Grade by Grade
E. Waves and Sound · Waves and Sound

Superposition and Interference

E3.3 E2.2 E2.1About 70 minutes
Learning goals
  • I can state the principle of superposition and use it to find the resultant displacement when waves overlap.
  • I can tell constructive interference from destructive interference.
  • I can add two waves graphically, point by point.
  • I can use path difference to predict loud and quiet spots near two sources.

Words to know

principle of superposition
When two or more waves overlap, the resultant displacement at any point is the sum of the individual displacements.
interference
What happens when two or more waves overlap in the same medium.
constructive interference
Interference in which the displacements are in the same direction, producing a larger resultant displacement.
destructive interference
Interference in which the displacements are in opposite directions, producing a smaller (or zero) resultant displacement.
resultant
The single wave (or displacement) produced by adding the overlapping waves.
node / nodal line
A point (or a line of points) that stays at rest because of continuous destructive interference.
antinode / antinodal line
A point (or line) where constructive interference makes the largest vibrations.
path difference
The difference between the distances from a point to two wave sources: |PS₁ − PS₂|.

When waves collide

Two snowballs thrown at each other collide and bounce off. Two waves do something stranger: they pass right through each other. While they overlap, they combine, and then each carries on as if the other had never been there. This lesson is about what happens during that overlap. It explains noise-cancelling headphones, dead spots in a gym and, in the next lesson, the notes a guitar string plays.

The principle of superposition

When two or more waves overlap, the displacement of the medium at each point is the sum of the displacements the waves would cause on their own.

Use signs: displacement above the rest position is positive, below is negative.

  • A +3 cm crest meets a +2 cm crest: the resultant is +3 + 2 = +5 cm. This is constructive interference.
  • A +3 cm crest meets a −2 cm trough: the resultant is +3 + (−2) = +1 cm. This is destructive interference.
  • A +3 cm crest meets a −3 cm trough of the same shape: the rope is flat for an instant. This is complete destructive interference. The energy is not destroyed: the rope is still moving at that instant (it has kinetic energy), and both pulses reappear afterward.

For sound waves, the same idea applies to the pressure: compressions can add to compressions, or a compression can cancel a rarefaction.

Constructive (two crests)Destructive (crest + equal trough)beforeoverlapaftercrest heights addflat for an instant
Pulses travelling toward each other (arrows) on a rope. While they overlap, their displacements add. Afterward each pulse continues unchanged, as if the other was never there.

Adding periodic waves

When two periodic waves with the same wavelength overlap:

  • If they are in phase (crest on crest), they interfere constructively all along. The resultant amplitude is A₁ + A₂.
  • If they are half a cycle out of phase (crest on trough), they interfere destructively. The resultant amplitude is |A₁ − A₂|, which is zero if the amplitudes are equal.

To add waves graphically, pick several positions, read each wave's displacement there, add them with signs, and plot the sums. Join the points with a smooth curve.

In phase: A = 1 + 1 = 2Half a cycle out of phase: A = 1.5 − 1 = 0.5wave 1wave 2sum (resultant)
Graphical addition. Left: two in-phase waves of amplitude 1 unit add to a wave of amplitude 2. Right: waves of amplitude 1.5 and 1, half a cycle apart, add to a wave of amplitude only 0.5.

Interference from two sources

Drop two pebbles into a calm pond at the same moment, or let two dippers tap the water together in a ripple tank. Each makes circular waves. Where they overlap, a fixed pattern appears:

  • Antinodal lines: where crest meets crest and trough meets trough. The water moves the most.
  • Nodal lines: where crest meets trough. The water stays nearly still.

What decides which happens at a point P? The path difference: how much farther P is from one source than from the other. If the sources vibrate in phase:

  • constructive (antinode) when the path difference is a whole number of wavelengths: |PS₁ − PS₂| = nλ (n = 0, 1, 2, …)
  • destructive (node) when it is a whole number plus a half: |PS₁ − PS₂| = (n + ½)λ

The line straight down the middle (equal distances, path difference 0) is always antinodal.

S₁S₂antinodal line (constructive)nodal line (destructive)
Two sources S₁ and S₂ in phase. Arcs show crests spreading from each source (seen from above). Where arcs from both sources cross, crest meets crest: those points lie on antinodal lines (solid): the centre line (path difference 0) and two lines where it is 1λ. On the nodal lines (dashed) the path difference is ½λ, so crest meets trough and the water stays nearly still.
Worked example

Example 1: Adding pulses

Pulse A has a maximum displacement of +4.0 cm and pulse B has −1.5 cm. The pulses have the same shape and meet exactly. What is the maximum displacement during overlap? What kind of interference is this? What do the pulses look like afterward?

  1. Add with signs: +4.0 + (−1.5) = +2.5 cm.
  2. The displacements are in opposite directions, so this is destructive interference (partial, since it does not cancel completely).
  3. Afterward each pulse continues with its original shape: a +4.0 cm crest and a −1.5 cm trough, moving apart.

+2.5 cm, destructive (partial); both pulses continue unchanged.

Worked example

Example 2: Loud or quiet?

Two speakers side by side play the same 343 Hz tone in phase. Sound travels at 343 m/s. Mira stands 4.0 m from one speaker and 5.5 m from the other. Does she hear a loud spot or a quiet spot?

  1. Find the wavelength: λ = v/f = 343 ÷ 343 = 1.00 m.
  2. Path difference = 5.5 − 4.0 = 1.5 m.
  3. In wavelengths: 1.5 m ÷ 1.00 m = 1.5λ, which is (1 + ½)λ.
  4. A whole number plus a half means crest meets trough: destructive interference.

