A progressive transverse wave traveling in Medium 1 is described by the equation y = 0.04 … — JAMB UTME

A progressive transverse wave traveling in Medium 1 is described by the equation $y = 0.04 \sin(200\pi t - 5\pi x)$ , where $x$ and $y$ are in meters and $t$ is in seconds. If the wave enters Medium 2 where its propagation speed drops to $20\text{ m/s}$ , what is the wavelength of the wave in Medium 2?

A
$0.40\text{ m}$
$0.20\text{ m}$ Answer
C
$0.10\text{ m}$
D
$0.80\text{ m}$

Answer

0.20\text{ m}

When a wave propagates from one medium into another, its frequency remains unchanged because frequency is determined solely by the periodic source. From the wave equation in Medium 1, the angular frequency

\omega = 200\pi\text{ rad/s}

, which corresponds to a frequency

f = \frac{\omega}{2\pi} = 100\text{ Hz}

. Using

v = f\lambda

in Medium 2 with

v = 20\text{ m/s}

, the new wavelength is

\lambda = \frac{20}{100} = 0.20\text{ m}

Step-by-Step Solution

Extract the angular frequency from the wave equation.

Comparing

y = 0.04 \sin(200\pi t - 5\pi x)

y = A \sin(\omega t - k x)

gives

\omega = 200\pi\text{ rad/s}

The coefficient of

t

inside the sine term represents the angular frequency

\omega

Calculate the wave frequency.

f = \frac{\omega}{2\pi} = \frac{200\pi}{2\pi} = 100\text{ Hz}

Frequency is related to angular frequency by

\omega = 2\pi f

Apply the boundary condition for wave refraction.

Frequency in Medium 2 is

f = 100\text{ Hz}

Frequency is determined strictly by the wave source and remains invariant when passing between different media.

Calculate the wavelength in Medium 2 using the wave speed equation.

\lambda_2 = \frac{v_2}{f} = \frac{20\text{ m/s}}{100\text{ Hz}} = 0.20\text{ m}.

The fundamental wave equation

v = f\lambda

allows finding wavelength from speed and frequency.

Key Concept

Frequency Invariance Across Media Boundaries

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