A progressive wave traveling along a stretched string is represented by the equation y = 0… — JAMB UTME

A progressive wave traveling along a stretched string is represented by the equation $y = 0.02 \sin(50\pi t - 4\pi x)$ , where $x$ and $y$ are in meters and $t$ is in seconds. What is the speed of the wave in meters per second ( $\text{m/s}$ )?

Answer: 12.5 m/s

Answer

The speed of the wave is

12.5\text{ m/s}

Comparing the wave equation

y = 0.02 \sin(50\pi t - 4\pi x)

with the standard form

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

shows that the angular frequency is

\omega = 50\pi\text{ rad/s}

and the wave number is

k = 4\pi\text{ rad/m}

. Using the relation

v = \frac{\omega}{k}

, the speed of the wave is calculated as

v = \frac{50\pi}{4\pi} = 12.5\text{ m/s}

Step-by-Step Solution

Extract angular frequency and wave number from the equation

Comparing

y = 0.02 \sin(50\pi t - 4\pi x)

with

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

gives

\omega = 50\pi\text{ rad/s}

and

k = 4\pi\text{ rad/m}

Matching corresponding terms yields the wave parameters.

Calculate the wave propagation speed

v = \frac{\omega}{k} = \frac{50\pi}{4\pi} = 12.5\text{ m/s}

Wave speed equals the ratio of angular frequency to wave number.

Key Concept

Calculating wave speed from a mathematical wave equation

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