E3.1 Explain two-dimensional diffraction and interference of water waves

Ontario Grade 12 Physics

The Wave Nature of Light

How openings and overlapping waves shape a water surface

In SPH3U, you learned that a wave transfers energy without carrying matter along with it overall. A water wave can travel across a surface, so its motion is two-dimensional: it spreads in two directions across the surface. In this lesson, the physical system is the water surface and the waves travelling on it. We describe positions from above, using a fixed reference frame attached to the ripple tank or pool. Choose right as the positive

x

direction and up as the positive

y

direction. Direction is described with vectors; wavelength, frequency, and wave speed are scalar quantities. The focus is how waves spread at openings and what happens when spreading waves meet.

What you will learn

Describe diffraction as waves spreading after passing an opening or obstacle.
Explain how two-dimensional water waves overlap to produce interference.
Use path difference to identify constructive and destructive interference.
Connect wavelength, frequency, and wave speed in a water-wave situation.

1. Wave features and diffraction

A wavefront is a line joining points on a wave that are in the same part of their repeating motion. In a top view of water waves, wavefronts are often drawn as parallel lines or as circles centred on a source. The spacing between neighbouring wavefronts is the wavelength, represented by

\lambda

. Wavelength is measured in metres. Frequency,

f

, is the number of complete waves produced each second, measured in hertz. Wave speed,

v

, is measured in metres per second.

For a steady wave travelling through the same water, wave speed, frequency, and wavelength are related by

v=f\lambda

. If the source frequency stays fixed and the wave speed does not change, the wavelength stays fixed too. This relationship describes the spacing and travel of the waves; it does not by itself describe their spreading at an opening.

Diffraction is the spreading of waves as they pass through an opening or around an obstacle. A straight wavefront approaching a gap can spread beyond the gap. The effect is easy to notice when the gap width is about the same size as the wavelength. If the gap is much wider than the wavelength, the waves mostly continue forward, although some spreading can still occur at the edges.

A point-like opening produces approximately circular wavefronts. A wider opening can be viewed as producing waves from different parts of the opening. The opening does not create a new kind of wave. It changes the directions in which the existing water waves travel.

v=f\lambda

Diffraction is wave spreading at an opening or obstacle.
The relative sizes of gap width and wavelength affect how noticeable spreading is.
Use metres for wavelength and gap width, hertz for frequency, and metres per second for speed.

2. Two-dimensional interference

Interference occurs when two or more waves overlap in the same region. Water displacement is the vertical motion of the surface above or below its undisturbed level. At a point where waves overlap, their displacements combine. This is called superposition. A crest meeting a crest gives a larger upward displacement. A crest meeting a trough can reduce the displacement.

Two sources are coherent when they produce waves of the same frequency and maintain a steady phase relationship. Phase describes where a wave is in its repeating cycle. Two in-phase sources produce crests at the same time. Coherent sources make a stable pattern of reinforcement and cancellation easier to describe.

At a point reached by waves from two coherent sources, compare the distances travelled from the sources. The path difference is the difference between those distances. If the path difference is a whole-number multiple of the wavelength, the waves arrive in step and reinforce. This is constructive interference. The surface motion there is relatively large.

If the path difference is half a wavelength, or a half-wavelength plus any whole number of wavelengths, a crest from one source meets a trough from the other. This is destructive interference. For equal-amplitude waves, they can cancel at that point. In a real water pattern, the cancellation may not be complete if the waves have different amplitudes.

A top-view sketch helps locate the pattern. Imagine two in-phase sources,

S_1

and

S_2

, below one another. Their circular wavefronts spread outward. Where crest lines from both sources meet, mark constructive interference. Where a crest line from one meets a trough line from the other, mark destructive interference. The resulting regions form lines across the surface, not just a single point. These are often called antinodal lines for reinforcement and nodal lines for cancellation.

