Waves and sound

NUST NET (Engineering) · Physics · Oscillations, waves and light. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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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

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?

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

Answer: 100 Hz and 300 Hz

Practice questions

Try each one, then open the answer.

1. Beats are heard when two sound waves are superposed that have

  1. A
    the same frequency but opposite phase
  2. B
    slightly different frequencies
  3. C
    very different frequencies
  4. D
    the same frequency but different amplitudes
Show answer

Answer: B. Two waves of slightly different frequencies drift in and out of phase, so the loudness rises and falls f1 − f2 times per second. Very different frequencies give no beats the ear can follow.

2. A pipe closed at one end can produce

  1. A
    only the odd harmonics
  2. B
    only the even harmonics
  3. C
    all the harmonics
  4. D
    only its fundamental
Show answer

Answer: A. The closed end must be a node and the open end an antinode, so the pipe holds λ/4, 3λ/4, 5λ/4 ... giving f, 3f, 5f: odd harmonics only. A pipe open at both ends gives all harmonics.

3. A listener moves directly away from a stationary source of sound at one-fifth of the speed of sound. The frequency he hears is

  1. A
    20% lower than the source frequency
  2. B
    about 17% lower than the source frequency
  3. C
    20% higher than the source frequency
  4. D
    25% higher than the source frequency
Show answer

Answer: A. For a moving listener f′ = f(v − uo)/v = f(1 − 1/5) = 0.8f, a 20% drop. About 17% lower (f × 5/6) is what a source moving away at the same speed would give.

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