) \,dX \end{align}