Stationary waves
Section: Superposition | Syllabus: Cambridge AS Level Physics (9702)
The Principle of Superposition
When two or more waves travel through the same medium simultaneously, their effects combine. This can occur with waves travelling in the same direction, different directions, or opposite directions. Superposition occurs between waves of the same type only (e.g., sound cannot superpose with electromagnetic waves).
Principle of Superposition At any point in the medium, the resultant displacement is the vector sum of the individual displacements of the overlapping waves. FIG 8.1: Superposition of Wave PulsesShow two wave pulses travelling towards each other on a rope. In sequence: (a) pulses approaching, (b) pulses overlapping with combined amplitude, (c) pulses continuing unchanged after passing through each other. Label the individual displacements and resultant.
Waves vs ParticlesWhen two wave pulses meet, they pass through each other and continue unchanged. Particles collide and bounce or combine. Waves can occupy the same space simultaneously-particles cannot.
Constructive and Destructive Interference
Constructive Interference Occurs when two waves are in phase (phase difference = 0 or 2π). The resultant amplitude equals the sum of the individual amplitudes. Destructive Interference Occurs when two waves are in antiphase (phase difference = π or 180°). For waves of equal amplitude, the resultant amplitude is zero. FIG 8.2: Constructive and Destructive InterferenceFour displacement-time graphs at a single point: (a) two waves in phase → resultant with doubled amplitude, (b) two waves in antiphase → zero resultant, (c) waves with arbitrary phase difference, (d) waves with different frequencies (f and 3f). Show Wave 1, Wave 2, and Sum clearly labelled.
Formation of Stationary Waves
A stationary wave (standing wave) forms when two progressive waves of the same frequency, wavelength, and amplitude travel in opposite directions through the same medium and superpose.
Stationary Wave A wave pattern that does not progress through the medium. It results from superposition of two progressive waves of equal frequency and amplitude travelling in opposite directions.Graphical Formation
Consider two identical waves travelling in opposite directions:
FIG 8.3: Formation of Stationary Wave - Time SequenceShow five displacement-distance graphs at times t = 0, T/4, T/2, 3T/4, and T. Each graph shows: orange wave (travelling right), purple wave (travelling left), and red resultant wave. Mark node positions (N) where displacement is always zero. Show how the resultant oscillates between maximum positive and maximum negative positions while nodes remain fixed.
- Where crests continuously meet crests (in phase): antinodes form with maximum amplitude
- Where crests continuously meet troughs (antiphase): nodes form with zero amplitude
- The wave profile oscillates but does not travel-it appears to "stand still"
Nodes and Antinodes
Node A point on a stationary wave where the amplitude is zero at all times. Antinode A point on a stationary wave where the amplitude is maximum. Located midway between adjacent nodes. FIG 8.4: Stationary Wave with Nodes and AntinodesShow a stationary wave at multiple time instants superimposed. Clearly label: nodes (N) at positions of zero displacement, antinodes (A) at positions of maximum displacement. Mark distances: node-to-node = λ/2, antinode-to-antinode = λ/2, node-to-antinode = λ/4.
Distance Relationships
Key Distance Relationships:- Distance between adjacent nodes = (λ/2)
- Distance between adjacent antinodes = (λ/2)
- Distance between a node and nearest antinode = (λ/4)
Phase Relationships
- All particles between adjacent nodes oscillate in phase with each other (but with different amplitudes)
- Particles on opposite sides of a node are in antiphase (180° phase difference)
Progressive Waves vs Stationary Waves
| Property | Progressive Wave | Stationary Wave |
|---|---|---|
| Energy | Transfers energy in direction of travel | Stores energy; does not transfer it |
| Wave profile | Moves through the medium | Oscillates in place; does not move |
| Amplitude | Same for all particles | Varies: zero at nodes, maximum at antinodes |
| Phase | Varies continuously along wave | Same between nodes; opposite across nodes |
| Wavelength | Distance between adjacent in-phase points | Twice the node-to-node distance |
Stationary waves do not transfer energy outside the bounded system. Energy oscillates between kinetic (at equilibrium) and potential (at maximum displacement), remaining localised. This distinguishes them from progressive waves.
Stationary Waves with Microwaves
Microwaves from a transmitter reflect off a metal plate, creating two coherent waves travelling in opposite directions.
FIG 8.5: Microwave Stationary Wave ApparatusShow: microwave transmitter (left) → microwaves → metal plate reflector (right). Include movable probe/detector between them connected to an output meter or loudspeaker. Label the direction of incident and reflected waves.
Observations: Moving the detector between transmitter and reflector reveals alternating maxima (antinodes) and minima (nodes). The spacing between consecutive maxima equals λ/2.
Application: Using measured λ and known frequency f, verify v = fλ = 3.0 × 10^8 m s^-1 (speed of light).
Practical NoteA node occurs at the metal plate (boundary). An antinode occurs at approximately λ/4 from the plate.
Stationary Waves on Stretched Strings
A string fixed at both ends, driven by a mechanical vibration generator, produces stationary waves at resonant frequencies.
FIG 8.6: String Vibration ApparatusShow: signal generator connected to mechanical vibration generator, string passing over a pulley, hanging mass providing tension. Label: vibration generator, string length L, pulley, mass m.
Observations: At certain frequencies, stable patterns of loops appear. Nodes occur at the fixed ends and at intermediate points. A strobe light can "freeze" the motion to reveal the pattern clearly.
First Harmonic (Fundamental) The lowest frequency that produces a stationary wave in a bounded system. For strings: one antinode between two nodes.Each harmonic is a whole-number multiple of the fundamental frequency.
Harmonics on Strings (Fixed at Both Ends)
FIG 8.7: String HarmonicsShow four patterns for a string of length L: 1st harmonic (1 loop, nodes at ends), 2nd harmonic (2 loops), 3rd harmonic (3 loops), 4th harmonic (4 loops). Label nodes and antinodes. Show wavelength relationship for each.
| Harmonic | Loops | Wavelength | Frequency |
|---|---|---|---|
| 1st (fundamental) | 1 | _1 = 2L | f_1 = (v/2L) |
| 2nd | 2 | _2 = L | f_2 = 2f_1 |
| 3rd | 3 | _3 = (2L/3) | f_3 = 3f_1 |
| nth | n | _n = (2L/n) | f_n = nf_1 |
For a string fixed at both ends: number of nodes = number of harmonic + 1. The nth harmonic has n loops (antinodes) and (n + 1) nodes.
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