Principle of superposition
When two or more waves pass through the same point, the total displacement is the sum of the separate displacements:
y = y₁ + y₂ + …
- Crest meets crest: bigger crest (constructive).
- Crest meets trough: they cancel (destructive).
After crossing, each wave carries on unchanged. Superposition explains standing waves, beats and interference.
Reflection of waves
When a wave reaches a boundary, some or all of it comes back.
- Rigid (fixed) boundary: the end cannot move, so the reflected wave is inverted: a phase change of π (180°). A crest returns as a trough. A node forms at the end.
- Free (open) boundary: the end is free to move, so the wave returns upright, with no phase change. An antinode forms at the end.
If the incident wave is y = A sin(kx − ωt), the wave reflected from a rigid end at x = 0 is y = −A sin(−kx − ωt) = A sin(kx + ωt): same size, opposite direction, flipped so the end stays still.
Standing waves: nodes and antinodes
Two identical waves moving in opposite directions add up to
y = 2A sin(kx) cos(ωt)
This is a standing (stationary) wave: the shape does not move along. Each point does SHM with its own amplitude 2A sin(kx).
- Nodes: points that never move (sin kx = 0), at x = 0, λ/2, λ, …
- Antinodes: points of biggest motion, at x = λ/4, 3λ/4, …
- Node to next node = λ/2; node to next antinode = λ/4.
- A standing wave does not carry energy along; energy stays trapped between nodes.
Standing waves on a stretched string
Both ends are fixed, so both ends are nodes. Only whole half-waves fit:
L = nλ/2 → λₙ = 2L/n → fₙ = n·v/2L, n = 1, 2, 3, …, with v = √(T/μ).
- n = 1: fundamental (first harmonic), f₁ = v/2L = (1/2L)√(T/μ).
- n = 2, 3, …: second, third harmonics (overtones). A string gives all harmonics.
- The n-th mode has n + 1 nodes and n antinodes.
Organ pipes: closed and open
In a pipe, the air column vibrates. A closed end is a displacement node; an open end is an antinode.
Closed pipe (one end closed)
L = nλ/4 with n odd → fₙ = n·v/4L, n = 1, 3, 5, … Only odd harmonics. Fundamental f₁ = v/4L.
Open pipe (both ends open)
L = nλ/2 → fₙ = n·v/2L, n = 1, 2, 3, … All harmonics. Fundamental f₁ = v/2L, twice that of a closed pipe of the same length.
(For accurate work an end correction of about 0.3 × diameter is added to L at each open end.)
Try it: a bottle organ
Take a glass bottle and blow across its mouth: you hear a low note. The bottle is a closed pipe (bottom closed). Pour in some water: the air column L gets shorter and the note goes higher, since f = v/4L. Predict: half the air column should give about double the frequency. In the 3D, pick “closed pipe” in step 5 and shorten L to check.
Key formulas and definitions
- y = y₁ + y₂ (superposition)
- Standing wave: y = 2A sin(kx) cos(ωt); nodes λ/2 apart
- String / open pipe: fₙ = n·v/2L, n = 1, 2, 3…
- Closed pipe: fₙ = n·v/4L, n = 1, 3, 5…
- String: f₁ = (1/2L)√(T/μ)
Worked examples
1. A 0.5 m string has wave speed 200 m/s. Find its fundamental and third harmonic.
f₁ = v/2L = 200/1 = 200 Hz. f₃ = 3 × 200 = 600 Hz.
2. A 1 m string of mass 10 g is under 100 N tension. Find the fundamental frequency.
μ = 0.01 kg/m, v = √(100/0.01) = 100 m/s. f₁ = v/2L = 100/2 = 50 Hz.
3. An open pipe is 0.85 m long. Speed of sound is 340 m/s. Find the first three frequencies.
f₁ = v/2L = 340/1.7 = 200 Hz; then 400 Hz and 600 Hz.
4. A pipe of the same length (0.85 m) is closed at one end. Find its first three frequencies.
f₁ = v/4L = 340/3.4 = 100 Hz; only odd harmonics: 100, 300, 500 Hz.
5. The distance between two nearest nodes of a standing wave is 15 cm. The frequency is 1000 Hz. Find the wave speed.
Node to node = λ/2 → λ = 0.30 m. v = fλ = 1000 × 0.3 = 300 m/s.
6. A string vibrates in 4 loops at 480 Hz. What is its fundamental?
4 loops means n = 4, so f₁ = 480/4 = 120 Hz.
7. The fundamental of a closed pipe equals the first overtone of an open pipe. Find the ratio of their lengths L_c : L_o.
v/4L_c = 2v/2L_o → v/4L_c = v/L_o → L_o = 4L_c, so L_c : L_o = 1 : 4.
8. A string’s tension is increased by 44%. By what percent does its fundamental change?
f ∝ √T. √1.44 = 1.2, so f rises by 20%.
Common mistakes
- Thinking a closed pipe has even harmonics. It has only odd ones: 1, 3, 5…
- Taking node-to-node distance as λ. It is λ/2.
- Forgetting the phase change of π when a wave reflects from a fixed end.
- Using v = 340 m/s for a string. On a string use v = √(T/μ); 340 m/s is for sound in air.