A plane mechanical wave propagating through a fluid is represented by the equation y = 0.0… — JAMB UTME

A plane mechanical wave propagating through a fluid is represented by the equation $y = 0.02 \sin \left(250\pi t - \frac{5\pi}{2} x\right)$ , where $x$ and $y$ are in meters and $t$ is in seconds. When the wave passes into a secondary fluid medium, its speed decreases to $60\text{ m/s}$ . Assuming the frequency remains constant, what is the wavelength of the wave in the secondary medium?

$0.48\text{ m}$ Answer
B
$0.80\text{ m}$
C
$0.24\text{ m}$
D
$1.33\text{ m}$

Answer

The wavelength of the wave in the secondary medium is

0.48\text{ m}

Comparing the given equation to the general progressive wave equation

y = A \sin(\omega t - kx)

, we find the angular frequency

\omega = 250\pi\text{ rad/s}

. The wave frequency is

f = \frac{\omega}{2\pi} = 125\text{ Hz}

. Because wave frequency is invariant across media boundaries,

f

remains

125\text{ Hz}

in the secondary medium. Using the wave equation

v = f \lambda

, the wavelength in the secondary fluid is

\lambda = \frac{v}{f} = \frac{60\text{ m/s}}{125\text{ Hz}} = 0.48\text{ m}

Step-by-Step Solution

Extract angular frequency and wave number from the progressive wave equation

From

y = 0.02 \sin \left(250\pi t - \frac{5\pi}{2} x\right)

\omega = 250\pi\text{ rad/s}

and

k = \frac{5\pi}{2}\text{ rad/m}

Standard progressive wave equations follow the format

y = A \sin(\omega t - kx)

Determine the frequency of the wave

f = \frac{\omega}{2\pi} = \frac{250\pi}{2\pi} = 125\text{ Hz}

Frequency is related to angular frequency by

\omega = 2\pi f

Apply boundary transition rules to determine the new wavelength

Frequency

f

remains constant at

125\text{ Hz}

across media. Therefore,

\lambda_2 = \frac{v_2}{f} = \frac{60\text{ m/s}}{125\text{ Hz}} = 0.48\text{ m}

When a wave travels across different media boundaries, its frequency depends only on the source and remains constant, whereas speed and wavelength adjust accordingly.

Key Concept

Wave equation parameter extraction and frequency invariance during refraction

Estimated Time:1m 30s

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