OCR Specification focus:
‘Compare similarities and differences between stationary and progressive waves.’
Waves transport energy through oscillations. Stationary and progressive waves are two fundamental types, differing in energy transfer and oscillation behaviour, yet sharing essential wave characteristics such as frequency and wavelength.
Nature of Stationary and Progressive Waves
Progressive Waves
A progressive wave (also known as a travelling wave) is one that transfers energy from one point to another through the medium, without any net transfer of matter. Each particle of the medium oscillates about its equilibrium position, passing the disturbance along.
Progressive Wave: A wave in which energy and information are transferred through the medium as successive particles vibrate with a constant amplitude and phase difference.
Progressive waves can be transverse or longitudinal, depending on the direction of particle oscillation relative to energy propagation. In transverse waves, particles vibrate perpendicular to the direction of wave travel, while in longitudinal waves, they vibrate parallel to it.
Stationary Waves
A stationary wave (or standing wave) is formed when two waves of equal frequency and amplitude, travelling in opposite directions, superpose. The interference of these waves produces a pattern of nodes and antinodes, where particles either remain still or oscillate with maximum amplitude.
Diagram showing two counter-propagating waves and their resultant standing wave. The fixed nodes and oscillating antinodes emerge from continuous superposition, clarifying why the overall pattern does not propagate even though the constituent waves do. Source.
Stationary Wave: A wave pattern formed by the superposition of two identical waves moving in opposite directions, producing regions of zero and maximum displacement (nodes and antinodes)
Unlike progressive waves, stationary waves do not transfer energy along the medium; energy oscillates within fixed positions between nodes and antinodes.
Formation and Characteristics
Formation Process
Stationary waves occur under specific boundary conditions, such as reflection at a fixed end. For example, when a wave on a string reflects back upon reaching a boundary, it interferes with the incident wave, creating a standing pattern.
Key conditions for formation:
Two waves must have the same frequency, amplitude, and wavelength.
They must travel in opposite directions.
Coherent sources are necessary to maintain a constant phase relationship.
Nodes and Antinodes
The interference between the two waves leads to destructive interference at some points and constructive interference at others.
Node: A point on a stationary wave where displacement is always zero due to complete destructive interference.
Antinode: A point on a stationary wave where displacement reaches maximum amplitude due to complete constructive interference.
The distance between adjacent nodes (or antinodes) equals half the wavelength (λ/2) of the original waves.
A labelled stationary wave showing nodes (zero displacement) and antinodes (maximum displacement) along the medium. The fixed positions of nodes and antinodes illustrate that a stationary wave does not transport energy along the medium. Source.
Comparison of Stationary and Progressive Waves
Energy Transfer
Progressive waves transfer energy through the medium; each particle passes energy to the next.
Stationary waves do not transfer energy along the medium; instead, energy oscillates locally between kinetic and potential forms.
Amplitude Variation
In a progressive wave, all particles have the same amplitude (in an ideal medium) and the amplitude does not depend on position.
In a stationary wave, amplitude varies along the wave, from zero at nodes to maximum at antinodes.
Phase Relationship
In a progressive wave, adjacent particles have a constant phase difference that depends on their separation.
In a stationary wave, all particles between adjacent nodes oscillate in phase, but with different amplitudes. On opposite sides of a node, particles are in antiphase (a phase difference of 180°).
Wavelength and Frequency
Both stationary and progressive waves share the same wavelength and frequency as determined by their sources. However, for stationary waves, the wavelength is determined by the distance between nodes according to the boundary conditions.
EQUATION
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Wave Speed (v) = Frequency (f) × Wavelength (λ)
v = speed of the wave (m s⁻¹)
f = frequency of the wave (Hz)
λ = wavelength (m)
—-----------------------------------------------------------------
In stationary waves on strings or air columns, this relationship still holds for the constituent travelling waves that form the pattern, even though the overall standing pattern does not propagate.
Behaviour of Particles
Particle Displacement
In progressive waves:
Every particle oscillates with the same amplitude (for a uniform medium).
There is a continuous transfer of energy from one region to another.
The phase of vibration changes smoothly along the wave.
In stationary waves:
Particle displacement depends on position relative to nodes and antinodes.
Some particles (nodes) never move, while others (antinodes) oscillate with maximum amplitude.
Energy alternates between potential and kinetic forms within each loop.
