From two oscillators to the vibrating string

The step from two coupled oscillators to any number is natural. With $N$ identical masses connected by $N+1$ springs to the walls and $N-1$ coupling springs, one obtains $N$ normal modes, each with its own frequency.

In the limit $N \to \infty$, with masses and springs becoming continuous densities, the system converges to a vibrating string: its normal modes are the standing-wave harmonics we shall study in the chapter on waves. The discrete chain of masses and springs and the continuous string are the same physics, seen at two different resolutions.

This is where Fourier analysis is born

Every complicated periodic motion is a superposition of pure harmonic motions: this is the content of Fourier analysis, and it arises exactly from this passage. The same logical pattern recurs for the phonons of a solid (quantised lattice vibrations), for the fields of quantum field theory, for the modes of gravitational waves. Physics is much more single-themed than it seems: once you have learned the harmonic oscillator and its modes, you have learned a piece of almost everything.

Summary of the thread

• 2 coupled oscillators $\to$ 2 normal modes (symmetric, antisymmetric).
• Superposition of nearby modes $\to$ beats, energy migrating back and forth.
• Generalisation to $N$ masses $\to$ vibrating string and, in the continuum limit, waves and Fourier.

Links

Topics: Oscillations and harmonic motion Concepts: Normal modes · Standing waves Objects: Vibrating string

Related exercises: Problem — Two coupled swings · Worked exercise — Two pendulums coupled by a string · Problem — Identifying the third normal mode