Standing waves and resonance

The fixed pattern made when **two identical waves travel in opposite directions** and superpose — it does not move along.

When two waves overlap, you **add their displacements** at every point to get the total wave.

A point on a standing wave that **never moves** (zero displacement) — the two waves always cancel there.

A point on a standing wave that swings with the **largest amplitude**, halfway between two nodes.

**Half a wavelength (λ/2).** So λ = 2 × the node-to-node spacing. (Not in the data booklet — remember it.)

**No** — there is no net energy transfer along a standing wave; the energy stays stored in place.

They move **in phase** (all together). Points on opposite sides of a node move in **antiphase** (exactly opposite).

Standing: points are only ever **in phase or antiphase**. Travelling: the phase shifts **smoothly** from point to point.

A wave **reflects off a fixed end** and meets itself coming back — two identical opposite waves that superpose.

Microwaves reflect off the walls and form a **standing wave**; the field is strongest at the **antinodes**, so it melts there and stays solid at the nodes.

Spots are one antinode apart = λ/2, so λ = 2 × 0.060 = 0.12 m.

Thinking it **carries energy along** the string, or halving (instead of doubling) the node spacing to get the wavelength.

Two **identical waves travelling in opposite directions** superpose; where they always cancel are the nodes.

**Antinodes at both ends, a node in the middle** (like a pipe open at both ends).

At the **displacement nodes**, where the air does not move.

A point on a standing wave that **never moves** (zero amplitude).

A point on a standing wave that swings with the **largest** amplitude.

The **lowest** frequency at which a string or air column resonates — the standing-wave pattern with the fewest loops.

When a system is driven at one of its **natural frequencies** and vibrates with a large amplitude — that is what makes a harmonic loud.

**λ = 2L/n** for n = 1, 2, 3, … — n half-wavelengths fit into the length L.

**λ = 4L/n** with **n = 1, 3, 5, …** (odd harmonics only — node at the closed end, antinode at the open end).

**No** — you must memorise them. Only the wave equation v = fλ is given.

Its ends are different (node at the closed end, antinode at the open end), so only odd numbers of quarter-wavelengths fit: λ = 4L/n, n = 1, 3, 5, …

**Half a wavelength.** So λ = 2 × the node-to-node spacing.

Use the given wave equation **v = fλ**, rearranged to **f = v ÷ λ** (v is the wave speed — the speed of sound for a pipe).

λ = 2L = 1.3 m; f = v/λ = 260/1.3 = 200 Hz.

The spots (antinodes) are half a wavelength apart; double the spacing for λ, then f = c/λ.

The maximum amplitude **falls**, the peak gets **broader**, and it moves to a **slightly lower** frequency.

Light: many oscillations, the amplitude dies slowly. Critical: back to equilibrium in the **shortest time** without oscillating. Heavy: no oscillation, a slow return.

Useful: a musical instrument sounds loud; tuning a radio. Destructive: a bridge or building shaken strongly at its natural frequency.