Interference Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Interference.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

The phenomenon that occurs when two or more waves overlap in space, combining their displacements at every point according to the principle of superposition.

When waves meet, they add up or cancel out at each point depending on whether their crests and troughs align.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Interference asks what oscillates, what travels, and which wave quantity is being measured.

Common stuck point: Students often know a formula related to interference but skip the recognition step: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition?

Worked Examples

Example 1

easy

Two speakers emit identical sound waves in phase. A listener stands equidistant from both speakers. Does the listener hear constructive or destructive interference? What about if one speaker is moved half a wavelength farther away?

Answer

Equidistant: constructive (louder); \frac{λ}{2} offset: destructive (quieter)

First step

When the path difference is zero (equidistant), the waves arrive in phase and interfere constructively — the listener hears a louder sound.

Full solution

2
If one speaker is moved $\frac{\lambda}{2}$ farther, the path difference becomes $\frac{\lambda}{2}$ .
3
A path difference of $\frac{\lambda}{2}$ means the waves arrive exactly out of phase, causing destructive interference — the sound is much quieter.

Interference occurs when two or more waves overlap. Constructive interference (waves in phase) increases amplitude; destructive interference (waves out of phase by half a wavelength) decreases amplitude.

Example 2

medium

In a double-slit experiment, light of wavelength

500 \text{ nm}

passes through slits

0.2 \text{ mm}

apart. A screen is

2 \text{ m}

away. What is the spacing between adjacent bright fringes?

Example 3

medium

In a double-slit experiment, light of wavelength

\lambda = 550\text{ nm}

hits slits separated by

d = 0.25\text{ mm}

. Find the angle of the second-order bright fringe (

m=2

Example 4

medium

Two in-phase sources produce identical waves of amplitude

A

. At point P the intensity is maximum. At point Q the path difference is

\tfrac{\lambda}{3}

. What is the intensity ratio

I_Q/I_P

? (Use

I \propto \cos^2(\phi/2)

Example 5

medium

Two waves of amplitude

3\text{ cm}

and

5\text{ cm}

superpose perfectly in phase. What are the maximum and minimum possible amplitudes of the resultant as their phase varies?

Example 6

hard

In a double-slit experiment with

\lambda=500\text{ nm}

d=0.20\text{ mm}

, and

L=1.0\text{ m}

, a thin glass slab (

n=1.5

, thickness

t=10\,\mu\text{m}

) is placed in front of one slit. By how many fringes does the central maximum shift?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Two coherent light sources create an interference pattern. The third bright fringe is observed at an angle where the path difference is

3\lambda

. If

\lambda = 600 \text{ nm}

, what is the path difference in meters?

Example 2

hard

In a double-slit experiment, the slit separation is

0.5 \text{ mm}

and the screen is

1.5 \text{ m}

away. If the fringe spacing is measured to be

1.9 \text{ mm}

, what is the wavelength of the light?

Example 3

easy

Two wave crests meet at a point. Is this constructive or destructive interference?

Example 4

easy

A wave crest meets a wave trough of equal size. What is the result?

Example 5

easy

What principle governs how overlapping waves combine?

Example 6

easy

For two identical waves to interfere constructively, their path difference should be how many wavelengths?

Example 7

easy

For two identical waves to interfere destructively, their path difference should be what?

Example 8

easy

Does destructive interference destroy energy?

Example 9

easy

After two waves overlap and interfere, what happens to them as they continue past the overlap region?

Example 10

easy

Two speakers play the same tone in phase. At a point equidistant from both, is the interference constructive or destructive?

Example 11

medium

Two waves of wavelength

4

m have a path difference of

8

m. Is the interference constructive or destructive?

Example 12

medium

Two waves of wavelength

4

m have a path difference of

6

m. Constructive or destructive?

Example 13

medium

In a double-slit setup, the path difference to a point is

\tfrac{1}{2}\lambda

. Bright or dark fringe?

Example 14

medium

Two in-phase sources are

3\lambda

from point P and

5\lambda

from point P respectively. Constructive or destructive at P?

Example 15

medium

Why is interference different from diffraction?

Example 16

medium

Two identical waves perfectly in phase each have amplitude

A

. What is the amplitude of the combined wave?

Example 17

medium

Two identical waves of amplitude

A

are exactly out of phase (half-wavelength offset). What is the combined amplitude?

Example 18

medium

Two waves of wavelength

3

m have a path difference of

9

m. Constructive or destructive?

Example 19

medium

Two waves of wavelength

3

m have a path difference of

4.5

m. Constructive or destructive?

