Two or more oscillators that interact are called coupled oscillators. In general, the motions of a system of coupled oscillators are very complex. However, coupled oscillators can also exhibit simple patterns in which individual oscillators move “in sync” with each other at the same frequency. These simple oscillatory patterns are called the normal modes of the system.

Normal modes play an important role in science and engineering—from the vibrations of molecules and motors to the generation of microwaves and laser pulses. The theory of normal modes is the subject of advanced courses, but we can illustrate the basic ideas in a simple situation with two coupled oscillators.

FIGURE 15.27 shows two oscillators, each a mass m on a spring with spring constant k, that are connected by a third spring with a different spring constant l. It’s this third spring that provides the coupling between the oscillators; without it they would oscillate independently of each other at angular frequency $\omega =\sqrt{k\text{/}m}$. We assume that the springs are massless and that no spring is stretched or compressed when the masses are at rest.

FIGURE 15.27 Two coupled oscillators.

Writing Newton’s second law for each mass is straightforward, although care must be taken with the signs. Suppose mass 1 is at $x_1$ and mass 2 at $x_2$, each measured from its equilibrium position. The left spring is stretched by $x_1$ while the middle spring is stretched by $x_2-x_1$. According to Hooke’s law, the x-component of the net force on mass 1 is

The minus sign with the first term indicates a pulling force to the left from the left spring if $x_1>0$ and a pushing force to the right if $x_1<0$. The plus sign with the second term shows that the middle spring pulls to the right if it is stretched $(x_2>x_1)$ but pushes to the left if compressed $(x_1<x_2)$. Similar considerations find that the net force on mass 2 is

For each mass, Newton’s second law is $F_x=m{a}_x=m{d}^{2}x/d{t}^2$. Dividing both sides of Equations 15.62 and 15.63 by m gives the equations of motion:

These are coupled differential equations because $x_1$ and $x_2$ appear in both. Keep in mind that $x_1$ and $x_2$ are not numbers but are functions of time—$x_1(t)$ and $x_2(t)$—that describe the motions of the two masses.

Sometimes coupled differential equations can be solved by a suitable change of variables. For example, let’s define $u=x_1+x_2$ and $v=x_1-x_2$. Thus u and v are also functions of time, not numbers. If we add Equations 15.64, the ($x_2-x_1$) term cancels and we find

This is now a differential equation for $u=x_1+x_2$. Similarly, subtracting the second equation from the first in Equations 15.64 gives a differential equation for $v=x_1-x_2$. The change of variables has changed the equations to

Not only are these now uncoupled differential equations, each independent of the other, but also they are differential equations that we know how to solve! They are none other than Equation 15.32 for the SHM of a single mass on a spring, so we can immediately write the solutions:

where the angular frequencies are

The four constants A, B, ${\varphi }_{\text{a}}$, and ${\varphi }_{\text{b}}$, which include two phase constants, must be determined by the four initial conditions: the initial position and initial velocity of each mass. The factors of 2 in the amplitudes are not essential but will lead to a simpler result at the next step of our analysis.