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Superposition overview

Topic 5 of 9

Coherence, phase and two sources

A stable interference pattern depends on the phase difference where the waves meet. That difference includes both the source relationship and the distances travelled.

Name the conditions and quantities

Interference is the combined effect of overlapping waves. Coherent waves have the same frequency and a constant phase difference. They can be in phase or have a fixed nonzero phase difference.

Phase difference compares stages of oscillation. Path difference is the difference between distances travelled from the sources to the observation point. An extra wavelength adds one cycle of propagation delay; an extra half-wavelength adds half a cycle.

Diffraction is spreading through an opening or around an edge. It can allow waves from separate slits to overlap. Its opening-width and wavelength dependence is developed in diffraction and a single slit.

Use a path rule only with its source-phase condition

For sources in phase at emission and the same propagation wavelength:

Constructive: path difference = nλ
Destructive: path difference = (n + 1/2)λ
n = 0, 1, 2, ... for a path-difference magnitude

Complete cancellation additionally requires equal amplitudes at the observation point. Unequal amplitudes can still interfere, but leave a nonzero minimum. Comparable amplitudes improve the contrast between large and small resultant oscillations.

Worked source and path phase

Two half-cycle differences

Suppose two equal-amplitude sound sources are in antiphase at emission, and their paths to P differ by λ/2.

The source relationship contributes half a cycle. The propagation difference contributes another half-cycle difference. Together they are equivalent to a whole cycle, so the arrivals are in phase and interfere constructively.

Applying the in-phase-source path rule without checking the source condition would give the wrong answer.

Optional check Two coherent sound sources of equal amplitude are in antiphase at emission. Their paths to P differ by half a wavelength. What happens at P?
Two coherent sound sources of equal amplitude are in antiphase at emission. Their paths to P differ by half a wavelength. What happens at P?

Recognise the four demonstrations

Water: two dippers driven from a common source generate overlapping ripples. Stable regions of large and small oscillation amplitude can be observed. Keep frequency and water depth fixed. A snapshot showing crests and troughs is a phase map at an instant; it is not itself a map of oscillation amplitude.

Sound: two loudspeakers driven by the same signal generator produce alternating louder and quieter regions as a microphone or listener moves through the overlap. State the relative phase, including any reversed connection. Room reflections and background sound can obscure minima.

Microwaves: one transmitter illuminates two apertures whose outputs overlap. Move an aligned detector through the pattern. Keep polarisation and detector orientation fixed, so a response change caused by rotating the detector is not misidentified as interference.

Identify the common source and the overlap region

Water: two dippers driven in phase

A common drive keeps two ripple sources in a fixed phase relationA top-view ripple-tank schematic contains two dippers linked to the same drive. Concentric solid crest lines and dashed trough lines have the same spacing and phase about each source. They overlap in the water. The lines show one instant of the component waves and are cropped at the drawing window; they are not the stationary curves of resultant amplitude. A fixed frequency and water depth are retained. The drive connections are schematic links identifying the shared driver, not water-particle trajectories.Common driveS1S2Solid: crests. Dashed: troughs.

This is an instantaneous phase-line map of the component waves. To locate persistent small- and large-amplitude regions, observe the resultant over time.

Sound: source phase and path phase both matter

Antiphase sources arrive in phase after a half-wavelength path differenceTwo coherent loudspeaker sources share a signal generator but have opposite source phases, zero and pi. In the geometric model, S1 and S2 are one wavelength apart. Observation point P lies three quarters of a wavelength perpendicularly above S1. Thus S1 to P is 0.75 wavelength and S2 to P is 1.25 wavelengths, forming an exact right triangle on equal horizontal and vertical scales. The longer path adds half a cycle of propagation-phase difference to the source half-cycle difference, so the arrivals are in phase. The diagram establishes phase, not equal received amplitudes or a common sound intensity.P0.75λ1.25λλS1: phase 0S2: phase πCommon signalgeneratorOpposite phase at emissionPath difference = λ/2

One source is phase-reversed. The extra half-wavelength of path supplies another half-cycle difference, giving constructive interference at P. Source phase cannot be omitted from the reasoning.

Microwaves: one transmitter illuminates two apertures

Two aperture outputs overlap at a detector whose orientation stays fixedOne transmitter illuminates two openings in a metal barrier. The openings are symmetrically placed about the source axis, making their incident paths equal in this schematic. Dashed geometric paths lead from the two apertures to a common receiving probe in their overlap. The probe can be moved across the pattern while retaining its vertical orientation, parallel to the detected electric-field component. The diagram labels both openings, the transmitter and the detector. Paths are schematic, not measured ray lengths or particle tracks.OnesourceS1S2Move probeTwo aperturesKeep probe parallel to the same E.

Hold source frequency and detector orientation fixed. Rotating the probe would change its response for a different reason. The light demonstration uses the separate double-slit apparatus below.

The water view shows coherent excitation and instantaneous wavefronts. The sound view states the antiphase source relationship and the two paths. The microwave apertures share one transmitter, and the detector keeps a fixed alignment. These are distinct apparatus arrangements, not particle trajectories.

Light: illuminate two slits with one coherent source and let their diffracted outputs overlap on a screen. A stable relative phase and compatible polarisation components allow visible fringes. The double-slit apparatus and screen pattern show this arrangement and its wavelength measurement. Ordinary independent lamps generally have rapidly changing relative phase, so their illumination does not normally produce a stable visible fringe pattern.

Separate a stable pattern from a perfectly dark minimum

Observable sustained fringes need overlapping waves of matching frequency with a sufficiently stable relative phase over the observation. For light, the interfering components must have compatible polarisation. The detector or screen must resolve the spatial variation, with enough contrast over background.

These conditions establish a stable pattern. Equal amplitudes and opposite arrival phase are the additional conditions for complete cancellation at a particular minimum. Coherence alone neither guarantees equal amplitudes nor means that every point is constructive.