For a struck string obeying \(u_{xx} - \ddot{u}/c^2 =0\), \(a_n = 0\) and* \begin{align} b_n &= \frac{2}{n^2\pi^2} [1 - (-1)^n] \label{eq:recover:struck:b} \end{align}
The analytical (gray) and numerical (blue) initial conditions and spectra are compared below:
*Blackstock 2000, p. 227