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 Pulses Show 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 Particles When 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 Interference Four 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 Sequence Show 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 Antinodes Show 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 = fraction Distance between adjacent antinodes = fraction Distance between a node and nearest antinode = fraction where λ = wavelength of the component progressive waves 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 Exam Insight: Energy 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 Apparatus Show: 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. Observ…
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