How to collapse trajectory to wave equation? I have a "discretized string" as in the figure whose motion I know can be mostly represented by a wave equation
$m\ddot{x}=k(x_{n+1}+x_{n-1}-2x_{n})$ where $x$ is the vertical distance. The horizontal distance is always constant.
Say the equilibrium position is at $x=0$ and I take measurements of the vertical positions and get 
$x(t=0)=(x_{1}(0),x_{2}(0),...,x_{n}(0))$,
$x(t=1)=(x_{1}(1),x_{2}(1),...,x_{n}(1))$, etc.
until $t=t_{final}$. 
How can I obtain an estimate of the value of $k/m$?
 
 A: The motion of the $i^{th}$ point mass on the string is sinusoidal in time :
$x_i=a_i \sin(\omega t+\phi_i)$
where $\omega^2=k/m$ is the common angular frequency of each discrete mass.
Take any point $i$ and plot the values of $x_i(t)$ against the values of $t$ then fit a sinusoidal function to it. This is a non-linear least-squares regression. 
If you have sufficiently sophisticated Data Analysis software,  you will be able to select a sinusoid as a non-linear fitting function. If not, you can do the fitting "by hand" by trial and improvement in a spreadsheet. Adjust the 3 parameters $a_i, \omega, \phi_i$ until $S=\Sigma_t (x_i - a_i \sin(\omega t+\phi_i))^2$ for all time values $t$, is minimised. You could speed up the search process by making use of a GoalSeek function - eg vary $a_i$ until $S$ is a minimum - invoking it in a cycle on the parameters in turn - ie $a_i \to \omega \to \phi_i \to a_i \to \omega \to \phi_1 \to$ etc.
You could repeat this exercise for each of the $i$ point masses to check that the value of $\omega$ is the same for each. Values of $a_i, \phi_i$ will differ.  