\Delta r=n\lambda\;\text{(constructive)},\qquad \Delta r=\left(n+\frac{1}{2}\right)\lambda\;\text{(destructive)}

Superposition means the displacements of overlapping waves combine.
Whole-wavelength path difference gives constructive interference.
Half-wavelength path difference gives destructive interference for equal-amplitude waves.

3. Reading and explaining a water-wave pattern

To explain a pattern, first identify the sources and the region where their waves overlap. State whether the sources are in phase and whether their frequencies match. Then compare the path lengths to a point of interest. A path difference is a scalar distance, not a direction. The wavefronts themselves spread in different directions, but the interference conditions depend on how much farther one wave travels than the other.

For the usual whole-number condition,

n

is an integer: zero, one, two, and so on. A path difference of zero means the waves travelled equal distances. A path difference of one wavelength means one wave travelled one full cycle farther. Both cases bring the waves to the point in step. For destructive interference, a difference of half a wavelength brings them to the point half a cycle apart.

A water surface does not have to remain still wherever destructive interference occurs. The cancellation condition describes the displacement at a location for the ideal equal-amplitude case. Waves continue travelling through and beyond the region. Interference describes the combined pattern while the waves overlap; it does not mean that the waves permanently disappear.

When comparing a diagram with a calculation, check that the labels agree. A crest-to-crest overlap should be labelled constructive, not destructive. A crest-to-trough overlap should be labelled destructive. Distances must use the same unit before comparing them with wavelength.

\Delta r=|r_1-r_2|

Interference depends on both waves reaching the same place.
Path difference is a distance difference and has units of metres.
Destructive interference is not permanent removal of the travelling waves.

Worked example

Recognizing diffraction at a gap

A water wave has wavelength

0.080\,\mathrm{m}

. It reaches an opening that is

0.090\,\mathrm{m}

wide. Predict whether spreading after the opening will be easy to notice.

Define the system
The system is the water wave and the opening. Use a top-view frame fixed to the opening, with right and up as the positive coordinate directions. The wave travels toward the opening; the unknown is how noticeable its spreading will be.
Compare the length scales
Diffraction is pronounced when the opening width is comparable to the wavelength. Both lengths are already in metres, so they can be compared directly.
$\frac{0.090\,\mathrm{m}}{0.080\,\mathrm{m}}=1.1$
Interpret the comparison
The opening is about the same size as one wavelength. The waves should spread noticeably after passing through it. The ratio is unitless, as expected for a comparison of two lengths.

Answer: Noticeable spreading is expected because the gap width is comparable to the wavelength.

Check: The prediction is reasonable: a gap nearly one wavelength wide does not confine the wave to a narrow forward path.

Worked example

Classifying a path difference

Two in-phase water-wave sources produce waves with wavelength

0.12\,\mathrm{m}

. At point

P

, the distances from the sources are

0.66\,\mathrm{m}

and

0.42\,\mathrm{m}

. Is the interference constructive or destructive?

Set the frame and known values
The system is the two-source water-wave pattern viewed from above in a frame fixed to the sources. The sources are in phase. Distances and wavelength are scalars, so no direction sign is needed. Find the path difference, then compare it with the wavelength.
$r_1=0.66\,\mathrm{m},\quad r_2=0.42\,\mathrm{m},\quad \lambda=0.12\,\mathrm{m}$
Find the path difference
The absolute difference of the two travel distances gives the extra distance travelled by one wave.
$\Delta r=|0.66\,\mathrm{m}-0.42\,\mathrm{m}|=0.24\,\mathrm{m}$
Compare with wavelength
The path difference is two wavelengths. The waves therefore arrive in step and reinforce at point $P$ .
$\frac{\Delta r}{\lambda}=\frac{0.24\,\mathrm{m}}{0.12\,\mathrm{m}}=2$

Answer: The interference at

P

is constructive.

Check: The quotient is dimensionless, and a path difference of two wavelengths satisfies the constructive condition.

Worked example

Finding the first destructive path difference

Two equal-amplitude, in-phase sources produce water waves of wavelength

0.20\,\mathrm{m}

. What is the smallest non-zero path difference for destructive interference?