Energy Distribution
In progressive waves:
Energy is spread along the medium.
As the wave travels, energy is continually passed forward.
In stationary waves:
Energy remains trapped between nodes.
At any instant, some segments possess maximum kinetic energy while others hold maximum potential energy, but the total energy in the system remains constant.
Graphical Representation
Displacement–Position Graphs
For a progressive wave, a snapshot graph shows a continuous oscillatory pattern where crests and troughs move over time. In contrast, a stationary wave’s displacement–position graph at an instant shows fixed nodes and alternating antinodes, which remain at constant positions.
Displacement–Time Graphs
For a fixed point on the medium:
In a progressive wave, the displacement–time graph is a simple sine or cosine curve identical for all points but with phase shifts between them.
In a stationary wave, the graph’s amplitude depends on the position: at a node it is a flat line (no motion), while at an antinode it shows the largest oscillation.
Summary of Key Differences
Energy transfer: Progressive waves transfer energy; stationary waves store it locally.
Amplitude distribution: Uniform for progressive; varies for stationary.
Phase difference: Gradual change for progressive; in-phase or antiphase in stationary.
Node–antinode pattern: Present only in stationary waves.
Propagation: Progressive waves move through the medium; stationary waves do not.
FAQ
At a fixed boundary, the reflected wave undergoes a phase change of 180°, meaning the reflected displacement is inverted relative to the incident wave. This allows a node to form at the boundary.
At a free boundary, there is no phase change, so the reflected wave reinforces the incident one, forming an antinode at the boundary.
These conditions determine the position of nodes and antinodes and influence the harmonic patterns that develop on strings and air columns.
In a stationary wave, nodes remain stationary because the incident and reflected waves at those points are exactly out of phase.
Their displacements are equal and opposite, leading to complete destructive interference. The energy of the two waves cancels at that location, meaning no oscillation occurs.
Every half wavelength (λ/2), this perfect cancellation repeats, fixing the node positions permanently along the medium.
The number depends on the wavelength of the waves and the length of the medium in which the wave forms.
For a string fixed at both ends, the distance between adjacent nodes equals λ/2.
The number of loops (antinodes) increases with frequency as shorter wavelengths fit more segments into the same length.
The fundamental frequency produces the simplest pattern (one loop), while higher harmonics show multiple nodes and antinodes.
Although the overall wave pattern does not propagate, energy oscillates locally.
At any given time:
Near an antinode, particles possess maximum kinetic energy when passing through equilibrium.
Near a node, the medium stores potential energy due to the tension or compression.
Energy continuously transfers between these forms, but because the motion in adjacent regions cancels overall propagation, no net energy travels through the medium.
Yes. Stationary waves can occur in any system where waves reflect and interfere under suitable boundary conditions.
Examples include:
Microwave cavities, where electromagnetic waves reflect between conducting surfaces.
Optical resonators in lasers, forming standing light waves between mirrors.
Seismic or acoustic systems, where reflected sound waves create resonance patterns.
In all cases, the principle remains the same: two identical waves moving in opposite directions produce fixed nodes and antinodes through superposition.
Practice Questions
Question 1 (2 marks)
State two differences between a stationary wave and a progressive wave.
Mark scheme:
1 mark for identifying that a stationary wave does not transfer energy along the medium, whereas a progressive wave does transfer energy.
1 mark for stating that in a stationary wave, amplitude varies from node to antinode, while in a progressive wave, all points have the same amplitude (in an ideal medium).
Question 2 (5 marks)
A stretched string is fixed at both ends and vibrates to form a stationary wave.
(a) Explain how the stationary wave is formed on the string.
(b) Describe the motion of the particles along the string, and explain the energy transfer within the stationary wave.
Mark scheme:
(a) Formation of stationary wave (3 marks total):
1 mark: Incident wave travels along the string and is reflected at the fixed end.
1 mark: The incident and reflected waves have the same frequency and amplitude and travel in opposite directions.
1 mark: These waves superpose, producing nodes (no displacement) and antinodes (maximum displacement) where destructive and constructive interference occur, respectively.
(b) Particle motion and energy (2 marks total):
1 mark: Particles between adjacent nodes oscillate in phase, with amplitude depending on position; particles on opposite sides of a node oscillate in antiphase.
1 mark: No net energy transfer occurs along the string; energy is stored alternately as kinetic and potential energy within each segment between nodes.
