Difficulty: Intermediate | Prerequisites: Basic wave properties (wavelength, frequency, speed), superposition principle
Two-source interference is one of the clearest demonstrations that waves carry energy and information differently from particles. When two coherent wave sources overlap, the resulting pattern of reinforcement and cancellation depends entirely on how the waves' crests and troughs line up at each point in space. This topic sits at the heart of PHYS 116A's wave unit, bridging single-source wave behaviour to the double-slit experiments and spectral analysis you will meet next. You should already be comfortable with wavelength, frequency, wave speed, and the superposition principle before diving in.
Two point sources vibrating in a ripple tank create an interference pattern of nodal lines (destructive) and anti-nodal lines (constructive). Whether the sources are in phase or out of phase determines what happens along the centre line. The number of nodal and anti-nodal lines depends on the ratio of source separation to wavelength.
Constructive interference (anti-nodal line)
Occurs when two waves arrive at a point perfectly in step, so their amplitudes add. The path length difference is an integer multiple of the wavelength: Δr = 0, λ, 2λ, 3λ, ...
In simple terms, the crests land on crests and the wave gets bigger.
Destructive interference (nodal line)
Occurs when two waves arrive exactly half a cycle out of step, so they cancel. The path length difference is a half-integer multiple of the wavelength: Δr = λ/2, 3λ/2, 5λ/2, ...
Think of it as crests landing on troughs, so the surface stays flat.
Path length difference (Δr)
The difference in distance from a given point to each of the two sources: Δr = |r₁ − r₂|.
It tells you how far "out of sync" the two waves are when they meet at that point.
In phase
Two sources are in phase when they emit crests at the same instant. The centre line between in-phase sources is always constructive.
Out of phase (by half a cycle)
Two sources are out of phase by half a cycle when one emits a crest at the exact moment the other emits a trough. The centre line between such sources is always destructive.
Nodal line
A line in the interference pattern along which destructive interference occurs everywhere. Shown as dashed lines in a typical ripple-tank diagram.
Anti-nodal line
A line along which constructive interference (maximum amplitude) occurs everywhere. Shown as solid lines in a typical ripple-tank diagram.
Look at the centre line (the perpendicular bisector of the line joining the two sources). On the centre line, Δr = 0.
If the centre line is a solid line (constructive), the sources are in phase: both emit crests simultaneously, so zero path difference means the waves reinforce.
If the centre line is a dashed line (destructive), the sources are out of phase by half a cycle: zero path difference but opposite displacements means cancellation.
If the pattern cannot be explained by either case, the sources are out of phase by some other fraction of a cycle.
For in-phase sources:
Constructive (solid): Δr = 0, λ, 2λ, 3λ, ...
Destructive (dashed): Δr = λ/2, 3λ/2, 5λ/2, ...
For sources out of phase by half a cycle:
Constructive (solid): Δr = λ/2, 3λ/2, 5λ/2, ...
Destructive (dashed): Δr = 0, λ, 2λ, 3λ, ...
To identify Δr for a specific line in the diagram, count how many lines out from the centre line it sits, and apply the pattern above.
The number of interference lines depends on the ratio d/λ, where d is the source separation and λ is the wavelength.
More lines appear when d/λ is larger (sources farther apart, or wavelength shorter).
Fewer lines appear when d/λ is smaller (sources closer together, or wavelength longer).
Increasing frequency (at constant wave speed) shortens the wavelength, increasing d/λ and producing more lines.
Decreasing frequency lengthens the wavelength, decreasing d/λ and producing fewer lines.
Replacing the medium with one that has a lower wave speed (at the same frequency) also shortens the wavelength (since v = fλ), increasing d/λ and producing more lines, not fewer.
Wave speed relation:
v = f \lambdaConstructive interference condition (in-phase sources):
\Delta r = m \lambda \quad (m = 0, \pm1, \pm2, \ldots)Destructive interference condition (in-phase sources):
\Delta r = \left(m + \tfrac{1}{2}\right) \lambda \quad (m = 0, \pm1, \pm2, \ldots)For sources out of phase by half a cycle, the conditions swap: constructive where destructive was, and vice versa.
