Mathematics › 🎓 University Year 1 › The Harmonic Oscillator: The Archetype of Oscillation

The Harmonic Oscillator: The Archetype of Oscillation

A spring, a swinging pendulum and an electrical circuit all obey the very same differential equation — a restoring pull proportional to displacement — and all of them answer with a sine wave.

University Year 1Calculus II / Linear Algebra

The Harmonic Oscillator: The Archetype of Oscillation — illustration

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The big idea: Whenever a system is pulled back toward equilibrium by a force proportional to how far it has strayed, it obeys the equation x″ + ω²x = 0. This one differential equation is the beating heart of springs, pendulums, LC circuits and molecular vibrations. Its solution is a sinusoid x(t) = A cos(ωt + φ): the amplitude A sets the size of the swing, the angular frequency ω sets how fast, and the phase φ sets where it starts. Understanding it once means understanding oscillation everywhere.

🎯 By the end, you'll be able to

Recognise simple harmonic motion from a restoring force proportional to displacement
Write the governing equation x″ + ω²x = 0 and verify its sinusoidal solution
Relate angular frequency ω, period T and frequency f
Compute ω for a spring (√(k/m)) and identify the pendulum and LC analogues
Interpret amplitude and phase, and describe how damping changes the picture

📎 You should already know

Derivatives of sine and cosine
Second derivatives
Radian angle measure

A pull that grows with distance

Stretch a spring and it pulls back; stretch it twice as far and it pulls back twice as hard. That is Hooke's law: the restoring force is proportional to the displacement and points back toward the resting position, F = −kx. The minus sign is the whole story — the force always fights the displacement.

Feed that force into Newton's second law, F = ma, and remember that acceleration is the second derivative of position. Out pops the equation that governs every simple oscillation.

🔑 The equation of simple harmonic motion

A restoring force proportional to displacement gives m x″ = −kx, which rearranges to x″ + ω²x = 0 with ω = √(k/m). Any system obeying this equation performs simple harmonic motion. The constant ω is the angular frequency — it alone sets the rhythm.

\[ x'' + \omega^2 x = 0, \qquad \omega = \sqrt{\tfrac{k}{m}} \]

The simple-harmonic-motion equation: a second derivative that is the negative of the function itself (up to the factor ω²).

🎮 Harmonic Oscillator LIVE

A spring or pendulum traces a sine wave in time — the archetype of oscillation.

The answer is a sinusoid

What function is the negative of its own second derivative? Sine and cosine. The general solution is x(t) = A cos(ωt + φ). Here A is the amplitude (how far it swings), ω is the angular frequency (how fast), and φ is the phase (where in the cycle it starts). Differentiate twice and you get back −ω² times the original — the equation is satisfied for any A and φ.

\[ x(t) = A\cos(\omega t + \varphi) \;\Rightarrow\; x''(t) = -\omega^2 A\cos(\omega t + \varphi) = -\omega^2 x(t) \]

Differentiating the sinusoid twice returns −ω²x, confirming it solves the equation for any amplitude and phase.

Frequency and period

The motion repeats every time ωt advances by 2π, so the period is T = 2π/ω — the time for one full cycle. Its reciprocal is the frequency f = 1/T = ω/2π, the number of cycles per second. Notice what is missing: none of these depend on the amplitude. A big swing and a small swing of the same oscillator take exactly the same time — the property that made pendulum clocks possible.

\[ T = \frac{2\pi}{\omega}, \qquad f = \frac{1}{T} = \frac{\omega}{2\pi} \]

Period and frequency depend only on ω, never on the amplitude — the hallmark of simple harmonic motion.

📝 Worked example: A mass of 0.5 kg on a spring with stiffness k = 2 N/m is pulled 0.1 m and released from rest. Find ω, the period, and the motion x(t).

Angular frequency: \( \omega = \sqrt{k/m} = \sqrt{2/0.5} = \sqrt{4} = 2 \) rad/s.
Period: \( T = 2\pi/\omega = 2\pi/2 = \pi \approx 3.14 \) s.
Released from rest at maximum displacement, so A = 0.1 m and it starts at a peak — take φ = 0 with a cosine.

✓ ω = 2 rad/s, T = π ≈ 3.14 s, and x(t) = 0.1 cos(2t) m.

✨ The same equation, three disguises

The oscillator is universal. A pendulum (small swings) obeys the same equation with ω = √(g/L). An LC circuit oscillates charge with ω = 1/√(LC). Molecular bonds, tuning forks and swaying bridges all reduce to x″ + ω²x = 0. Energy sloshes back and forth between two stores — kinetic and potential, or magnetic and electric — while the total E = ½kA² stays fixed.

⚠️ Real oscillators lose energy

The pure equation assumes no friction, so the swing never dies. Add a resistance proportional to velocity and you get the damped equation x″ + 2βx′ + ω²x = 0, whose solutions are sinusoids inside a decaying envelope. And the small-angle pendulum formula is an approximation: for large swings the motion is still periodic but no longer a simple sine.

Check your understanding

1. What defines simple harmonic motion?

SHM arises whenever the restoring force is proportional to displacement and directed back toward equilibrium: F = −kx.

2. Which function solves x″ + ω²x = 0?

A sinusoid A cos(ωt + φ) has second derivative −ω²x, so it satisfies the equation for any amplitude A and phase φ.

3. For a spring, how does the angular frequency depend on mass and stiffness?

From m x″ = −kx, ω = √(k/m): stiffer springs oscillate faster, heavier masses slower.

4. A mass–spring system has ω = 2 rad/s. What is its period?

T = 2π/ω = 2π/2 = π ≈ 3.14 s.

5. If you double the amplitude of an ideal harmonic oscillator, its period…

The period T = 2π/ω depends only on ω, not on amplitude — big and small swings take the same time.

✅ Key takeaways

Simple harmonic motion arises from a restoring force proportional to displacement, giving x″ + ω²x = 0.
The solution is a sinusoid x(t) = A cos(ωt + φ): amplitude A, angular frequency ω, and phase φ.
The period T = 2π/ω and frequency f = ω/2π depend only on ω, never on the amplitude.
For a spring ω = √(k/m); a pendulum uses √(g/L) and an LC circuit uses 1/√(LC) — one equation, many systems.
Real oscillators are damped (x″ + 2βx′ + ω²x = 0), giving a sinusoid inside a decaying envelope.