Mechanical Vibrations

Infinitely Many Masses — Standing Waves

Let the masses go to infinity and you get a string or beam; the modes are integer-multiple standing waves fₙ = n·f₁ (harmonics), pitch set by f₁ = (1/2L)√(T/μ)

Last lesson, N masses gave N modes. Let the masses become infinite — a single continuous string — and the modes become infinite too: each one a smooth standing wave with both ends pinned, its frequency an integer multiple of the fundamental. That is why a string makes music.

Slide N. A few masses give the jagged mode shape of the last lesson; as they multiply, the shape converges to one smooth curve — a continuous string. A continuous system is a multi-DOF system with infinitely many masses.

The simplest mode, the fundamental: the whole string rises and falls as a single arch with its ends pinned (nodes). Half a wavelength spans the string, so λ₁ = 2L, and the frequency is the lowest, f₁.

Pick a higher mode. The nth mode has n arches with n−1 nodes between them, and its frequency is fₙ = n·f₁ — an integer multiple of the fundamental. This integer ladder is the harmonic series, the reason a string sounds musical.

A5's superposition, now with infinitely many modes. Pluck the string at a point and that triangular shape is a sum of standing-wave modes. Add more harmonics — the corner sharpens and the sum converges to the plucked shape. The particular blend is the timbre.

What sets the pitch? f₁ = (12L)√(Tμ) — tension T, length L, linear density μ. Move all three: tighter (T↑), shorter (L↓, fretting), or thinner (μ↓) raises the pitch. It is why bass strings are thick — and why bridge cables and power lines hum the same way.

In PracticeA continuous system — string, beam, or plate — is a multi-DOF system with infinitely many masses, so it has infinitely many modes. Each mode is a standing wave between fixed nodes; on a string the frequencies are fₙ = n·f₁, integer multiples of the fundamental (the harmonics). Any vibration decomposes into a sum of these modes (the plucked shape = a sum of harmonics = the timbre), and the fundamental is f₁ = (12L)√(Tμ), set by tension, length, and linear density. The practical work is unchanged: find the modes' shapes and frequencies, and keep any driving force away from all of them. Bridges, wings, and buildings are continuous bodies with infinitely many modes too, and standing waves are where vibration meets waves and music.

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