Time-Varying Spectrum of the Random String

TL;DR

The paper addresses the time-varying spectrum of a finite string driven by white Gaussian noise and defines the spectrum via the ensemble-averaged Wigner distribution, yielding exact transient and steady-state results for the string's response starting from a finite time. It derives the spectrum using two approaches: transforming the string equations into a phase-space representation and an impulse-response method, and it provides closed-form expressions for the time-dependent spectrum $\overline{W}(t,\omega;x)$ and the associated variance $\sigma^2(t,x)$. The work covers damped and undamped cases, recovers the classical steady-state spectrum in the appropriate limit, and gives explicit expressions for the time evolution of energy among modes, with a detailed numerical example illustrating mode-specific dynamics and spatial symmetry. A complete set of appendices establishes the statistics of the modal forcing and proves that cross-spectral terms vanish, ensuring the self-term spectrum fully characterizes the nonstationary behavior.

Abstract

We consider the response of a finite string to white noise and obtain the exact time-dependent spectrum. The complete exact solution is obtained, that is, both the transient and steady-state solution. To define the time-varying spectrum we ensemble average the Wigner distribution. We obtain the exact solution by transforming the differential equation for the string into the phase space differential equation of time and frequency and solve it directly. We also obtain the exact solution by an impulse response method which gives a different form of the solution. Also, we obtain the time-dependent variance of the process at each position. Limiting cases for small and large times are obtained. As a special case we obtain the results of van Lear Jr. and Uhlenbeck and Lyon. A numerical example is given and the results plotted.

Figures (6)

• Figure 1: The first 3 modes of the Wigner spectrum $\overline{W}(t,\omega,x)$, as given by Eq. (\ref{['eq92']}) computed at $x=L/4$.
• Figure 2: The first 4 modes of the Wigner spectrum computed at $x=L/2$. The second and fourth mode are zero because of the $\sin^{2}$ term in Eq. (\ref{['eq92']}), which zeros all of the even modes.
• Figure 3: The Wigner spectrum at $x=L/\sqrt{5}$.
• Figure 4: The first 12 modes of the Wigner spectrum $\overline{W}(t,\omega,x)$, computed for $x=L/4$. The fourth, eighth, and twelfth mode are zero because of the $\sin^{2}$ term in Eq. (\ref{['eq92']}).
• Figure 5: The plot shows the first 12 modes of the Wigner spectrum $\overline{W}(t,\omega,x)$, computed for $x=L/10$.