A quiet spot (on a nodal line). Real rooms also have echoes, so the sound is quieter but rarely silent.

Worked example

Example 3: Graphical addition at chosen points

Wave 1 has displacements 0, +2, 0, −2, 0 cm at five equally spaced points. Wave 2 has displacements 0, +1, 0, −1, 0 cm at the same points. Find the resultant and describe it.

  1. Add point by point: 0 + 0 = 0; 2 + 1 = 3; 0 + 0 = 0; −2 + (−1) = −3; 0 + 0 = 0.
  2. The resultant displacements are 0, +3, 0, −3, 0 cm.
  3. The waves are in phase, so this is constructive interference with amplitude 2 + 1 = 3 cm.

Resultant: 0, +3, 0, −3, 0 cm (amplitude 3 cm).

Just for fun

Two waves walk into a bar. One says, "I'm feeling a bit down today." The other replies, "Same here. Let's get together, we'll be constructive!"

Hands-on activity

Interference you can hear

You need: a phone or computer that can play a steady tone (a free online tone generator), and two small speakers or a pair of earbuds held about 1 m apart, both playing the same channel (mono). A quiet room helps. Keep the volume low.

  1. Play a steady tone of about 500 Hz through both speakers. (λ = 343 ÷ 500 ≈ 0.69 m.)
  2. Cover one ear. Walk slowly in a line parallel to the speakers about 2 m in front of them. Listen for loud and quiet spots.
  3. Mark the quiet spots with tape. Measure how far each quiet spot is from each speaker and calculate the path difference.
  4. Try a higher frequency (about 1000 Hz). Are the quiet spots closer together or farther apart?

What to look for: quiet spots where the path difference is about ½λ (≈ 0.35 m), 1½λ and so on. At the higher frequency the wavelength is shorter, so the loud and quiet spots are closer together. Echoes off walls blur the pattern, so outside on a calm day or a room with soft furniture works best.

Your turn

1

State the principle of superposition.

2

A +5 cm pulse meets a +3 cm pulse. Find the resultant displacement where their peaks overlap.

3

A +5 cm pulse meets a −3 cm pulse. Find the resultant where their peaks overlap.

4

Two identical pulses, one a crest and one a trough, overlap exactly. What does the rope look like? Is the energy gone?

5

Two sources in phase produce waves with λ = 2.0 cm. Point P is 9.0 cm from one source and 13.0 cm from the other. Constructive or destructive?

6

Two waves of amplitudes 3 cm and 5 cm, half a cycle out of phase, overlap. Find the resultant amplitude.

7

What happens to each pulse after two pulses pass through each other?

More practice: warm-up

1

Crest meets crest. Which type of interference is this?

2

Crest meets trough. Which type of interference is this?

3

What is a node?

4

Two in-phase waves of amplitude 2 cm each overlap. Resultant amplitude?

5

What is the path difference for a point on the central line between two sources?

More practice: on your own

1

Two speakers in phase emit sound of wavelength 0.80 m. A listener is 3.0 m from one and 3.4 m from the other. Loud or quiet? Explain.

2

For the same speakers, give two path differences (in metres) that would give loud spots and two that would give quiet spots.

3

Wave 1 has displacements +1, +3, +1, −1, −3 cm at five points; wave 2 has −1, −1, +2, +2, 0 cm at the same points. Find the resultant at each point.

4

In a ripple tank, two dippers vibrate in phase. If the frequency is increased, what happens to the spacing of the nodal lines? Explain.

5

Explain why the nodal lines in a two-source pattern are not perfectly still in a real ripple tank.

Challenge

1

Two speakers 3.0 m apart play the same tone in phase. You stand 4.0 m directly in front of one speaker (so the line from you to that speaker is perpendicular to the line joining the speakers). The speed of sound is 340 m/s. Find the path difference, and find the lowest frequency that gives a quiet spot there.

2

Two pulses, each 20 cm wide, approach each other on a rope at 1.0 m/s each. One has height +6 cm; the other −6 cm (same shape). At t = 0 their centres are 60 cm apart. When is the rope completely flat, and what is the maximum displacement at the midpoint between them at all times?

3

Noise-cancelling headphones use a microphone to sample incoming noise and a speaker to produce a second sound. Use superposition to explain what that second sound must be, and suggest why this works better for steady low-frequency hum (like an airplane engine) than for sudden high-pitched sounds.

Check your understanding

Quiz

1.

When two waves overlap, the resultant displacement equals…

2.

A +2 cm crest meets a −2 cm trough of the same shape. At the instant they overlap exactly, the displacement is…

3.

After two pulses pass through each other, they…

4.

Two sources vibrate in phase with λ = 4 cm. A point with path difference 6 cm is on…

5.

Two in-phase waves of amplitudes 3 cm and 4 cm overlap. The resultant amplitude is…

6.

Which is an example of destructive interference being used on purpose?

For parents and teachers

Big idea: expectation E3.3 asks students to explain and illustrate superposition (this lesson, plus standing waves and beats in later lessons). E2.2 asks for investigations of interference with springs, strings or simulations.

How to help: The listening activity is short and memorable. Free online ripple-tank and wave-interference simulations (for example, PhET's "Wave Interference") are an excellent extra if your teen has a computer.

Common difficulties:

  • Forgetting signs: a trough is a negative displacement.
  • Thinking waves bounce off each other. They pass through.
  • Thinking destructive interference destroys energy. The energy is redistributed: less at nodes, more at antinodes.
  • Path difference: it is a difference of distances, then compared with λ (whole number → loud; half number → quiet).