§ 19. Standing Waves & String Overtones in 3D (Grade 9–11)
Normal modes of a fixed string, nodes & antinodes, and frequency vs. tension. Superposition of forward and reflected waves creates a standing wave with stationary nodes and vibrating antinodes.
Theory Summary & Scientific Mechanism
A standing wave forms from the superposition of two coherent counter-propagating waves of equal amplitude and frequency on a string of length L fixed at both ends: y(x,t) = y₊(x - vt) + y₋(x + vt) = Asin (kx)cos (ω t). Unlike a traveling wave, a standing wave does not transport net energy along the string; instead, kinetic and potential energy oscillate locally within each half-wavelength loop.
Boundary conditions y(0,t) = y(L,t) = 0 quantize the allowed wavelengths so that an integer number of half-waves fits along L: L = n(λₙ)/(2) (n = 1, 2, 3, …). With transverse wave speed v = √(T/μ) set by tension T and linear mass density μ, the natural frequencies form a discrete harmonic series: fₙ = (n)/(2L)√((T)/(μ)) = n f₁.
Mode n = 1 is the fundamental frequency, while n ≥ 2 are overtones (harmonics) featuring n + 1 stationary nodes and n vibrating antinodes. This boundary-value quantization supports musical acoustics and provides the classical wave analogy for quantized electron states in quantum mechanics.
Everyday Analogy
Pluck a guitar string or shake a jump rope with a friend: the wave reflects off the fixed end and overlaps itself, locking into stationary nodes (quiet points) and antinodes (wildly vibrating loops), the secret behind all musical instruments!
Interactive Question & Derivation
A guitar string of length L fixed at both ends is excited in its n-th harmonic (n = 3). How many antinodes form along the string, and how does the resonant frequency fₙ change if the string tension T is quadrupled?
- A: 3 antinodes form, and the frequency fₙ doubles because fₙ ∝ √(T)
- B: 1 antinode forms, and the frequency increases 4-fold
- C: The number of antinodes and frequency are independent of n and tension T
Study Hint: How many half-wavelengths λ/2 fit along string length L for harmonic mode n (L = nλ/2), and how does frequency fₙ scale with n?
Proof & Explanation: For a string fixed at both ends, standing waves satisfy L = n(λₙ)/(2), so the harmonic number n equals the number of antinodes (with n+1 total nodes including both endpoints). Since wave speed is v = √(T/μ), quadrupling tension T doubles the phase speed and doubles the natural frequency fₙ = (n)/(2L)√((T)/(μ)).
Key Equations & 3D Simulation Parameters
λₙ = (2L)/(n), v = √((T)/(μ)), fₙ = (n)/(2L)√((T)/(μ))
- Harmonic mode (antinodes) n: 1 … 6 (default: 3)
- String tension T: 15 … 90 N (default: 45 N)
- Linear mass density μ: 0.5 … 3.5 g/m (default: 1.5 g/m)
- Antinode amplitude A: 0.4 … 1.6 m (default: 1.1 m)