The Anharmonic Oscillator

Definition

Start with the workhorse of introductory physics — the simple harmonic oscillator. Its restoring force is strictly linear:

F = -kx     (Hooke's law)     U(x) = ½kx²

The potential is a perfect parabola, and the resulting motion is a single clean sinusoid whose angular frequency ω₀ = √(k/m) does not depend on how large the swing is. This special property is called isochronism, and it is a mathematical accident of the parabola.

An anharmonic oscillator is anything for which this is not exactly true. Expand any real potential in a Taylor series about its minimum at x = 0:

U(x) = ½kx² + (1/3)αx³ + (1/4)βx⁴ + …
F(x) = -dU/dx = -kx - αx² - βx³ - …

The terms beyond ½kx² are the anharmonic terms. The cubic (αx³) is an asymmetric correction — it tilts the well so one side is steeper than the other. The quartic (βx⁴) is a symmetric correction — it stiffens (β > 0, "hardening") or softens (β < 0, "softening") the well while keeping it symmetric. Because these terms grow faster than x, they are invisible at small amplitude and dominate at large amplitude.

Why anharmonicity matters

Simple harmonic motion is a lie we tell ourselves at small amplitude. The moment you look closely — or push hard — nature is anharmonic. This one idea unifies a startling range of physics:

• Thermal expansion. A symmetric parabola cannot expand a solid when heated; the cubic term in the interatomic potential is why bridges, rails and pistons grow with temperature.
• Molecular spectroscopy. The convergence of vibrational energy levels toward a dissociation limit, and the appearance of overtone bands, are direct fingerprints of the anharmonic Morse potential.
• Nonlinear optics. Electrons in an anharmonic potential re-radiate at doubled frequency — the basis of second-harmonic generation and the green laser pointer.
• Audio distortion. Overdriven loudspeaker cones and amplifiers add harmonics and intermodulation tones because their restoring behavior is no longer linear.
• Chaos. The driven double-well Duffing oscillator is one of the simplest systems to exhibit deterministic chaos and a strange attractor.
• MEMS and precision clocks. Micro-mechanical resonators become anharmonic at large drive, limiting their frequency stability and inviting the amplitude "jump" phenomenon.

How it works, step by step

1. A parabola makes the period constant. For F = -kx the equation of motion mẍ = -kx has the solution x(t) = A cos(ω₀t + φ) with ω₀ = √(k/m). Notice A cancels out of ω₀ — double the amplitude and the period is unchanged. That is isochronism.

2. A quartic well breaks isochronism. Add βx³ to the force (a symmetric Duffing potential U = ½kx² + ¼βx⁴). Now the effective stiffness felt by the mass depends on where it is. Averaged over a cycle, the frequency shifts with amplitude A. Multiple-scales / averaging perturbation theory gives, to leading order:

ω(A) ≈ ω₀ · ( 1 + (3β A²) / (8 ω₀²) )

For a hardening spring (β > 0) the frequency rises with amplitude; for a softening spring (β < 0, the pendulum case) it falls.

3. A cubic well makes the motion asymmetric. With an αx³ term the mass spends more time on the softer side of the well, so its time-averaged position <x> drifts away from x = 0. Heat the oscillator (increase its energy) and <x> drifts further — the microscopic origin of thermal expansion.

4. The response is no longer a pure sine. Because the restoring force distorts the motion, driving at frequency ω produces output at ω, 2ω, 3ω, … The Fourier spectrum sprouts harmonics whose amplitudes grow with drive strength.

5. Two tones mix. Drive at ω₁ and ω₂ simultaneously and the nonlinearity generates combination frequencies. A quadratic term (αx²) produces sum/difference tones ω₁ ± ω₂; a cubic term (βx³) produces ω₁ ± 2ω₂ and 2ω₁ ± ω₂. These are heterodyne and intermodulation products.