Noise-cancelling headphones exploit destructive interference: a speaker generates a wave that is half a cycle out of phase with incoming noise, producing cancellation at your ear. Sonar and radar systems use two-source interference patterns to locate objects by measuring how reflected waves from multiple transmitters combine at a detector.
Students often assume the centre line is always constructive. It is only constructive when the sources are in phase. If the sources are out of phase by half a cycle, the centre line is destructive.
Students sometimes confuse "fewer lines" with "weaker waves." Fewer nodal and anti-nodal lines means a larger wavelength-to-separation ratio, not a smaller amplitude.
A common error is thinking that lowering the wave speed (by changing the medium) at constant frequency will produce fewer lines. It actually shortens the wavelength (v = fλ), which increases d/λ and produces more lines.
Students often believe that decreasing source separation produces more lines. Moving the sources closer together decreases d/λ and produces fewer lines.
⚠️ You will be asked to determine phase from a ripple-tank diagram. Always check the centre line first.
⚠️ Expect questions that change one variable (frequency, wave speed, source separation) and ask how the number of lines changes. The key ratio is d/λ.
⚠️ Know the path length difference conditions cold. If the exam gives you a specific line and asks for Δr, count outward from the centre line and apply the correct formula for the source phase.
⚠️ Be careful with "trick" answer choices that combine two changes (e.g., changing both frequency and separation). The exam specifies a single change.
True or false: If the centre line of a two-source interference pattern is a nodal line, the sources must be in phase. (False: it means they are out of phase by half a cycle.)
Fill in the blank: For in-phase sources, constructive interference occurs when Δr = ____. (mλ, where m is an integer.)
True or false: Decreasing the frequency of both sources (at constant wave speed) increases the number of nodal lines. (False: it increases λ, decreasing d/λ, so there are fewer lines.)
Fill in the blank: The path length difference at the centre line is always ____. (Zero.)
True or false: Replacing the medium with one that has a lower wave speed (same frequency) produces fewer interference lines. (False: lower speed means shorter wavelength, increasing d/λ and producing more lines.)
Q: Two point sources in a ripple tank produce an interference pattern. The centre line is a dashed (nodal) line. Are the sources in phase or out of phase?
A: Out of phase by exactly half a cycle. On the centre line Δr = 0, so in-phase sources would produce constructive interference there. A nodal line at Δr = 0 means the sources are producing opposite displacements at the same moment.
Q: For in-phase sources, what is the path length difference Δr at the first anti-nodal line away from the centre?
A: Δr = λ. The centre line itself is the zeroth anti-nodal line (Δr = 0), and each successive constructive line adds one full wavelength.
Q: A single change to a two-source experiment produces fewer nodal and anti-nodal lines. The change was to decrease the frequency of both sources. Explain why this produces fewer lines.
A: Decreasing frequency increases the wavelength (since v = fλ and wave speed stays constant). A larger wavelength means d/λ is smaller, so fewer complete interference orders fit between the sources.
Q: Could replacing the water with a fluid of lower wave speed (same frequency) also produce fewer lines?
A: No. Lower wave speed at the same frequency gives a shorter wavelength (λ = v/f). A shorter wavelength increases d/λ, producing more lines, not fewer.
This connects directly to double-slit interference with light, which uses the same path-length-difference logic but with monochromatic light and a distant screen instead of water waves. The conditions for constructive and destructive interference carry over unchanged. It also connects to the hydrogen emission spectrum, where discrete energy-level transitions produce light at specific wavelengths, and those wavelengths determine the interference patterns you see through a diffraction grating.
Two-source interference, ripple tank, nodal line, anti-nodal line, constructive interference, destructive interference, path length difference, Δr, in phase, out of phase, superposition, wave interference pattern, PHYS 116A, UW Physics 116A, wave speed and wavelength, d over lambda ratio, source separation