Why do two masses connected to each other by a spring have the same frequency of oscillation?
Why do two masses, connected to each other by a spring, and each connected to a wall by a spring, have the same frequencies of oscillation when perturbed? In solving for the motion of the masses, under and assumption of SHM for each, the displacement of each can be guessed as $Ae^{i(wt + \theta)}$, where each mass has a different amplitude, and they have the same frequency. That the frequencies are the same seems important, since the next step is to lay out the equations of motion so as to get a matrix multiplication like this,
$m\ddot x_1 + \kappa x_1 + \beta x_1 -\beta x_2 = 0$
$m\ddot x_2 + \kappa x_2 + \beta x_2 -\beta x_1 = 0$
$$ \begin{pmatrix} w_1^2 - \frac \kappa m-\frac \beta m & \frac \beta m\ \frac \beta m & w_2^2 - \frac \kappa m-\frac \beta m\ \end{pmatrix} $$
where the top left and bottom right terms are identical. Then, finding the determinant to find a non-trivial solution, you can solve for omega. This works because $w_1$ = $w_2$, or the equation would be difficult, and maybe impossible, to solve. 37
You might get a solution for the product of the two but not for either. In the end, you get two $w$ values, one for a fast oscillation and one for a slow one; in one mode the masses move together, and in one they are half a phase apart. In both they have equal amplitude, and the motion of the masses, whatever is observed, is a combination of these two motions (I think this result is saying that they only motions the masses can undergo are those which are linear combinations of these two motions).
Visually, I am trying to picture the masses moving with different frequencies. I can picture it until I get to the region of spring which connects them, which could not move at both frequencies simultaneously. Is this disconnect in reality and picture why the frequencies are different? Or is it possible for the frequency of oscillation to vary along a spring? 453