Exercise 5.7: propagation speed in a linear chain

Another interesting property to analyze is the propagation of the energy. In this problem we'll disturb the system determine the time that takes for the disturbance to travel a given distance.

1. Consider a linear chain of coupled oscillators at rest with $k=k_{c}=1$. Create a disturbance giving particle 1 an initial displacement $u_{1}=1$. Determine the time it takes for particles $N/2$ and $N$ to satisfy the conditions $\vert u_{N/2}\vert\ge d$ and $u_{N}\ge d$. Choose $N=10$ and $d=0.3$ for your initial runs. Use your results to estimate $v$, the speed of propagation of the disturbance. Consider larger values of $N$ to ensure that your value is independent of $N$.

2. Do you expect the speed of propagation to be an increasing or decreasing function of the spring constant $k$? Do a simulation and estimate $v$ for different values of $k$.

3. Create a disturbance by applying an external force $F(t)=F_{0}\cos {% \omega t}$ to particle 1. Estimate the propagation speed of the disturbance as in part 1. Consider the value so $\omega =0.1$ and $\omega = 1$. Explain why the propagation speed depends on $\omega$. Can a disturbance propagate for $\omega = 4$? In what way does the system act as a mechanical filter? Explain the ``filtering'' property of the system in terms of the frequency of the normal modes.