A progressive transverse wave traveling through a primary medium is governed by the mathem… — JAMB UTME

A progressive transverse wave traveling through a primary medium is governed by the mathematical wave equation $y = 0.05 \sin\left(100\pi t - 2.5\pi x\right)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. Upon entering a secondary medium, the wave undergoes refraction such that its wavelength decreases by $20\%$ . What is the speed of the wave in the secondary medium in $\text{m/s}$ ?

Answer: 32 m/s

Answer

The speed of the wave in the secondary medium is

32\text{ m/s}

Comparing

y = 0.05 \sin\left(100\pi t - 2.5\pi x\right)

with

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

yields

\omega = 100\pi\text{ rad/s}

and

k = 2.5\pi\text{ rad/m}

. This gives a source frequency

f = \frac{\omega}{2\pi} = 50\text{ Hz}

and initial wavelength

\lambda_1 = \frac{2\pi}{k} = 0.8\text{ m}

. Because wave frequency is invariant across boundaries,

f

remains

50\text{ Hz}

in the second medium. The new wavelength is

\lambda_2 = 0.8\text{ m} \times 0.80 = 0.64\text{ m}

. Consequently, the wave speed in the second medium is

v_2 = 50\text{ Hz} \times 0.64\text{ m} = 32\text{ m/s}

Step-by-Step Solution

Extract angular frequency and wave number from the wave equation

\omega = 100\pi\text{ rad/s}

and

k = 2.5\pi\text{ rad/m}

Matching coefficients in

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

allows determination of temporal and spatial characteristics.

Determine wave frequency and original wavelength

f = 50\text{ Hz}

and

\lambda_1 = 0.8\text{ m}

Using fundamental relationships

f = \frac{\omega}{2\pi}

and

\lambda = \frac{2\pi}{k}

Calculate the refracted wavelength under constant frequency

\lambda_2 = 0.64\text{ m}

while

f

remains

50\text{ Hz}

Wave frequency depends solely on the wave source and remains unchanged during medium transitions.

Compute wave propagation speed in the new medium

v_2 = 32\text{ m/s}

Applying the wave equation

v = f\lambda

with the updated wavelength.

Key Concept

Wave equation parameter extraction and frequency invariance during refraction

Estimated Time:2m 0s

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