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Lesson 3 of 5Waves and Sound
Unit 4 · Lesson 3 of 5

Reflection, Superposition and Beats

What happens where two waves meet, why a reflected pulse sometimes flips, and the pulsing you hear from two nearly equal frequencies.

4 min read10 practice questionsInteractiveAsk AI about this →

Key things to remember

  1. 1Superposition: where waves overlap, the displacement is the sum of the individual displacements.
  2. 2A pulse reflecting from a fixed end is inverted; from a free end it is upright.
  3. 3Constructive interference: waves in phase, displacements add, larger amplitude.
  4. 4Destructive interference: waves out of phase, displacements cancel, smaller or zero amplitude.
  5. 5Waves pass through each other unchanged — interference only occurs while they overlap.
  6. 6
  7. 7Zero beats means the frequencies are identical, which is how instruments are tuned.
  8. 8The absolute value means beats cannot tell you which frequency is higher.

Two objects cannot occupy the same place, but two waves can. Where they overlap, the medium's displacement is simply the sum of what each wave would have produced alone. That rule is called the principle of superposition, and everything in this topic follows from it.

Reflection: fixed and free ends

Fixed end — the reflection is invertedA crest returns as a trough. The wall pullsback by Newton's third law — a node forms here.Free end — the reflection stays uprightA crest returns as a crest. The end is free tomove, so an antinode forms here instead.
Fig 1A fixed end inverts the pulse. A free end does not.
AFixed end
  • The end cannot move — clamped or attached to a wall
  • The reflected pulse is inverted (flipped upside down)
  • A crest returns as a trough
  • Creates a node at the boundary
BFree end
  • The end is free to move — a loose ring on a pole
  • The reflected pulse is upright, not inverted
  • A crest returns as a crest
  • Creates an antinode at the boundary
Why a fixed end inverts
When the pulse reaches the wall, the string pulls up on the wall. By Newton's third law the wall pulls down on the string with an equal force — and that downward pull is what launches the inverted pulse back along the string. The inversion is the third law made visible.

Constructive and destructive interference

Constructive — in phaseDisplacements add — a bigger amplitude.Destructive — out of phaseDisplacements cancel — momentarily flat.Superposition: add the displacements point by point, keeping their signs.The waves are not destroyed. An instant later they separate and continue exactly as they were —which is the proof they never actually stopped existing.
Fig 2Add the displacements point by point. That is the whole rule.

When two waves meet in phase — crest on crest — the displacements add and the result is larger. This is constructive interference. When they meet out of phase — crest on trough — the displacements partly or completely cancel, which is destructive interference. Two identical waves exactly out of phase cancel completely, producing nothing at all.

The waves pass through each other unchanged
Interference happens only while the waves overlap. Afterwards each carries on exactly as it was, with its original shape, amplitude and direction. Two pulses that momentarily cancel to a flat string do not destroy each other — an instant later both reappear, which is proof they never really stopped existing.
  • Noise-cancelling headphones generate a wave exactly out of phase with the incoming noise, so the two destructively interfere at your ear.
  • Dead spots in a hall occur where sound from two speakers arrives out of phase. Moving a metre can restore the sound entirely.
  • Anti-phase wiring — connecting one stereo speaker backwards makes the bass thin and weak, because the two speakers now cancel rather than reinforce at low frequencies.

Beats

Sound two notes of nearly the same frequency together and you hear a single tone that pulses slowly in loudness. Those pulses are beats. They happen because the two waves drift in and out of phase with each other: when they are in phase the sound is loud, and a moment later they are out of phase and it is quiet.

f₁f₂sumThe two waves drift in and out of step. The dashed envelope is the pulsing in loudness you hear.f_beat = |f₁ − f₂|. Tune until the beats vanish, and the frequencies match exactly.
Fig 3The two waves drift in and out of step, and the envelope of their sum is what you hear as beats.
The absolute value matters — a beat frequency is never negative.
How musicians actually tune
Play the string against a reference tone and listen for beats. Slow beats mean you are close; fast beats mean you are far off. Adjust until the beats disappear entirely — zero beat frequency means the two frequencies are identical. It is far more precise than trying to judge pitch directly, because the ear detects a slow pulsing much better than a small pitch difference.
Beats do not tell you which way to tune
A beat frequency of against a reference means your string is at or — the absolute value hides the direction. Musicians resolve it by tightening slightly and listening: if the beats speed up, they were going the wrong way.

Beats are only audible when the two frequencies are close — typically within about of each other. Beyond that the pulsing becomes too rapid for the ear to follow and you simply hear two separate notes instead.

Try it yourself

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Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1
Two tuning forks of and are sounded together. Find the beat frequency and how many beats are heard in .
Example 2
A guitar string is played against a reference and 12 beats are heard in . What are the possible frequencies of the string?
Example 3
Two pulses travel toward each other on a string. One is a crest of height , the other a trough of depth . What is the displacement at the instant they exactly overlap, and what happens next?

Where marks get lost

  • Adding amplitudes without signs, so a crest meeting a trough appears to grow instead of cancelling.
  • Thinking waves destroy each other during destructive interference. They pass through and re-emerge unchanged.
  • Getting the reflection rule backwards. Fixed ends invert; free ends do not.
  • Forgetting the absolute value on the beat frequency and reporting a negative answer.
  • Assuming a beat frequency identifies which note is higher. Both and beat at against .
  • Dividing the frequencies instead of subtracting them.
  • Expecting audible beats from two widely separated frequencies. Beyond roughly apart you just hear two notes.

Flashcards 10

Principle of superposition
Constructive interference
Destructive interference
Reflection from a fixed end
+ 6 more

Practice

1
Two tuning forks of and sound together. What is the beat frequency, in hertz?
Hz
2
A listener counts 15 beats in . What is the beat frequency, in hertz?
Hz
3
A pulse travelling along a string reflects from an end clamped firmly to a wall. What happens?
4
Two identical waves meet exactly out of phase, momentarily flattening the string. What happens next?
5
A guitar string beats 4 times per second against a reference. What could the string's frequency be?
6
What principle states that overlapping waves add their displacements?
7
Two waves of and sound together. How many beats are heard in ?
8
How do noise-cancelling headphones reduce unwanted sound?
9
Match each situation to what happens.
Crest meets crest
Crest meets an equal trough
Pulse reflects from a fixed end
Two nearly equal frequencies sound together
10
A string is tuned against a fork. Order these beat frequencies from most in tune to least.
  1. 11 beat per second
  2. 29 beats per second
  3. 35 beats per second
  4. 40 beats per second

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