Lesson 3.4 • Competency 3.4Complete Lesson
Back to UnitStationary Waves: Strings and Organ Pipes
Transverse standing waves on stretched strings, Mersenne’s laws of sonometer, longitudinal standing waves in closed and open air columns, harmonics, and end corrections.
Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.
Learning Objectives
- Derive fundamental and harmonic frequencies for stretched strings, closed pipes, and open pipes.
- Apply end corrections to explain resonance tube acoustic physics.
Core Theory & Concepts
Musical acoustics depends on boundary conditions. A fixed end reflects waves with a 180° phase inversion creating a displacement node, while an open end allows unrestricted oscillation creating an antinode. These boundary constraints select discrete resonant frequencies called harmonics.
Stretched String Fundamental
For string of length L, tension T, and linear density μ: f = (1 / 2L) √(T / μ).
f = (1 / 2L) · √(T / μ)
Common Misconceptions
Misconception
Assuming a closed organ pipe can produce even harmonics (like 2f or 4f).
Physical Clarification:Because a closed pipe requires a node at the closed base and an antinode at the open top, it can only support odd quarter-wavelength modes (λ/4, 3λ/4, 5λ/4), yielding only odd harmonics: f, 3f, 5f...
Syllabus Topic Outline
- 1Vibrations of stretched strings (f = (1/2L) √(T/μ))
- 2Closed organ pipe: odd harmonics f, 3f, 5f...
- 3Open organ pipe: all harmonics f, 2f, 3f, 4f...
- 4Acoustic end correction at tube open ends (e ≈ 0.3 d)