A longitudinal mechanical wave travels through Medium X with a propagation speed of 340\te… — JAMB UTME

A longitudinal mechanical wave travels through Medium X with a propagation speed of $340\text{ m/s}$ . The distance between two consecutive compressions in Medium X is $0.68\text{ m}$ . When the wave propagates across a boundary into Medium Y, the distance between a compression and the immediately adjacent rarefaction is measured as $1.70\text{ m}$ . What is the speed of propagation of the wave in Medium Y?

$1700\text{ m/s}$ Answer
B
$850\text{ m/s}$
C
$340\text{ m/s}$
D
$680\text{ m/s}$

Answer

The speed of propagation of the wave in Medium Y is

1700\text{ m/s}

In Medium X, the distance between consecutive compressions gives a full wavelength of

0.68\text{ m}

, yielding a wave frequency of

f = \frac{340\text{ m/s}}{0.68\text{ m}} = 500\text{ Hz}

. Because frequency depends solely on the source, it remains

500\text{ Hz}

in Medium Y. In Medium Y, the distance between a compression and the adjacent rarefaction is half a wavelength, making

\lambda_Y = 2 \times 1.70\text{ m} = 3.40\text{ m}

. Multiplying frequency by the new wavelength gives the speed in Medium Y as

v_Y = 500\text{ Hz} \times 3.40\text{ m} = 1700\text{ m/s}

Step-by-Step Solution

Determine the frequency of the wave using parameters from Medium X.

In longitudinal waves, the distance between consecutive compressions equals one wavelength, so

\lambda_X = 0.68\text{ m}

. Using

v = f \cdot \lambda

, frequency

f = \frac{340\text{ m/s}}{0.68\text{ m}} = 500\text{ Hz}

Frequency is determined by the wave source and remains unchanged when a wave transitions between different media.

Calculate the wavelength of the wave in Medium Y.

The distance from a compression to the immediately adjacent rarefaction is half of a wavelength (

\frac{\lambda_Y}{2} = 1.70\text{ m}

). Therefore,

\lambda_Y = 2 \times 1.70\text{ m} = 3.40\text{ m}

A full wavelength spans from compression to compression or rarefaction to rarefaction.

Calculate the wave propagation speed in Medium Y.

v_Y = f \cdot \lambda_Y = 500\text{ Hz} \times 3.40\text{ m} = 1700\text{ m/s}

Applying the wave speed formula with constant frequency.

Key Concept

Wave Propagation across Boundaries and Longitudinal Wave Characteristics

Estimated Time:2m 0s

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