Stationary vs Progressive Waves

Spec mapping (OCR H556 Module 4.5): Compare and contrast stationary and progressive waves with respect to energy transfer, amplitude profile, phase relations across the wave, formation, and applications. The comparison-paragraph is a recurring OCR mark-scheme item and the synoptic glue that knits the rest of the module together.

Throughout this module we have distinguished the two fundamentally different types of wave: the progressive (or travelling) wave, which carries energy through space, and the stationary (or standing) wave, which stores energy locally in the form of oscillations. This final lesson draws together every difference and similarity, provides a full quantitative comparison, and explores applications that illustrate the concepts in real contexts. By the end you should be able to explain — in a single examiner-friendly paragraph — exactly how the two kinds of wave differ.

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Side-by-Side Comparison

Property	Progressive wave	Stationary wave
Energy transfer	Yes — energy is carried along the wave	No — energy is stored in the oscillation and does not propagate
Waveform movement	The pattern of crests and troughs moves through space	The pattern is fixed in space; only the amplitudes oscillate in time
Amplitude	All particles oscillate with the same amplitude (A)	Amplitude varies from zero (nodes) to maximum (antinodes)
Phase	Neighbouring particles are in continuously varying phase — phase increases smoothly along the wave	All particles between two adjacent nodes are exactly in phase; particles on opposite sides of a node are in antiphase (π rad different)
Formation	A single source emits the wave	Requires superposition of two progressive waves of equal amplitude and frequency travelling in opposite directions (usually a wave and its reflection)
Wavelength measurement	Distance between two adjacent points in phase	Twice the distance between two adjacent nodes (= twice the distance between two adjacent antinodes)
Examples	Sound from a speaker, EM radiation, surface water waves, seismic waves	Vibrating guitar string, organ pipe, resonating wine glass, microwave oven

Commit this table to memory. Any OCR question asking to "compare stationary and progressive waves" should draw on at least three or four of these rows in the answer.

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Energy Transfer — The Key Physical Difference

The most fundamental distinction is about energy.

In a progressive wave, energy continually flows through each cross-section of the medium. A loudspeaker broadcasts sound energy into the air; the waves carry that energy outward to the listener; the energy is delivered and absorbed. Mathematically, the intensity I (power per unit area) is a non-zero quantity:

I ∝ A² for a progressive wave

In a stationary wave, the energy stays in the region where the wave is set up. Each element of the medium oscillates about its equilibrium position — so kinetic and potential energy slosh back and forth within each half-wavelength cell — but no net power flows through any fixed plane. A plucked guitar string vibrates audibly for seconds, slowly losing energy to the air and to friction, but the energy does not travel along the string; it is local to the part of the string that is vibrating.

This is why musical instruments can be played repeatedly without needing to replenish their strings — the energy you put in with each pluck or bow stroke stays in the string until it is gradually radiated (as sound) or damped (by friction in the instrument body). The audible sound itself is a progressive wave, radiated by the string into the surrounding air.

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Amplitude Variation

Another diagnostic distinction is amplitude as a function of position.

In a progressive wave, every particle has the same maximum displacement from equilibrium. If you mark two points on a wave-carrying rope, both will eventually reach exactly the amplitude A.

In a stationary wave, the amplitude varies from point to point. At an antinode, the oscillation amplitude is maximum (2A, where A is the amplitude of the constituent travelling waves). At a node, the amplitude is zero — the particle at that point never moves. Between nodes and antinodes, intermediate amplitudes occur.

You can actually see this distinction by freezing-frame imagination: a progressive wave on a rope, stopped in time, looks like a sinusoid of uniform amplitude running along the rope; a stationary wave, stopped, looks like a sinusoid that is modulated — smaller in some places, larger in others — with zeros at regular intervals.

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Phase Relationships

The phase differences between particles also distinguish the two types of wave.

In a progressive wave, particles separated by a distance d have phase difference φ = 2πd/λ. Neighbouring particles are slightly out of phase with each other (with a small, smooth phase shift); particles λ/2 apart are in antiphase; particles λ apart are back in phase; and so on.

In a stationary wave, the phase relationships are discrete:

• All particles in the same "segment" (between two adjacent nodes) are in phase. They all move upwards together, reach their maxima together, move downwards together, and so on.
• Particles in adjacent segments (either side of a node) are in antiphase. When one segment is fully up, the other is fully down.

This is a striking feature of stationary waves and can be demonstrated visually with a slow-motion video of a vibrating string.

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The Wavelength of a Stationary Wave

Since there are no "progressing" wavefronts in a stationary wave, the idea of "wavelength" needs careful definition. The standard convention is that the wavelength λ of a stationary wave is the wavelength of the underlying travelling waves that superposed to create it. In practice this means:

λ (stationary wave) = 2 × (node-to-node distance) = 2 × (antinode-to-antinode distance)

If you measure 25 cm between two adjacent nodes on a rope in a lab demonstration, the wavelength is 50 cm.

