Coupled Oscillators

Consider two idential simple harmonic oscillators of mass $m$ and spring constant $k$ as shown in Figure 5.1 (a.). The two oscillators are independent with

$$x_0(t)=a_0 \, \cos (\omega t + \phi_0)$$ (5.1)

and

$$x_1(t)=a_1 \, \cos (\omega t + \phi_1)$$ (5.2)

where they both oscillate with the same frequency $\omega=\sqrt{\frac{k}{m}}$. The amplitudes $a_0, a_1$ and the phases $\phi_0,\phi_1$ of the two oscillators are in no way interdependent. The question which we take up for discussion here is what happens if the two masses are coupled by a third spring as shown in Figure 1 (b.).

Figure 5.1: This shows two identical spring-mass systems. In (a.) the two oscillators are independent whereas in (b.) they are coupled through an extra spring.

The motion of the two oscillators is now coupled through the third spring of spring constant $k^{'}$. It is clear that the oscillation of one oscillator affects the second. The phases and amplitudes of the two oscillators are no longer independent and the frequency of oscillation is also modified. We proceed to calculate these effects below.

The equations governing the coupled oscillators are

$$m \frac{d^2 \, x_0}{dt^2} = - k x_0 - k^{'} (x_0 - x_1)$$ (5.3)

and

$$m \frac{d^2 \, x_1}{dt^2} = - k x_1 - k^{'} (x_1- x_0)$$ (5.4)