Two identical coupled pendulums: what are the two normal mode frequencies?

Mode 1 (in-phase): both pendulums swing together at ω₀; Mode 2 (anti-phase): slightly higher ω.

Start one pendulum and watch the other. What happens? Why?

Energy slowly transfers between them — a beat pattern; the motion is a mix of both normal modes.

How do normal modes relate to a guitar string's harmonics?

Each harmonic is a normal mode of the string with nodes at both ends; frequency = n×fundamental.

Pro 🔒~25 min

Normal Modes

Coupled oscillators and standing wave patterns

How it works

Normal modes are the independent oscillation patterns of a coupled system — each mode oscillates at a distinct frequency with all parts moving in phase or exactly anti-phase. Any arbitrary motion can be written as a superposition of normal modes. For a string, normal modes are harmonics (standing waves). For coupled pendulums, the two modes are in-phase (lower frequency) and out-of-phase (higher frequency). This concept extends to quantum mechanics (phonons) and molecular vibrations.

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Step-by-step

Click to excite specific normal modes using the mode buttons.
The masses will oscillate in the characteristic pattern.
Switch to 'mix' mode and observe how two modes combine into a complex beating pattern.
For a string view, observe standing wave patterns for each harmonic.

Key formulas

$\omega_n = 2\omega_0\sin\left(\frac{n\pi}{2(N+1)}\right)$ Normal mode frequencies
$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$ String harmonics

Frequently asked questions

Two identical coupled pendulums: what are the two normal mode frequencies?: Mode 1 (in-phase): both pendulums swing together at ω₀; Mode 2 (anti-phase): slightly higher ω.
Start one pendulum and watch the other. What happens? Why?: Energy slowly transfers between them — a beat pattern; the motion is a mix of both normal modes.
How do normal modes relate to a guitar string's harmonics?: Each harmonic is a normal mode of the string with nodes at both ends; frequency = n×fundamental.