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Formation of Stationary Waves — Reflection and Superposition

Stationary waves are almost always formed when a progressive wave is reflected back on itself — the incoming wave and the reflected wave superpose. For this to give a clean stationary pattern, the reflection must happen at a well-defined end with well-defined boundary conditions:

• A fixed end (e.g. a string fixed to a wall) reflects with a π phase change (the reflected wave is inverted). This produces a node at the fixed end.
• A free end (e.g. a string attached to a very loose ring) reflects without phase change. This produces an antinode at the free end.
• A closed pipe end reflects sound with a pressure antinode (displacement node).
• An open pipe end reflects sound with a pressure node (displacement antinode).

Musical instruments exploit these boundary conditions to control which frequencies are supported. A guitar string (both ends fixed) supports nodes at both ends; a flute (both ends open) supports antinodes at both ends; a clarinet (closed at the mouthpiece, open at the bell) supports a node at the mouthpiece and an antinode at the bell — which is why clarinets produce only odd harmonics.

flowchart LR
    subgraph Progressive["Progressive wave"]
        P1[Source emits wave]
        P2[Wave travels outward]
        P3[Energy delivered to distant receiver]
    end
    subgraph Stationary["Stationary wave"]
        S1[Wave emitted]
        S2[Reflected at fixed boundary]
        S3[Superposition of forward and reflected]
        S4[Nodes and antinodes form]
        S5[Energy remains trapped in oscillation]
    end

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Observing Stationary Waves in the Lab

The classic A-Level laboratory demonstration uses a Melde's apparatus: a string is attached to a mechanical oscillator at one end and passed over a pulley at the other, with a weight hanging from the free end to provide tension. The oscillator drives the string at a chosen frequency; when that frequency matches one of the harmonics (see Lesson 11), the string vibrates visibly in a clear stationary-wave pattern of nodes and antinodes.

The nodes are easily identified: they are the points where the string appears not to move. In between are the segments oscillating at visible amplitude. By measuring node-to-node distance, you can directly infer the wavelength — and, knowing the tension and mass per unit length, verify the formula f = (1/2L)√(T/μ).

For sound waves, a similar experiment can be done using a speaker pointed at a reflecting wall, with a microphone in between. As the microphone is moved along the line, it detects alternating loud and quiet regions — the pressure antinodes (where the microphone records peak sound amplitude) and pressure nodes (where the signal is minimum). The distance between adjacent minima is λ/2, giving a direct wavelength measurement.

For microwaves, a microwave source and a metal plate can set up a standing-wave pattern of microwave intensity. A detector moved along the line registers alternating signals at node-antinode spacing equal to λ/2 of the microwave.

All three experiments — with string, sound and microwaves — are excellent ways of measuring wavelength from first principles, and of demonstrating that the same physics applies to very different wave types.

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Applications of Stationary Waves

1. Musical instruments (strings, pipes, drumheads, xylophones). Every pitched instrument works by setting up a stationary wave at a chosen harmonic; the frequencies are determined by the length, tension, mass and boundary conditions. The characteristic timbre of an instrument is determined by the relative amplitudes of its higher harmonics.
2. Resonance in tuned circuits — not strictly stationary waves, but closely analogous: an LC circuit supports a standing wave of electric and magnetic energy at its natural frequency.
3. Microwave ovens use stationary microwaves inside a metal cavity to heat food. The fixed pattern of nodes and antinodes is the reason that food does not heat evenly — pouring water into a sink of the oven would reveal hot and cold spots corresponding to microwave antinodes and nodes. The turntable is designed to rotate food through these spots.
4. Acoustic resonance — your ear canal (closed at one end by the eardrum) has a fundamental resonance around 3 kHz, making it particularly sensitive to speech frequencies.
5. Lasers work by setting up a stationary electromagnetic wave between two mirrors inside the laser cavity, confined by stimulated emission amplification.
6. Standing-wave accelerometers and sensors exploit how the positions of nodes and antinodes shift in response to environmental changes.

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Applications of Progressive Waves

1. All communication — radio, TV, mobile phones, Wi-Fi, optical fibre, ordinary speech — relies on progressive waves carrying information from source to receiver.
2. All imaging — photography, medical ultrasound, MRI, X-rays — uses progressive waves to probe structure.
3. All energy-delivery by wave — solar radiation, microwave heating (by absorbed progressive waves, distinct from oven cavity standing waves), laser cutting, sonar, seismology.
4. All perception — vision and hearing — depends on progressive waves reaching the sensory organs.

Progressive waves are how the universe transmits information and energy across space.

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Synoptic Links

• Module 4.4 (Wave motion, wave parameters): both wave types obey $v = f\lambda$v=fλ and the principle of superposition; the contrast is in energy transfer and boundary conditions, not in the underlying wave physics.
• Module 4.5 (Superposition, Lesson 7): stationary waves are the limiting case of two-source superposition when the two sources are travelling in opposite directions at the same frequency.
• Module 6.5 (Quantum physics): the standing-wave logic of stationary waves is exactly what gives rise to quantised energy levels in atoms — boundary conditions on the Schrödinger wavefunction enforce discrete energies.

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Specimen question modelled on the OCR H556 paper format