--- title: "Normal Modes of a Bounded System" description: "Boundary conditions select the normal modes of an ideal uniform string or air column. Fixed-end strings and pipes open at both ends allow all whole-number harmonics. Pipes closed at one end allow odd " canonical: https://lightmysky.com/learn/science/normal-modes-of-a-bounded-system-mt_650O-YrqJs source: https://lightmysky.com/learn/science/normal-modes-of-a-bounded-system-mt_650O-YrqJs.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Normal Modes of a Bounded System Boundary conditions select the normal modes of an ideal uniform string or air column. Fixed-end strings and pipes open at both ends allow all whole-number harmonics. Pipes closed at one end allow odd harmonics when end corrections are neglected. Subject: Science · Area: Waves, Light & Sound · Ages 18 to 20 Page: https://lightmysky.com/learn/science/normal-modes-of-a-bounded-system-mt_650O-YrqJs ## Ready when they can - Finds allowed wavelengths and frequencies for an ideal uniform string fixed at both ends - Compares ideal open-open and open-closed pipes using displacement end conditions, neglecting end corrections - Explains how relative harmonic strengths contribute to an instrument's timbre ## Lesson: Normal modes of strings and air columns A normal mode is a standing-wave pattern allowed by the boundaries. On an ideal uniform string, displacement nodes stay at zero. Antinodes lie midway between neighbouring nodes and have the greatest oscillation amplitude, but pass through zero each cycle. Fixed ends require nodes, so L = nλ_n/2, λ_n = 2L/n and f_n = nv/(2L) = n f_1. Here L is length, v is travelling-wave speed and n = 1, 2, 3, ... . The lowest mode frequency f_1 is the fundamental. *(drawing: The fundamental has endpoint nodes and one central antinode. These cells list positions, not the moving string's shape.)* **Example.** A string has fundamental frequency 120 Hz. Its second harmonic is 2 × 120 = 240 Hz, and its third is 3 × 120 = 360 Hz. A harmonic's number multiplies the fundamental frequency. The bars compare frequencies in Hz; their heights do not show loudness. For an ideal uniform pipe, neglect end corrections. A rigid closed end is an air-displacement node; an open end is a displacement antinode. Pressure variation has the opposite end pattern. A pipe open at both ends fits whole numbers of half-wavelengths: λ_n = 2L/n and f_n = nv/(2L), for n = 1, 2, 3, ... . Its fundamental wavelength is 2L and frequency is v/(2L). A pipe closed at one end needs a displacement node there and an antinode at its open end. Node-to-antinode spacing is λ/4. Only odd numbers of quarter-wavelengths join those endpoints: λ_n = 4L/n and f_n = nv/(4L), for n = 1, 3, 5, ... . An overtone is a mode above the fundamental. This pipe's first overtone is the third harmonic at 3f_1; the second harmonic is absent. Compare equal-length pipes at the same sound speed, ignoring end corrections. The open-open fundamental has wavelength 2L; the open-closed fundamental has wavelength 4L. Since f = v/λ, the open-closed frequency is half as large, one octave lower. These are natural-mode predictions. A driven system can still move at other frequencies. Instruments can sustain the same fundamental while their harmonics have different relative strengths. That spectral mix contributes to timbre, the quality that helps us distinguish their sounds. How a sound begins and fades also affects timbre. Allowed mode frequencies alone do not tell us how strongly each mode is excited. **Recap.** Ideal boundaries select normal modes. Fixed-end strings and open-open pipes allow all harmonics; open-closed pipes allow odd ones. Relative harmonic strengths contribute to timbre. *Normal-mode lesson, questions, worked solutions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Superposition and Stationary Waves](https://lightmysky.com/learn/science/superposition-and-stationary-waves-mt_AIxA2TYGJN)