Revise Standing waves and resonance for IB Physics (C.4)

A standing wave is produced by the superposition of two identical progressive waves — same frequency, same amplitude, same speed — travelling in opposite directions through the same medium. In practice the second wave is usually the reflection of the first from a boundary. Where the two waves are permanently in step they reinforce, and where they are permanently in opposition they cancel. Because the two waves keep passing through each other, those regions of reinforcement and cancellation stay put, and the pattern appears frozen in space: the crests no longer march along, they just grow and shrink on the spot.

It is worth being precise about how a standing wave differs from the travelling waves that build it, because examiners test exactly this distinction. A travelling wave transports energy along the medium; a standing wave does not — its energy stays put, sloshing between kinetic and potential within each loop but never flowing past a node. The phase behaviour differs too. In a travelling wave, neighbouring points lag one another, so a disturbance sweeps along. In a standing wave, all the points within one loop (between two adjacent nodes) move in phase — they reach their peaks together — while points in adjacent loops move in exact antiphase. Finally, in a travelling wave every point has the same amplitude; in a standing wave the amplitude depends on position, running from zero at the nodes to a maximum at the antinodes.

Consider a string of length $L$ fixed at both ends, as on a guitar or violin. Because the ends are clamped they cannot move, so each end must be a node. This boundary condition allows only certain wavelengths to fit: the string length must contain a whole number of half-wavelengths. The fundamental (first harmonic) is a single loop, so $L = \lambda_1/2$; the second harmonic is two loops, $L = \lambda_2$; and so on. These allowed frequencies are the natural frequencies, or harmonics, of the string.

For a string of length $L$ fixed at both ends carrying waves of speed $v$: $\lambda_n = \frac{2L}{n}, \qquad f_n = \frac{nv}{2L}, \qquad n = 1, 2, 3, \ldots$ The $n = 1$ case is the fundamental (first harmonic); ALL integer harmonics are allowed.

Standing sound waves form in the air inside pipes, which is how wind instruments make their notes. The boundary conditions come from how freely the air can move at each end. A closed end forces the air to be stationary, so it is a displacement node. An open end lets the air move most freely, so it is a displacement antinode. Two cases matter for C.4.

Open pipe (antinode–antinode): $\lambda_n = \frac{2L}{n}, \qquad f_n = \frac{nv}{2L}, \qquad n = 1, 2, 3, \ldots$ Closed pipe (node at the closed end, antinode at the open end): $\lambda_n = \frac{4L}{n}, \qquad f_n = \frac{nv}{4L}, \qquad n = 1, 3, 5, \ldots$

Many exam marks are earned simply by reading a standing-wave pattern correctly: counting loops, locating nodes and antinodes, and turning that count into a wavelength. The key facts are that boundaries fix the pattern (node at a fixed or closed end, antinode at an open end) and that the node-to-node distance is always $\lambda/2$.

Left to itself after a disturbance, any bounded oscillator vibrates at one of its natural frequencies — for a string or air column, these are exactly the harmonics above. If instead a periodic driving force is applied, the system undergoes forced oscillations at the driving frequency. When the driving frequency matches a natural frequency, energy is transferred from driver to system with maximum efficiency and the amplitude builds to a large value: this is resonance. It is why a wine glass shatters at the right sung note, why pushing a swing in time makes it soar, and why an instrument's body amplifies the string's vibration.

Real oscillators lose energy to resistive forces — friction, air resistance, or the radiation of sound — and this loss is called damping. The amount of damping controls both how a free oscillation dies away and how sharply a driven system resonates. Three regimes are named. With light damping the amplitude falls gradually over many cycles, so the system oscillates many times before stopping; its resonance peak is tall and narrow. With critical damping the system returns to equilibrium in the shortest possible time WITHOUT overshooting or oscillating — the target behaviour for car suspensions, analogue meters and door closers. With heavy (over) damping the resistive force is so large that the system creeps back to equilibrium slowly, again without oscillating, taking longer than the critical case. Increasing damping always lowers the height of the resonance peak and broadens it.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A) — and error-carried-forward (ECF) means a wrong number early on need not cost you the marks that follow. That protection only exists if your method is on the page. Study how each mark below is earned by a specific line, especially where the second harmonic is built from the first.

Standing waves are superposition made visible, and the same boundary-condition thinking reappears throughout physics: in the resonance of electrical circuits, in the modes of a laser cavity, and — in quantum theory — in the standing electron waves that fix the allowed energy levels of an atom. Master the routine here — identify the boundary conditions, sketch the harmonic, read off $\lambda$ from the length, then apply $v = f\lambda$, showing every line — and these later, more abstract standing-wave problems become variations on a method you already own.