Interference

The video lesson for this topic is coming soon.

Essential Question

What happens when two waves meet?

When two balls collide, they bounce off each other. Waves behave very differently. By the end of this lesson, you should be able to add overlapping waves, explain constructive and destructive interference, and describe how noise-cancelling headphones work.

Superposition

Waves can overlap
Displacements add together
Up is +, down is −

Top row, labeled before: a pulse 2 cm tall moves right and a pulse 3 cm tall moves left along a blue rope. Bottom row, labeled overlapping: a single green pulse 5 cm tall, labeled 2 + 3 = 5 cm.

When two waves overlap in the same place, the principle of superposition says that their displacements simply add. If a 2 cm pulse and a 3 cm pulse (both upward) overlap, the rope rises 2 + 3 = 5 cm at that moment. Displacements above equilibrium count as positive and displacements below count as negative. This adding of overlapping waves is called interference.

Constructive Interference

Crest meets crest
Trough meets trough
Result: a bigger wave

A blue wave plus a red wave that lines up with it crest to crest equals a green wave with twice the amplitude. Caption: crest plus crest makes a bigger crest.

In constructive interference, the waves line up crest to crest and trough to trough. Their displacements add, so the result is a wave with a larger amplitude. Two identical waves that line up perfectly make a wave with twice the amplitude.

Destructive Interference

Crest meets trough
Displacements cancel
Equal amplitudes: flat line

A blue wave plus a red wave that is flipped, so each blue crest lines up with a red trough, equals a flat green line. Caption: crest plus trough cancel.

In destructive interference, the crest of one wave lines up with the trough of the other. One displacement is positive and the other is negative, so they partly or completely cancel. If the two waves have equal amplitudes, they cancel completely and the medium stays flat where they overlap.

Waves Pass Through Each Other

Pulses overlap, then separate
Each keeps its shape
Each keeps its direction

Three snapshots of a rope. Before: an upward pulse moves right (blue arrow) and a smaller downward pulse moves left (red arrow). During: the pulses overlap and partly cancel, leaving a small bump. After: the upward pulse is on the right still moving right, and the downward pulse is on the left still moving left, both unchanged.

Waves are not objects, so they don’t bounce off each other. Two pulses overlap and interfere for a moment, then pass right through each other. Afterward, each pulse has exactly the same shape and size it had before, and it keeps moving in its original direction, as if the other pulse had never been there.

Noise-Cancelling Headphones

Mic hears outside noise
Speaker plays the flipped wave
Destructive interference: quiet

Three rows. A red, bumpy noise wave, labeled noise. Below it, a blue wave that is the exact upside-down copy, labeled speaker. Below that, a flat green line labeled you hear. Caption: the flipped wave cancels the noise.

Noise-cancelling headphones use destructive interference on purpose. A tiny microphone on the outside of each ear cup picks up the incoming noise. A chip creates the exact upside-down copy of that wave, and the speaker plays it. Every crest of the noise meets a trough from the speaker, so the two cancel and you hear much less noise. Your music is added separately, so it isn’t cancelled.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Adding Two Crests

Two pulses travel toward each other on a rope. Both are crests. One is 3.0 cm tall, and the other is 5.0 cm tall. How tall is the rope when the pulses completely overlap?

A rope with an upward pulse labeled 3.0 cm moving right and a taller upward pulse labeled 5.0 cm moving left.

Knowns: y₁ = +3.0 cm, y₂ = +5.0 cm (both crests, so both positive)
Superposition: y = y₁ + y₂
Substitute: y = (+3.0 cm) + (+5.0 cm)
Answer: y = +8.0 cm (8.0 cm above equilibrium). This is constructive interference.

Example 2: Crest Meets Trough

A 6.0 cm crest and a 4.0 cm trough travel toward each other on a rope. What is the displacement of the rope when they completely overlap?

A rope with an upward pulse labeled 6.0 cm crest moving right and a downward pulse labeled 4.0 cm trough moving left.

Knowns: y₁ = +6.0 cm (crest), y₂ = −4.0 cm (trough)
Superposition: y = y₁ + y₂
Substitute: y = (+6.0 cm) + (−4.0 cm)
Answer: y = +2.0 cm (2.0 cm above equilibrium). This is destructive interference, since the waves partly cancel.

Example 3: After They Pass

The 6.0 cm crest and the 4.0 cm trough from the last example pass through each other. Describe each pulse after they separate.

Key idea: waves pass through each other without changing
Crest: still a 6.0 cm crest, still moving in its original direction
Trough: still a 4.0 cm trough, still moving in its original direction
Answer: both pulses are unchanged. The 2.0 cm displacement only existed while they overlapped.

Example 4: Canceling Noise

At one instant, the noise wave at your ear has a displacement of +0.40 µm. What displacement should the headphone speaker’s wave have at that instant to cancel the noise?

A red noise wave with one point marked by a dot and labeled +0.40 micrometers.

Goal: total displacement = 0
Superposition: y_noise + y_speaker = 0
Rearrange: y_speaker = −y_noise = −(+0.40 µm)
Answer: y_speaker = −0.40 µm, a trough exactly as deep as the noise crest is tall

Big Ideas

Overlapping displacements add
Crest + crest: constructive, bigger
Crest + trough: destructive, cancel
Waves pass through each other unchanged

Back to the essential question: when two waves meet, their displacements add together. Crest on crest builds a bigger wave (constructive interference), and crest on trough cancels (destructive interference). Then the waves pass through each other unchanged. Next up, we’ll see what happens when a wave interferes with its own reflection in Standing Waves.