Oscillations and waves

Damped and driven oscillations; resonance

Damping reduces amplitude; periodic forcing produces driven oscillation, with a large response when the drive frequency is near the natural frequency.

In real systems, friction and drag dissipate mechanical energy, so amplitude decays over time. If a periodic external force continuously acts on the system, it oscillates at the forcing frequency.

m\ddot x+b\dot x+kx=F_0\cos(\omega t),\qquad \omega_0=\sqrt{\frac{k}{m}}

Definition: Natural frequency and resonance

$\omega_0=\sqrt{k/m}$ is the undamped natural angular frequency. In steady state, the driven amplitude is $A(\omega)=F_0/\sqrt{(k-m\omega^2)^2+(b\omega)^2}$ . With weak damping the peak lies near $\omega_0$ ; with appreciable damping, the resonance frequency is below $\omega_0$ .

Vary drive frequency and damping; observe the steady-state amplitude peak near the natural frequency.

Damping and sustained motion

A freely damped oscillator loses energy to its surroundings. In steady driven motion, the external source replenishes energy dissipated each cycle; amplitude depends on drive frequency, damping, and forcing strength. Resonance does not mean energy is created spontaneously.

Example: Estimate the resonance frequency

A system has $m=0.50$ kg, $k=200$ N/m, and weak damping. Estimate its natural frequency in Hz to choose a drive frequency.

Solution

$\omega_0=\sqrt{k/m}=\sqrt{400}=20$ rad/s, so $f_0=\omega_0/(2\pi)\approx3.18$ Hz. With weak damping, the response peaks near this value.

Quick check

In steady state, the oscillator responds mainly at the driving frequency, not its original natural frequency. Amplitude depends on drive frequency and damping; damping lowers and broadens the resonance peak. Resonance can usefully amplify motion, but bridges and machines must be designed to avoid excessive vibration.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear.

In steady forced oscillation, at which frequency does the system oscillate?

What typically happens as the drive approaches resonance with weak damping?

References

Young, Freedman (2019). University Physics