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Difficulty: HardProduction, Propagation, and Classification of Waves

A mechanical longitudinal wave of frequency 250 Hz250\text{ Hz} propagates from Medium 1 into Medium 2. In Medium 1, the distance between two consecutive compressions is 1.40 m1.40\text{ m}. Upon entering Medium 2, the wave speed increases by 40%40\%. What is the distance, in meters, between a compression and the immediately adjacent rarefaction in Medium 2?

Answer: 0.98 m

Answer

The distance between a compression and the adjacent rarefaction in Medium 2 is 0.98 m0.98\text{ m}.
When a mechanical wave travels between media, its frequency remains unchanged. The wave speed equation v=fλv = f\lambda indicates that wavelength is directly proportional to wave speed. A 40%40\% increase in speed increases the wavelength in Medium 2 from 1.40 m1.40\text{ m} to 1.40×1.40 m=1.96 m1.40 \times 1.40\text{ m} = 1.96\text{ m}. In a longitudinal wave, the distance between a compression and an adjacent rarefaction is half a wavelength, yielding 1.96 m2=0.98 m\frac{1.96\text{ m}}{2} = 0.98\text{ m}.

Step-by-Step Solution

1
Identify the wavelength in Medium 1 from the compression spacing.
λ1=1.40 m\lambda_1 = 1.40\text{ m}
The distance between two successive compressions in a longitudinal wave corresponds to one complete wavelength.
2
Calculate the wavelength in Medium 2 using the constant frequency principle across media boundaries.
λ2=1.40×1.40 m=1.96 m\lambda_2 = 1.40 \times 1.40\text{ m} = 1.96\text{ m}
The frequency of a wave is determined by the source and does not change upon entering a new medium. Since v=fλv = f\lambda, a 40%40\% increase in wave speed results in a proportional 40%40\% increase in wavelength.
3
Find the distance between a compression and the adjacent rarefaction in Medium 2.
d = \frac{\lambda_2}{2} = \frac{1.96\text{ m}}{2} = 0.98\text{ m}
In any longitudinal wave, a compression and its adjacent rarefaction are out of phase by half a cycle, corresponding to half a wavelength.

Key Concept

Wave propagation across boundaries and spatial separation of compressions and rarefactions in longitudinal waves.
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Topics

Production, Propagation, and Classification of Waves

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