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Difficulty: EasyProduction, Propagation, and Classification of Waves

A sound wave of frequency 250 Hz250\text{ Hz} propagates through a metal rod at a speed of 5000 m s−15000\text{ m s}^{-1}. What is the distance between a compression and the adjacent rarefaction in the rod?

  1. A
    20 m20\text{ m}
  2. B
    0.05 m0.05\text{ m}
  3. 10 m10\text{ m}Answer
  4. D
    1.25×106 m1.25 \times 10^6\text{ m}

Answer

The distance between a compression and the adjacent rarefaction is 10 m10\text{ m}.
First, find the wavelength using λ=vf=5000 m s−1250 Hz=20 m\lambda = \frac{v}{f} = \frac{5000\text{ m s}^{-1}}{250\text{ Hz}} = 20\text{ m}. In a longitudinal mechanical wave propagating through a medium, one complete wavelength is the distance between two successive compressions or two successive rarefactions. The distance from a compression to the adjacent rarefaction is half a wavelength, giving 20 m2=10 m\frac{20\text{ m}}{2} = 10\text{ m}.

Step-by-Step Solution

1
Calculate the wavelength (\lambda) of the wave using the wave equation.
\lambda = \frac{v}{f} = \frac{5000\text{ m s}^{-1}}{250\text{ Hz}} = 20\text{ m}
The fundamental wave relationship is v=fλv = f\lambda, so wavelength is the ratio of wave speed to frequency.
2
Determine the distance between a compression and the adjacent rarefaction.
\text{Distance} = \frac{\lambda}{2} = \frac{20\text{ m}}{2} = 10\text{ m}
In a longitudinal wave, one full wavelength is the distance between two consecutive compressions. The distance between a compression and the immediate next rarefaction is half of one wavelength.

Key Concept

Distance between consecutive compression and rarefaction in a longitudinal wave
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Topics

Production, Propagation, and Classification of Waves

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