Waves and sound

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

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The idea

Every wave obeys v = fλ. The speed is fixed by the medium, so changing the frequency changes the wavelength, not the speed.

Two equal waves travelling in opposite directions form a stationary wave with fixed nodes (no motion) and antinodes (greatest motion). A fixed end of a string or the closed end of a pipe is a node; an open end is an antinode. This decides which frequencies, called harmonics, can exist.

Beats are the regular rise and fall in loudness when two notes of slightly different frequency sound together. The Doppler effect is the change in the frequency heard when the source or the listener moves.

Rules to remember

v = fλ; speed of a wave on a string v = √(T/μ), T = tension, μ = mass per unit length
String fixed at both ends, or pipe open at both ends: f_n = nv/(2L), n = 1, 2, 3 … (all harmonics)
Pipe closed at one end: f_n = nv/(4L), n = 1, 3, 5 … (odd harmonics only)
Distance from node to next node = λ/2; from node to nearest antinode = λ/4
Speed of sound in a gas ∝ √(absolute temperature); in air about 332 m s⁻¹ at 0 °C, rising about 0.61 m s⁻¹ per °C
Beat frequency = f₁ − f₂ (the difference); time between beats = 1 ÷ beat frequency
Doppler: f′ = f(v ± v_o)/(v ∓ v_s); upper signs when moving towards each other (v_o listener, v_s source)

Common mistake

Treating a moving source and a moving listener the same way. The listener's speed goes in the top of the fraction and the source's speed in the bottom, and the answers differ.

Worked example

A pipe closed at one end is 0.85 m long. The speed of sound is 340 m s⁻¹. What are its two lowest resonant frequencies?

Fundamental: f₁ = v/(4L) = 340/(4 × 0.85) = 340/3.4 = 100 Hz.
A closed pipe gives odd harmonics only, so the next is 3f₁.
3 × 100 = 300 Hz.

Answer: 100 Hz and 300 Hz

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