Define the system and unknown
The system is the overlapping waves from the two sources. Use a frame fixed to the sources and describe the pattern from above. The unknown is the smallest positive distance difference that makes the waves arrive half a cycle apart.
Use the destructive condition
For destructive interference, the path difference is a half-wavelength plus any whole number of wavelengths. The smallest non-zero value uses no added whole wavelengths.
$\Delta r=\left(0+\frac{1}{2}\right)(0.20\,\mathrm{m})$
Calculate and interpret
Half of the wavelength is the required path difference. Equal amplitudes allow the crest and trough displacements to cancel at that location.
$\Delta r=0.10\,\mathrm{m}$

Answer: The smallest non-zero destructive path difference is

0.10\,\mathrm{m}

.

Check: The answer has units of metres and is half of the stated wavelength, so it matches the destructive condition.

Common mistakes and how to avoid them

Assuming waves only diffract when an opening is smaller than the wavelength.

Correction: Spreading is most noticeable when opening width and wavelength are comparable. A wider opening can still produce some spreading.

Calling every overlap destructive interference.

Correction: Crest meeting crest reinforces. Crest meeting trough cancels partly or, for equal amplitudes, completely.

Using the sum of the two source distances as the path difference.

Correction: Subtract the distances and use the magnitude of their difference.

Thinking a destructive-interference location permanently removes the waves.

Correction: The waves continue travelling. Destructive interference describes their combined displacement where they overlap.

Lesson summary

Water waves travel across a surface, so their spreading and overlap are two-dimensional.
Diffraction is spreading at an opening or around an obstacle, and is most noticeable when opening width is comparable to wavelength.
Overlapping waves combine by superposition.
Whole-wavelength path differences produce constructive interference; half-wavelength path differences produce destructive interference for equal-amplitude waves.
Check units, source conditions, and whether the pattern shows crest-to-crest or crest-to-trough overlap.

Check your understanding

Question 1

Two in-phase sources make waves with wavelength

0.30\,\mathrm{m}

. A point has a path difference of

0.45\,\mathrm{m}

. What type of interference occurs for equal-amplitude waves?

Constructive, because the path difference is one and a half wavelengths.
Destructive, because the path difference is one and a half wavelengths.
Constructive, because any non-zero path difference reinforces.
No interference, because the waves have different path lengths.

Show answer and explanation

Destructive, because the path difference is one and a half wavelengths.

The path difference is

0.45\,\mathrm{m}/0.30\,\mathrm{m}=1.5

wavelengths. This is a half-wavelength plus one whole wavelength, so the waves arrive out of step and interfere destructively.

Question 2

Which opening is expected to show the most noticeable diffraction for water waves of wavelength

0.10\,\mathrm{m}

?

An opening $0.11\,\mathrm{m}$ wide.
An opening $1.0\,\mathrm{m}$ wide.
An opening $2.0\,\mathrm{m}$ wide.
All three openings must show identical spreading.

Show answer and explanation

An opening

0.11\,\mathrm{m}

wide.

The first opening is closest in size to the wavelength, so spreading is expected to be most noticeable there.

Key terms

Wavefront: A line joining points on a wave that are at the same part of their repeating motion.
Diffraction: The spreading of waves as they pass through an opening or around an obstacle.
Superposition: The combining of wave displacements where waves overlap.
Path difference: The difference between the distances travelled by two waves to the same point.
Constructive interference: Interference that reinforces the wave displacement at a location.
Destructive interference: Interference that reduces the wave displacement at a location.
Coherent sources: Sources that produce waves of the same frequency and maintain a steady phase relationship.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation E3.1. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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E3.1 · Explain two-dimensional diffraction and interference of water waves

Ontario Grade 12 Physics

How openings and overlapping waves shape a water surface

What you will learn

1. Wave features and diffraction

2. Two-dimensional interference

3. Reading and explaining a water-wave pattern

Recognizing diffraction at a gap

Classifying a path difference

Finding the first destructive path difference

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through SPH4U

About this lesson and its review