A forced oscillation occurs when an oscillator is driven by an external force. This force must also have a periodic form and can be expressed, for example, as:
$$
F(t) = F_0 \sin(\omega t)
$$
Assuming the oscillator is also subject to damping, the equation of motion extends the damped oscillator equation to:
$$
m\ddot{x}(t) + 2\gamma\dot{x}(t) + \omega_0^2 x(t) = F_0 \sin(\omega t)
$$
As with damped oscillations, we focus here on the solution after the transient response has died out. The steady-state amplitude of the resulting oscillation is:
$$
A(\omega) = \frac{F_0 / m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}}
$$
This equation describes how the amplitude depends on the driving frequency $\omega$. The parameters $\omega_0$ and $\omega$ are the natural frequency (or resonance frequency) of the oscillator and the driving frequency, respectively.
An important observation is that for any driving frequency $\omega$, the system eventually reaches a steady-state oscillation with a constant amplitude. A particularly interesting case occurs when the system is in resonance, i.e., when $\omega = \omega_0$. In this case, the first term under the square root vanishes, and the amplitude reaches its maximum.
If the system were undamped (i.e., $\gamma = 0$), the amplitude would theoretically become infinite. This is known as a resonance catastrophe. Such scenarios can have catastrophic consequences—for instance, in bridges:
Strong winds or vibrations can synchronize with the bridge's natural frequency, leading to excessive oscillations that may cause structural failure. For this reason, traffic laws in some countries prohibit marching in unison over bridges, as synchronized footfalls can accidentally match the resonance frequency and lead to damage.