Example 20

challenge

In a double slit, bright fringes satisfy

d\sin\theta = m\lambda

. With

d = 0.2

mm and

\lambda = 600

nm, find

\sin\theta

for the first-order (

m = 1

) bright fringe.

Example 21

challenge

Two coherent sources produce a path difference of

2.5\lambda

at a point. Classify the interference and state the phase difference in wavelengths.

Example 22

challenge

Two equal-amplitude

A

waves combine. At one point they are in phase; at another they are exactly out of phase. Find the ratio of intensities (intensity proportional to amplitude squared) at the two points.

Example 23

easy

Two coherent waves arrive at a point with a path difference of

0

. What type of interference occurs?

Example 24

easy

Two in-phase sources produce waves of wavelength

\lambda

. The path difference at point P is

1.5\lambda

. Is the interference constructive or destructive at P?

Example 25

easy

Two identical pulses traveling in opposite directions on a string meet. At the instant they fully overlap, the string is flat. Were the pulses in phase or out of phase?

Example 26

medium

Two coherent radio antennas, in phase, are

30\text{ m}

apart and emit at

\lambda = 12\text{ m}

. A receiver on the perpendicular bisector hears a strong signal. As the receiver walks along a line parallel to the antenna baseline, what is the first path difference at which the signal vanishes?

Example 27

medium

Two waves of equal amplitude

A

meet with a phase difference of

60°

. The resultant amplitude is

2A\cos(\phi/2)

. Find the resultant amplitude.

Example 28

medium

In a double-slit experiment,

\lambda = 600\text{ nm}

d = 0.3\text{ mm}

, screen distance

L = 1.2\text{ m}

. Find the fringe spacing.

Example 29

medium

A double-slit experiment uses sodium light (

\lambda = 589\text{ nm}

) with

d = 0.5\text{ mm}

and

L = 2.0\text{ m}

. Find the distance from the central maximum to the 5th bright fringe.

Example 30

medium

In a double-slit setup, the central bright fringe is at

y=0

. The first dark fringe (minimum) is at

y = 0.6\text{ mm}

on a screen

1.5\text{ m}

away with

d = 0.4\text{ mm}

. Find

\lambda

Example 31

medium

Two speakers

4.0\text{ m}

apart emit a

680\text{ Hz}

tone in phase (sound speed

340\text{ m/s}

). A listener walks along a line

10\text{ m}

in front of the speakers, parallel to the line joining them. What is the spacing between adjacent quiet spots?

Example 32

hard

A double-slit experiment uses white light. The central maximum is white, but the higher-order fringes show color separation. Why does the first-order red fringe appear farther from center than the first-order blue?

Example 33

hard

Two coherent sources

S_1

and

S_2

each have intensity

I_0

. At a point where they interfere constructively, what is the resultant intensity (use

I \propto A^2

Example 34

hard

A thin oil film (

n=1.4

) on water (

n=1.33

) appears bright when viewed at near-normal incidence in

\lambda_{\text{air}} = 560\text{ nm}

light. What is the smallest non-zero film thickness

t

? (Hint: only the top reflection has a

\pi

phase shift.)

Example 35

hard

In a double-slit experiment, immersing the apparatus in water (

n=1.33

) changes the fringe spacing. By what factor does the spacing change?

Example 36

hard

Two coherent sources are

90°

out of phase. The path difference at point P is

\tfrac{\lambda}{4}

. Is the interference at P constructive, destructive, or partial?

Example 37

hard

Two coherent sources emit identical waves. If the slit separation

d

is doubled while

\lambda

and

L

stay fixed, what happens to the fringe spacing?

Example 38

challenge

In Newton's rings (plano-convex lens on flat glass, viewed in reflected

\lambda=600\text{ nm}

light), the radius of the 10th dark ring is

r_{10} = 5\text{ mm}

. Find the radius of curvature

R

of the lens. Use

r_m^2 \approx mR\lambda

for dark rings in reflection.

Example 39

challenge

A Michelson interferometer is illuminated with

\lambda = 632.8\text{ nm}

(HeNe). One mirror is moved slowly and

1500

bright fringes pass a reference point. By how much was the mirror moved?

Example 40

challenge

Two coherent sources of equal intensity produce a pattern with fringe visibility

V = (I_{\max}-I_{\min})/(I_{\max}+I_{\min}) = 1

. If one source is now attenuated so its amplitude is

A/2

while the other stays at

A

, find the new visibility.

← Back to Interference Practice Mode

Related Concepts

Waves Amplitude Standing Waves

Background Knowledge

These ideas may be useful before you work through the harder examples.

wavesamplitude