Stationary waves — OCR A-Level Physics

A stationary wave forms when two progressive waves of the same frequency and similar amplitude travel in opposite directions through the same medium and superpose. Energy is not transferred along the wave; instead it is stored, oscillating between kinetic and potential forms. The pattern has fixed nodes, where displacement is always zero, and antinodes, where displacement varies with maximum amplitude. Adjacent nodes are separated by half a wavelength, λ/2, and the node-to-antinode distance is λ/4. For example, a stretched string fixed at both ends reflects waves so that the reflected and incident waves superpose, producing a stationary wave whose allowed frequencies depend on the string length and tension.

Stationary waves form by the superposition of two progressive waves of the same frequency travelling in opposite directions. With microwaves, a transmitter directs waves at a metal plate; the reflected wave superposes with the incident wave, and a probe detects nodes and antinodes spaced λ/2 apart. On a stretched string fixed at both ends, boundary conditions dictate that nodes form at the ends. The allowed wavelengths depend only on the string's length (L) and harmonic number (n), given by λ = 2L/n. The string's tension and mass per unit length determine the wave speed and frequency, but not the wavelength of a given mode. In air columns, closed ends form displacement nodes and open ends form antinodes.

A stationary wave can be represented graphically in two ways. A displacement–position graph at a fixed instant shows the wave envelope: a sine or cosine curve with regularly spaced nodes and antinodes, where the curve crosses the axis at nodes and reaches maxima at antinodes. A displacement–time graph for a single point shows simple harmonic oscillation whose amplitude depends on the point's position: zero at a node, maximum at an antinode. Adjacent points between nodes oscillate in phase, while points on opposite sides of a node are in antiphase. For example, on a string fixed at both ends, the fundamental mode has nodes at the ends and one antinode at the centre, and its displacement–position graph is a single half-sine curve.

Stationary and progressive waves share several features: both arise from oscillations, both have frequency, wavelength and amplitude, and both can be described by wave equations. They differ in key ways. A progressive wave transfers energy through the medium and its waveform moves; a stationary wave stores energy and its pattern stays in fixed positions. In a progressive wave all points oscillate with the same amplitude and neighbouring points have a continuously changing phase difference; in a stationary wave amplitude varies from zero at nodes to maximum at antinodes, and points between adjacent nodes oscillate in phase. For example, a travelling sound wave from a loudspeaker is progressive, whereas the pattern formed by reflection in a tube is stationary.

A stationary wave forms when two identical progressive waves travelling in opposite directions superpose. At a node the two displacements always cancel, so the resultant displacement stays zero and the point never moves. At an antinode the two displacements always reinforce, so the resultant displacement oscillates with maximum amplitude. Nodes and antinodes are fixed in space, spaced half a wavelength apart, and adjacent nodes and antinodes are separated by one quarter of a wavelength. Energy is not transferred past a node; it is stored between nodes. For a string fixed at both ends, the ends must be nodes, so the shortest pattern has one antinode at the centre.

This row is a specification sub-heading marker rather than a taught statement, so treat it as a signpost to the practical work that follows. Read it as the point where the stationary-wave theory of nodes, antinodes and patterns is applied experimentally. The learner should use this heading to organise revision: first secure the theory of stationary waves, then study the techniques and procedures for determining the speed of sound in air using a resonance tube. When reading the specification, note that the sub-heading itself carries no separate content to memorise; the assessable material sits in the statements beneath it. Use the heading to build a checklist linking theory to practical method.

Stationary-wave patterns depend on the boundary conditions. A stretched string fixed at both ends has nodes at both ends, so its wavelengths are given by L = nλ/2, where n is a whole number of half-wavelengths. An open tube has antinodes at both open ends, giving the same relationship L = nλ/2. A closed tube has a node at the closed end and an antinode at the open end, so the shortest pattern is a quarter wavelength: L = (2n − 1)λ/4. These patterns determine the resonant frequencies, and the end correction must be added to the measured length for real tubes.

A resonance tube is filled with water so the air column length can be varied. A tuning fork of known frequency f is held over the open end, and the air column is adjusted until a loud resonance is heard. At resonance a stationary wave forms with a node at the water surface and an antinode near the open end, so the first resonance length L₁ ≈ λ/4 and the second L₂ ≈ 3λ/4. The difference L₂ − L₁ = λ/2 removes the end correction, giving λ = 2(L₂ − L₁). The speed of sound follows from v = fλ. Repeating with several tuning forks and plotting L against 1/f allows a more reliable value.

A stationary wave forms when two progressive waves of the same frequency and similar amplitude travel in opposite directions through the same medium. The pattern has fixed nodes, where displacement is always zero, and antinodes, where displacement varies with maximum amplitude. The key relationship is that the distance between one node and the next node is half a wavelength, λ/2. The same spacing applies between adjacent antinodes. For example, if a microwave stationary wave has nodes 1.5 cm apart, then λ/2 = 1.5 cm, so λ = 3.0 cm. This spacing arises because the wave must travel one full wavelength to return to the same phase, and a node-to-node path corresponds to half that cycle. Students should measure across several node spacings and divide to reduce uncertainty.

A stationary wave on a stretched string or in a pipe can vibrate in several distinct patterns called harmonics. The simplest pattern is the fundamental mode, also called the first harmonic. For a string fixed at both ends, the fundamental has one antinode in the middle and nodes at each end, so the string length L equals λ/2. The second harmonic has two antinodes and L = λ, the third harmonic has three antinodes and L = 3λ/2, and so on. In general, for a string fixed at both ends, L = nλ/2 where n is the harmonic number. The frequency of each harmonic is an integer multiple of the fundamental frequency: f_n = nf_1. For a pipe open at both ends, the pattern is similar, but a pipe closed at one end has only odd harmonics. Students should be able to identify the harmonic number from a diagram by counting antinodes or half-wavelengths.