JEE/NEET Physics · Oscillations & Waves series · Part 6 of 8 · All parts →
- Two identical waves travelling opposite ways lock into a standing pattern
- Nodes: points that never move · Antinodes: points of maximum shake
- A string fixed at both ends fits whole numbers of half-wavelengths
- The harmonic series: f₁ = v/2L, then 2f₁, 3f₁… — the note menu of every instrument
- Open and closed pipes: different rules, same ‘whole-number’ idea
Pluck a guitar string and it doesn’t look like a travelling wave — it blurs into a fixed shape vibrating in place. That locked pattern is a standing wave, and the ‘allowed shapes’ it can take are exactly why a guitar string offers specific notes and not others. Part 6 of the Oscillations & Waves series.
- How standing waves form
- Nodes and antinodes
- The string’s allowed notes (harmonics)
- Pipes: open vs closed
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
How Standing Waves Form
Send a wave down a string; it reflects from the fixed end; the reflected wave travels back through the incoming one. When identical opposite-direction waves overlap, superposition locks them into a standing pattern: some points never move (nodes), some shake maximally (antinodes) — the wave stands and vibrates in place.
Nodes and Antinodes
| Feature | What it does | Where |
|---|---|---|
| Node | never moves — destructive interference always | fixed ends; interior points spaced λ/2 apart |
| Antinode | maximum oscillation — constructive always | midway between nodes, spaced λ/2 apart |
Node-to-node distance is always λ/2 — the ruler hidden inside every standing-wave question.
The String’s Allowed Notes (Harmonics)
A string fixed at both ends must have nodes AT the ends — so only whole numbers of half-wavelengths fit. Each fit is a harmonic:
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| L | string (or pipe) length | m |
| v | wave speed on the string (√(T/μ), Part 5) | m/s |
| n | harmonic number (1, 2, 3…) | f_n = n·f₁ |
Every string instrument is this table: press a guitar string at the 12th fret (L halves) → f₁ doubles → the octave. Shorter, tighter, lighter = higher (why violins sing higher than cellos).
Pipes: Open vs Closed
| Pipe | Ends behave like | Allowed harmonics | Fundamental |
|---|---|---|---|
| Open at BOTH ends | antinodes at both | ALL: f₁, 2f₁, 3f₁… | f₁ = v/2L |
| Closed at ONE end | node at closed, antinode at open | ODD only: f₁, 3f₁, 5f₁… | f₁ = v/4L |
The closed pipe sounds an octave lower (L/4 vs L/2 pattern) and misses the even harmonics — that’s the darker timbre of a clarinet versus a flute.
Solved Examples
Whole-number menu: 440 Hz and 660 Hz. The octave (440) is literally harmonic 2. ✔
Answer: f₂ = 440 Hz; f₃ = 660 Hz
f₁ = v/2L = 400/2 = 200 Hz. 3rd harmonic = 600 Hz, fitting 3 half-wavelengths: 4 nodes (including ends), 3 antinodes.
λ₃ = 2L/3 ≈ 0.67 m ✔
Answer: f₁ = 200 Hz; f₃ = 600 Hz (4 nodes, 3 antinodes)
Odd only: next is 3f₁ = 300 Hz (200 Hz is forbidden!).
Length: L = v/4f₁ = 340/400 = 0.85 m.
The clarinet’s dark sound is precisely the missing even harmonics. ✔
Answer: 300 Hz next; L = 0.85 m
- Giving the closed pipe all harmonics. Closed-one-end = odd only (f₁, 3f₁, 5f₁). The 2nd harmonic does not exist there — a favourite MCQ.
- Using L/2 patterns for closed pipes. Open pipe: f₁ = v/2L. Closed: v/4L — a full octave apart at the same length.
- Miscounting nodes/antinodes. Fixed ends ARE nodes — count them in. Nth harmonic on a string: n antinodes, n+1 nodes.
- Changing f instead of λ in a new medium. Same rule as Part 5: frequency holds.
This Physics in Your Daily Life
- Every fret on a guitar shortens L to jump to a chosen harmonic menu — the fretboard is a standing-wave calculator you play.
- Flutes (open) vs clarinets (closed) sound different because of the odd-harmonic rule — same physics, different timbre.
- Wind players ‘overblow’ to jump from f₁ to the next allowed harmonic — octave on flute, twelfth (3f₁) on clarinet: the odd-only rule, audible.
- Microwave ovens set up standing waves of ~12 cm — the rotating plate exists because nodes cook cold; rotation averages them out. Turntable = node management.
- Noise-cancelling headphones inject the opposite wave to create destructive interference — engineered ‘anti-standing’ patterns at your ear.
Practice set (answers hidden — try first)
(NEET-level) String fundamental 100 Hz. Fifth harmonic:
(JEE Main-level) Closed pipe fundamental 150 Hz. Next allowed:
(NEET-level) Node-to-node distance in wavelengths:
(JEE Main-level) A 0.5 m open pipe, v = 340. Fundamental:
(Concept) Pressing a guitar string midway (12th fret) changes f₁ how?
- 🧠 Chant: ‘nodes never move; ends are nodes; spacing λ/2’.
- 🧠 Pipe rule: ‘open = all; closed = odd’ (and an octave lower).
- 🏠 Daily: guitar frets and overblown flutes are this card performed live.
- 🏠 Daily: your microwave’s turntable exists to outrun standing-wave cold spots.
- 🔁 standing waves: superposition of opposite identical waves
- 🔁 nodes (still) and antinodes (max), λ/2 apart
- 🔁 string harmonics: f₁ = v/2L, then n·f₁
- standing wave = two identical opposite waves locked in place
- nodes: still; antinodes: max shake; spacing λ/2
- string (fixed ends): fits n half-wavelengths; f_n = n·v/2L
- open pipe: all harmonics (v/2L); closed pipe: odd only (v/4L)
- shorter/tighter/lighter → higher notes
Quick revision
- Two identical waves travelling opposite ways lock into a standing pattern
- Nodes: points that never move · Antinodes: points of maximum shake
- A string fixed at both ends fits whole numbers of half-wavelengths
- The harmonic series: f₁ = v/2L, then 2f₁, 3f₁… — the note menu of every instrument
- Open and closed pipes: different rules, same ‘whole-number’ idea
- The string’s allowed notes (harmonics)
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