When two identical sources send out waves together, the overlap is not random, it organises itself into a fixed pattern of loud-and-quiet, or bright-and-dark, lines that stay put. These maxima and minima come straight from superposition: where the two waves arrive in step you get reinforcement, where they arrive a half-wavelength out of step you get cancellation. The whole pattern is decided by one quantity: path difference.
You stand exactly the same distance from two in-phase speakers playing an identical tone.
Before reading: at a point equally far from both speakers, do the two sound waves arrive in step or out of step, and is it loud or quiet there? Write your prediction and one sentence explaining why.
Two coherent sources overlap to make a stable grid of reinforcement and cancellation
Two identical sources sending out waves together create overlapping ripples whose crossings settle into a permanent pattern of strong spots and dead spots that does not move.
When two coherent sources (same frequency, constant phase difference) send out waves into the same region, superposition produces a fixed interference pattern: lines of maxima where the waves always reinforce (constructive) alternate with lines of minima where they always cancel (destructive).
Two coherent sources (same frequency, constant phase relationship) produce a fixed interference pattern by superposition: lines of maxima where the waves reinforce (constructive) alternate with lines of minima where they cancel (destructive).
Pause, copy the highlighted description into your book before moving on.
Did you get this? True or false: a stable interference pattern requires the two sources to be coherent (same frequency, constant phase relationship).
Whole wavelengths give maxima; odd half-wavelengths give minima
We just saw that the pattern has fixed maxima and minima. That raises a question: how do we know which points are which? This card answers it → compare the extra distance one wave travels, the path difference, with the wavelength.
At any point, whether you get a maximum or a minimum depends only on how much further one wave has travelled than the other, measured in wavelengths.
The path difference is $\Delta = |S_1P - S_2P|$. For in-phase sources:
Maximum (constructive): $\Delta = m\lambda$ ($m = 0, 1, 2, \dots$)
Minimum (destructive): $\Delta = \left(m + \tfrac{1}{2}\right)\lambda$ ($m = 0, 1, 2, \dots$)
Two in-phase speakers emit sound of wavelength $0.40\ \text{m}$. At a point P, the distance to one speaker is $3.00\ \text{m}$ and to the other is $3.60\ \text{m}$. Determine whether P is a maximum or a minimum.
Path difference $\Delta = |S_1P - S_2P|$ decides the point. For in-phase sources, a maximum occurs when $\Delta = m\lambda$ (a whole number of wavelengths) and a minimum when $\Delta = (m + \tfrac{1}{2})\lambda$ (an odd number of half-wavelengths).
Add the highlighted path-difference conditions to your notes before the check below.
Quick check: Two in-phase sources produce waves of wavelength $2\ \text{cm}$. At a point the path difference is $4\ \text{cm}$. This point is a:
Pattern A, Calculate then classify
Two in-phase dippers in a ripple tank make waves of wavelength $1.0\ \text{cm}$. For point X the path difference is $3.0\ \text{cm}$; for point Y it is $2.5\ \text{cm}$. In your book: (1) express each path difference as a number of wavelengths; (2) classify each point as a maximum or minimum; (3) explain in one sentence why point Y is what it is.
Odd one out: Three of these conditions are needed for a steady, observable interference pattern. Which one is the odd one out?
Dot point: Explain how the superposition of waves produces interference patterns with observable maxima and minima
Connecting the path-difference rule back to superposition and coherence
We just learned the path-difference conditions. That raises a question: why does this produce a steady pattern rather than a flicker, and why is the central line always a maximum? This card answers it → because the sources are coherent and the centre has zero path difference.
The pattern holds still because the two sources keep a constant phase relationship, so each point's path difference, and therefore whether it reinforces or cancels, never changes.
Along the central line, every point is equidistant from both sources, so $\Delta = 0$: the waves always arrive in phase and superpose to a permanent maximum (this is the equal-distance point from the Think-First). Moving sideways, $\Delta$ grows steadily, passing through $\tfrac{1}{2}\lambda$ (first minimum), $\lambda$ (first side maximum), $\tfrac{3}{2}\lambda$ (next minimum), and so on, producing the alternating bands.
Because the sources are coherent, this geometry does not drift, so the maxima and minima stay locked in place. If the sources had different frequencies (incoherent), the path-difference relationship would change moment to moment and no stable pattern would form.
The interference pattern is steady because coherent sources keep each point's path difference fixed. The central line has $\Delta = 0$, always a maximum; moving outward, $\Delta$ increases through $\tfrac{1}{2}\lambda, \lambda, \tfrac{3}{2}\lambda, \dots$, giving alternating minima and maxima. Incoherent sources give no stable pattern.