6. The resonance peak bends and jumps. Solve the driven, damped Duffing equation and the amplitude-vs-frequency curve leans over (to higher ω for hardening, lower ω for softening). Over a range of drive frequencies three steady-state amplitudes coexist; as you sweep the drive the response snaps discontinuously between branches — a hysteretic amplitude jump with no counterpart in linear resonance.

The Duffing equation and its resonance

The complete damped, driven anharmonic oscillator is written compactly as the Duffing equation:

ẍ + 2γ ẋ + ω₀² x + β x³ = F₀ cos(ω t)

with symbols and SI units:

Symbol	Meaning	Units
x	displacement from equilibrium	m
ẋ, ẍ	velocity, acceleration	m/s, m/s²
γ	damping rate (½ the coefficient of ẋ)	s⁻¹
ω₀ = √(k/m)	natural (small-amplitude) angular frequency	rad/s
β	cubic nonlinearity per unit mass (+ hardening, − softening)	m⁻² s⁻²
F₀	drive acceleration amplitude (force per unit mass)	m/s²
ω	drive angular frequency	rad/s

Set β = 0 and this collapses to the ordinary driven, damped harmonic oscillator. Keep β ≠ 0 and you get the bent backbone curve, the amplitude jump, and — if ω₀² < 0 so the potential has two wells — period-doubling routes to chaos. Duffing published this in his 1918 monograph Erzwungene Schwingungen bei veränderlicher Eigenfrequenz ("Forced oscillations with variable natural frequency").

Harmonic vs. anharmonic at a glance

Property	Harmonic (SHO)	Anharmonic
Restoring force	F = -kx (exactly linear)	F = -kx - αx² - βx³ - … (nonlinear)
Potential shape	Perfect parabola U = ½kx²	Non-parabolic (tilted / stiffened / softened)
Period vs. amplitude	Independent (isochronous)	Amplitude-dependent
Waveform	Pure sinusoid, single frequency	Distorted — contains 2ω, 3ω, …
Two-tone drive	Just ω₁ and ω₂	Sum/difference & intermodulation tones
Resonance curve	Symmetric Lorentzian peak	Bent peak, hysteretic jump
Quantum levels	Evenly spaced ℏω(n+½)	Converging (Morse): ℏω(n+½) − ℏωxₑ(n+½)²
Thermal expansion	Exactly zero	Nonzero (from asymmetric term)
Long-time behavior	Always regular	Can be chaotic (double well, driven)

Worked example — the pendulum as a softening oscillator

The simple pendulum is the most familiar anharmonic oscillator. Its exact equation of motion is

θ̈ + (g/L) sin θ = 0

Expanding sin θ ≈ θ − θ³/6 shows the restoring "force" gains a negative cubic term — a softening spring. Because of it, the period lengthens with amplitude θ₀ according to the series

T(θ₀) = T₀ · ( 1 + (1/16)θ₀² + (11/3072)θ₀⁴ + … )

where T₀ = 2π√(L/g) is the small-angle period. Plug in numbers for a few amplitudes:

Amplitude θ₀	Fractional period increase
5° (0.087 rad)	+0.048%
10° (0.175 rad)	+0.19%
23° (0.40 rad)	+1.0%
45° (0.785 rad)	+3.99%
90° (1.571 rad)	+18.0%

This is exactly the anharmonic effect Christiaan Huygens confronted in the 1650s–1670s: because a wide-swinging pendulum runs slow, it is not perfectly isochronous, which spoils timekeeping. His cycloidal cheeks were an engineering fix that forced the bob onto a truly isochronous (tautochrone) path — an early acknowledgment that the real pendulum is anharmonic.

