Consider the famous coupled oscillation problem of 2 spring pendulum:
In a special case the solution can be given as follows:
$x_1 = \displaystyle \frac{C_1}{2}\,\cos(\omega_1\cdot t)+\frac{C_1}{2}\,\cos(\omega_2\cdot t)$
$x_2 = \displaystyle \frac{C_1}{2}\,\cos(\omega_1\cdot t)-\frac{C_1}{2}\,\cos(\omega_2\cdot t)$
where the starting conditions where be chosen like:
$x_1(0) = 2\,C_1 \quad x_2(0) = 0 \quad\dot{x}_1(0) = 0 \quad \dot{x}_2(0) = 0$
Notably the frequencies are expressed by:
$\omega_1 = \displaystyle \sqrt{\frac{D_1}{m}}$ and $\omega_2 = \displaystyle \sqrt{\frac{D_1+2\,D_2}{m}}$
But here is my problem: under these conditions the phenomena is called beating effect.
Assuming $D_1 = D_2 = 1\,\frac{N}{m}$ and plotting the graph I do not get the typical patterns:
Only if I chose $\omega_1$ fairly close to $\omega_2$, thus $D_2$ to be small, the model emerges:
Classical Question: why is that only occurring under certain constraints?