Drive an oscillator at its own natural frequency and each push arrives just when it helps most. Without friction the amplitude then grows without bound — the particular solution is t times a sine, the modification rule of undetermined coefficients in physical form. Close to the natural frequency the motion beats, swelling and fading; with friction the amplitude stays finite but peaks sharply near the natural frequency.

Pushing a swing. Push at random moments and little happens; push once a swing, at the top, and even small pushes build a large swing. A wineglass shattered by a sung note and a footbridge set swaying by walkers in step are the same arithmetic.

$x''+x=\cos t$, driven at its own frequency and started at rest: $x=\tfrac t2\sin t$, an oscillation whose amplitude grows in proportion to time.

$x''+4x=\cos1.8t$ from rest: $x=\tfrac{1}{0.76}(\cos1.8t-\cos2t)$, a fast oscillation inside a slow envelope that swells and fades — beats, with a loud-and-quiet cycle every .

For $x''+\omega_{0}^{2}x=F_{0}\cos\omega t$ with , , whose amplitude blows up as . At the plain guess solves the homogeneous equation, the modification rule applies, and grows without bound. With damping the amplitude is finite for every , and largest near when is small.

• Near resonance, from rest, $x=\tfrac{F_{0}}{\omega_{0}^{2}-\omega^{2}}(\cos\omega t-\cos\omega_{0}t)$, a fast oscillation inside a slow envelope: beats.

The steady amplitude of $x''+cx'+x=\cos\omega t$ against the driving frequency , for and . Both peak near the natural frequency , and the lighter the damping the taller and narrower the peak.

The plain guess solves the homogeneous equation, so the modification rule multiplies it by t; the cosine part of the modified guess turns out to have coefficient zero, so try t times a sine. Differentiating twice by the product rule gives twice omega times a cosine, minus omega squared times t times a sine. Adding omega squared times the trial cancels the terms in t and leaves twice omega times B times the cosine, which must equal the forcing. That fixes B, and the amplitude grows in proportion to t.

The modification rule: the plain guess solves the homogeneous equation, so multiply by t.

Differentiate twice by the product rule.

The terms in t cancel.

Matching the forcing fixes the coefficient.

$x''+\omega_{0}^{2}x=F_{0}\cos\omega t$ resonates when : the forcing then solves the homogeneous equation.

Great work! You now know how to:

• recognise resonance and find the growing particular solution
• explain beats as two close frequencies adding
• say why damping keeps the amplitude finite and how the peak depends on it
• read an amplitude curve against the driving frequency