Why Can a Fixed String Only Vibrate at Certain Resonant Frequencies?
Waves & SHM: Standing Waves on a Fixed String
Standing waves form when two identical traveling waves move in opposite directions. Their superposition gives Y(x,t) = 2A sin(kx) cos(ωt) — the sin(kx) factor sets the spatial pattern; cos(ωt) drives the time oscillation.
Boundary conditions force nodes at x = 0 and x = L (fixed ends), restricting allowed wavelengths to λₙ = 2L/n and resonant frequencies to fₙ = nv/(2L). Every harmonic is an integer multiple of f₁.
Nodes (n + 1) are positions of permanent zero displacement. Antinodes (n) oscillate with peak amplitude 2A. Increasing n adds more half-loops — shorter λ, higher frequency. The violet markers show node positions on the t-axis.
Y(x,t) = 2A sin(kx) cos(ωt) | fₙ = nv/(2L) | λₙ = 2L/n
© The Science CubeStanding Waves on a Fixed String — Concept Description (IIT JEE)
Concept Overview
When two sinusoidal waves of identical frequency, amplitude, and speed travel in opposite directions along a string, their superposition produces a standing wave — a pattern that oscillates in place rather than propagating.
The resultant displacement at any point is:
Y(x, t) = 2A sin(kx) cos(ωt)
where k = 2π/λ is the wave number and ω = 2πf is the angular frequency. The term sin(kx) is purely spatial — it determines which points oscillate and which stay fixed. The term cos(ωt) is purely temporal — it drives the oscillation at every point simultaneously.
Key Equations
Quantity Formula Allowed wavelengths λₙ = 2L/n Resonant frequencies fₙ = nv / 2L Fundamental frequency f₁ = v / 2L nth harmonic fₙ = n·f₁ Wave speed on string v = √(T/μ) Number of nodes n + 1 Number of antinodes n where L = string length, v = wave speed, T = string tension, μ = linear mass density, n = 1, 2, 3, … (harmonic number).
Nodes are positions where sin(kx) = 0 — permanent zero displacement. They occur at x = 0, L/n, 2L/n, … L.
Antinodes are positions where |sin(kx)| = 1 — maximum displacement 2A. They lie midway between consecutive nodes.
How to Use the Simulation
String Animation panel shows the standing wave Y(x, t) = 2A sin(kx) cos(ωt) in real time. White dots mark nodes (N); cyan dots mark antinodes (AN) on the oscillating string.
Harmonic Series graph plots fₙ against n for harmonics 1–8. The selected harmonic is highlighted — observe that fₙ increases linearly with n, confirming fₙ = n·f₁.
Controls:
- Harmonic Number n — switch between the 1st (fundamental) through 8th harmonic. Watch the number of loops, nodes, and antinodes update instantly.
- String Length L — increasing L lowers all resonant frequencies (fₙ ∝ 1/L). The wavelength of each harmonic stretches accordingly.
- Wave Speed v — proportional to fₙ at fixed L. Physically, v = √(T/μ), so this slider simulates changing tension or linear density.
- Amplitude A — scales the peak displacement (2A at antinodes). Does not affect frequency or node positions.
- Slow Motion — reduces animation speed to 15% — useful for tracking phase at individual points.
- Step — advances time by one frame when paused. Use this to freeze the wave at a chosen phase and study the spatial pattern.
- Reset — restores all parameters to defaults (n = 1, L = 1.0 m, v = 50 m/s, A = 0.20).
JEE-Relevant Observations to Make
- Set n = 1 (fundamental). Note 2 nodes, 1 antinode, λ₁ = 2L.
- Double n to 2. Frequency doubles, wavelength halves — verify λ₂ = L.
- Keep n fixed, halve L. Observe fₙ doubles — confirms fₙ ∝ 1/L.
- Use Slow Motion at n = 4 or higher to clearly see that nodes never move regardless of the phase of cos(ωt).
- Note that the Harmonic Series graph is always linear through the origin — the slope equals f₁ = v/2L.