Worked example — thermal expansion from an asymmetric well

Model an interatomic bond by U(x) = ½kx² − gx³ (a Morse/Lennard-Jones potential expanded to third order, with g > 0 making the well softer on the stretch side). Using classical statistical mechanics, the thermal average displacement at temperature T is, to leading order,

⟨x⟩ ≈ (3 g / k²) · k_B T

where k_B = 1.381 × 10⁻²³ J/K. The average bond length grows linearly with temperature, and the linear thermal-expansion coefficient α_L = (1/L)(d⟨x⟩/dT) comes out positive and roughly constant — matching observation. Representative values: aluminum 23 × 10⁻⁶ K⁻¹, steel 12 × 10⁻⁶ K⁻¹, fused silica 0.5 × 10⁻⁶ K⁻¹, and the anomalous Invar alloy ≈ 1 × 10⁻⁶ K⁻¹. Crucially, if you drop the −gx³ term, ⟨x⟩ = 0 for all T and the expansion coefficient is exactly zero — a purely harmonic solid never expands.

Quantum anharmonicity and molecular spectra

Quantize a real bond and the evenly spaced ladder of the quantum harmonic oscillator, Eₙ = ℏω(n + ½), gives way to the Morse spectrum:

Eₙ = ℏω(n + ½) − ℏω xₑ (n + ½)²

The anharmonicity constant xₑ is small and positive, so successive levels crowd together and eventually reach a dissociation limit rather than continuing forever. This is why infrared and Raman spectra show overtone bands (n = 0 → 2, 0 → 3) at frequencies slightly below exact integer multiples of the fundamental, and why the fundamental itself is a hair below the harmonic value. For hydrogen H₂ the vibrational constant is ω ≈ 4401 cm⁻¹ and ωxₑ ≈ 121 cm⁻¹; for carbon monoxide CO, ω ≈ 2170 cm⁻¹ and ωxₑ ≈ 13 cm⁻¹.

Common misconceptions

• "Anharmonic just means nonlinear damping." No — anharmonicity is a nonlinear restoring force (a non-parabolic potential). Nonlinear damping (like drag ∝ ẋ²) is a separate effect. A frictionless anharmonic oscillator is still anharmonic.
• "Real springs obey Hooke's law until they break." Hooke's law is only the first term of a Taylor expansion; springs deviate continuously, and at large strain a coil stiffens (coil binding) then plastically yields past its elastic limit.
• "Amplitude-dependent frequency violates conservation of energy." It does not. Energy is perfectly conserved in an undriven, undamped anharmonic oscillator; the period simply depends on the total energy (equivalently, on amplitude).
• "A symmetric anharmonic term causes thermal expansion." Only the asymmetric (odd-power, e.g. cubic) terms shift ⟨x⟩ and drive expansion. A purely quartic, symmetric well changes the frequency but keeps ⟨x⟩ = 0, so it does not expand.
• "Harmonics in the output mean the drive contained them." A single-frequency drive of a linear system gives a single-frequency output. Harmonics in the response are generated by the nonlinearity itself — evidence the system is anharmonic.
• "The Duffing amplitude jump is a resonance like any other." The jump is a genuine bistability with hysteresis: the amplitude you get depends on the sweep direction. Linear resonance has a single, direction-independent Lorentzian peak.

Where anharmonicity shows up

• Solid-state physics. Phonon–phonon scattering (a purely anharmonic process) gives crystals finite thermal conductivity and causes thermal expansion; harmonic crystals would have infinite conductivity and zero expansion.
• Nonlinear optics. The anharmonic electronic potential produces second- and third-harmonic generation, sum-frequency mixing, and the optical Kerr effect.
• Molecular and IR spectroscopy. Morse anharmonicity sets overtone positions and dissociation energies.
• Acoustics and audio. Total-harmonic-distortion (THD) and intermodulation distortion in speakers, strings and reeds are anharmonic effects.
• Nonlinear dynamics and chaos. The driven double-well Duffing oscillator is a textbook route to a strange attractor.
• MEMS/NEMS resonators and clocks. Drive them hard and they exhibit the Duffing bent peak and amplitude jump, limiting stability.
• Josephson junctions. A current-biased junction behaves as an anharmonic oscillator; its non-uniform level spacing is what makes a superconducting qubit addressable.