Superposition, interference, beats and Doppler effect

Physics · Waves. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Physics topics

The idea

When two waves meet at a point, the resultant displacement is the sum of the displacements of each wave. This is the principle of superposition. Waves of the same frequency give interference: they add where they arrive in phase and cancel where they arrive out of phase. Waves of slightly different frequencies give beats, a regular rise and fall of loudness.

A wave reflected from a denser medium (a fixed end) is turned over, a phase change of 180°. Reflected from a rarer medium (a free or open end), it is not.

The Doppler effect is the apparent change in frequency when the source and the observer move relative to each other. Moving closer raises the pitch. Moving apart lowers it.

Rules to remember

Constructive interference: path difference = nλ (n = 0, 1, 2, ...)
Destructive interference: path difference = (n + ½)λ
beat frequency = difference of the two frequencies (f₁ − f₂)
Denser medium: crest returns as trough (180° change). Rarer medium: crest returns as crest; for sound, a compression returns from an open end as a rarefaction
Observer moving, source at rest: f′ = f(v + u_o)/v towards, f′ = f(v − u_o)/v away
Source moving, observer at rest: f′ = fv/(v − u_s) towards, f′ = fv/(v + u_s) away
Uses: radar speed checks, sonar, red shift of light from stars and galaxies moving away, blue shift for those approaching

Common mistake

Putting the speed of a moving source in the top of the fraction. A moving observer changes the top (v ± u_o); a moving source changes the bottom (v ∓ u_s). Check at the end: approaching must give a higher frequency.

Worked example

A siren of frequency 720 Hz moves away from a stationary listener at 20 m s⁻¹. The speed of sound is 340 m s⁻¹. What frequency does the listener hear?

The source moves away, so f′ = fv/(v + u_s).
f′ = (720 × 340) ÷ (340 + 20) = 244 800 ÷ 360.
f′ = 680 Hz, lower than 720 Hz as expected.

Answer: 680 Hz.

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Read the full notes

Chapter 8

Keep going

← Progressive waves and the speed of sound Stationary waves in strings and air columns →Take a Physics test