A progressive transverse wave traveling along a taut string is governed by the displacemen… — JAMB UTME

A progressive transverse wave traveling along a taut string is governed by the displacement equation $y(x,t) = 0.05 \sin(200\pi t - 10\pi x)$ , where $x$ and $y$ are measured in meters and $t$ in seconds. Calculate the distance, in meters, traveled by the wave front during the time taken for a single particle on the string to complete $15$ full oscillations.

Answer: 3 m

Answer

The distance traveled by the wave front during 15 full particle oscillations is

3.0\text{ m}

Comparing

y(x,t) = 0.05 \sin(200\pi t - 10\pi x)

to the standard wave equation

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

gives

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

and

k = 10\pi\text{ rad/m}

. The wave speed

v = \frac{\omega}{k} = 20\text{ m/s}

. The time for one full oscillation is

T = \frac{2\pi}{\omega} = 0.01\text{ s}

, so 15 full oscillations take

t = 15 \times 0.01\text{ s} = 0.15\text{ s}

. The distance traveled by the wave front is

d = v \times t = 20\text{ m/s} \times 0.15\text{ s} = 3.0\text{ m}

. Alternatively, because a wave travels a distance of one wavelength

\lambda = \frac{2\pi}{k} = 0.2\text{ m}

during each period (1 full oscillation), in 15 full oscillations the wave travels

15 \times \lambda = 15 \times 0.2\text{ m} = 3.0\text{ m}

Step-by-Step Solution

Extract wave parameters from the progressive wave equation

Angular frequency

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

and wave number

k = 10\pi\text{ rad/m}

Matching the given equation

y = 0.05 \sin(200\pi t - 10\pi x)

with the standard form

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

identifies

\omega

and

k

Calculate the wave propagation velocity

v = 20\text{ m/s}

Wave speed is given by the relation

v = \frac{\omega}{k} = \frac{200\pi}{10\pi} = 20\text{ m/s}

Find the period of oscillation and total elapsed time

Period

T = 0.01\text{ s}

, total time

t = 0.15\text{ s}

The period

T = \frac{2\pi}{\omega} = \frac{2\pi}{200\pi} = 0.01\text{ s}

. For 15 complete oscillations,

t = 15 \times 0.01\text{ s} = 0.15\text{ s}

Compute the total distance traveled by the wave

Distance

d = 3.0\text{ m}

Using linear motion at constant wave speed,

d = v \times t = 20\text{ m/s} \times 0.15\text{ s} = 3.0\text{ m}

Key Concept

Wave equation parameters, particle oscillation period, and wave propagation distance

Estimated Time:2m 0s

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