We could now use $x_1=(u+v)/2$ and $x_2=(u-v)/2$ to write general expressions for the motions of the two masses, but the result is not very illuminating. We would find that each mass follows a complex motion that’s the sum of two oscillations at the two different frequencies $\omega$ and $\text{Ω}$.

It’s more useful to see what happens if we choose the initial conditions carefully. Suppose we choose initial conditions such that $B=0$ and ${\varphi }_{\text{a}}={\varphi }_b=0$. This makes $u=2A\text{cos}\omega t$ and $v=0$. Then, using $x_1=(u+v)/2$ and $x_2=(u-v)/2$, we find

This is a solution, shown in FIGURE 15.28, in which two masses have exactly the same motion, both in SHM at angular frequency $\omega$. This simple pattern, where the masses oscillate together at the same frequency, is a normal mode of the system. It’s called the symmetric normal mode because the two masses are doing the same thing at the same time. You can see that it’s not hard to create the initial conditions in which three of the four constants are zero: Simply pull both masses the same distance A to the right and let go.

FIGURE 15.28 The symmetric normal mode.

It’s interesting that the oscillation angular frequency $\omega =\sqrt{k\text{/}m}$ is exactly the same as for independent, uncoupled oscillations. A careful look at Figure 15.28 shows why: If both masses have exactly the same motion, then the middle spring is never stretched or compressed. It never exerts a force on either mass and has no effect on their motion. For all intents and purposes, the middle spring simply isn’t there for the symmetric normal mode.

Now consider the slightly different initial conditions with $A=0$ and ${\varphi }_{\text{a}}={\varphi }_b=0$. In this case $u=0$ and $v=2B\text{cos}\text{Ω}t$. Another use of $x_1=(u+v)/2$ and $x_2=(u-v)/2$ finds

This motion is shown in FIGURE 15.29. Both masses are again undergoing SHM at the same angular frequency $\text{Ω}$, but this time 180° out of phase with each other. This is called the antisymmetric normal mode. You could create this normal mode by displacing the masses equal distances in opposite directions and then releasing them.

FIGURE 15.29 The antisymmetric normal mode.

We noted in Equation 15.68 that $\text{Ω}>\omega$, so the antisymmetric mode frequency $\text{Ω}$ is higher than the symmetric mode frequency $\omega$. This is not surprising because now the middle spring is very much involved, increasing the restoring forces on the masses.