Pause, copy the highlighted explanation into your book before the check below.
Last check: Decide if the statement is true or false.
A point exactly equidistant from two in-phase sources has zero path difference and is always a maximum.
Match them up: Match each path difference to what is observed at that point.
Dot point: Explain how the superposition of waves produces interference patterns with observable maxima and minima
A fresh set drawn from this lesson's question bank, feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence, that tells the system what to drill next.
UnderstandBand 5(4 marks) 1. Explain how the superposition of waves from two coherent sources produces an interference pattern of maxima and minima. Refer to path difference in your answer.
1 mark: waves from both sources overlap and superpose · 1 mark: maxima where path difference $= m\lambda$ (in phase, reinforce) · 1 mark: minima where path difference $= (m+\tfrac{1}{2})\lambda$ (out of phase, cancel) · 1 mark: coherent sources keep the pattern fixed
ApplyBand 5(3 marks) 2. Two loudspeakers driven by the same signal generator emit sound of wavelength $0.50\ \text{m}$. At a certain point the path difference from the two speakers is $0.75\ \text{m}$. Determine whether a maximum or a minimum is heard there, showing your reasoning.
1 mark: path difference in wavelengths $= 0.75/0.50 = 1.5\lambda$ · 1 mark: identifies $1.5\lambda$ as an odd number of half-wavelengths · 1 mark: concludes a minimum is heard
Dot point: Explain how the superposition of waves produces interference patterns with observable maxima and minima
AnalyseBand 6(3 marks) 3. Explain why an interference pattern from two sources is steady and fixed in space rather than flickering, and state what would happen to the pattern if one source drifted slightly in frequency.
1 mark: the sources have the same frequency and a constant phase relationship, so the path difference at each point corresponds to a fixed phase difference · 1 mark: each point therefore stays a maximum or a minimum · 1 mark: a frequency drift makes the phase relationship change with time, so the maxima and minima move and the pattern washes out
Dot point: Explain how the superposition of waves produces interference patterns with observable maxima and minima
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set from the lesson bank.
Q1 (4 marks): Where waves from two coherent sources overlap, they superpose, the resultant displacement at each point is the sum of the two displacements. Whether a point is a maximum or a minimum depends on the path difference, $\Delta = |S_1P - S_2P|$. At points where the path difference is a whole number of wavelengths ($\Delta = m\lambda$), the waves arrive in phase and reinforce, giving a maximum (constructive interference). At points where the path difference is an odd number of half-wavelengths ($\Delta = (m+\tfrac{1}{2})\lambda$), the waves arrive out of step and cancel, giving a minimum (destructive interference). Because the sources are coherent, each point's path difference stays constant, so the maxima and minima remain fixed, producing a stable interference pattern of alternating bands.
Q2 (3 marks): Express the path difference as a number of wavelengths: $0.75/0.50 = 1.5\lambda$. A maximum occurs where the path difference is a whole number of wavelengths, because the waves then arrive in phase and their displacements add. A minimum occurs where the path difference is an odd number of half-wavelengths, because the waves then arrive exactly out of phase and their displacements cancel. Here $1.5\lambda$ is $3 \times \tfrac{1}{2}\lambda$, which is an odd number of half-wavelengths, so the two waves arrive out of phase and a minimum is heard: the sound is very quiet or, for equal amplitudes, silent at that point.
Q3 (3 marks): The pattern is fixed because the two sources are coherent: they emit at the same frequency and maintain a constant phase relationship with one another. Under those conditions the phase difference between the two waves arriving at any given point depends only on the path difference to that point, which is a fixed geometric quantity. So a point where the path difference is a whole number of wavelengths stays a maximum permanently, and a point where it is an odd number of half-wavelengths stays a minimum permanently. The pattern is therefore stationary in space and can be observed, measured and photographed. If one source drifted slightly in frequency, the two waves would no longer keep a constant phase relationship: the phase difference at every point would then change continuously with time. The positions of the maxima and minima would drift, and if the drift were fast compared with the observer's response the pattern would blur into a uniform average intensity and effectively disappear.
You can now explain how two coherent sources build an interference pattern and use path difference to classify maxima and minima. Next lesson looks at what happens when a wave interferes with its own reflection, a standing wave.
↩ Back to focus-area hubAt the start you predicted whether the equal-distance point between two speakers is loud or quiet. It is loud: the path difference there is zero, so the waves arrive in phase and reinforce, a maximum. If you predicted quiet, revisit Card 3, zero path difference means perfect constructive interference.
A short spaced-retrieval set, mixing this lesson with earlier ones. It gets harder to guess and easier to remember.
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