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

A transverse progressive wave traveling along a taut string is represented by the equation $y(x, t) = 0.04 \sin(200t - 8x)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. What is the maximum transverse speed of a particle on the string in meters per second?

Answer: 8 m/s

Answer

The maximum transverse speed of a particle on the string is

8.0\text{ m/s}

The maximum speed of a particle executing simple harmonic motion as part of a progressive wave is given by

v_{p,\text{max}} = A\omega

. From the wave equation

y(x, t) = 0.04 \sin(200t - 8x)

, the amplitude is

A = 0.04\text{ m}

and the angular frequency is

\omega = 200\text{ rad/s}

. Multiplying these values yields

v_{p,\text{max}} = 0.04 \times 200 = 8.0\text{ m/s}

Step-by-Step Solution

Compare the given wave displacement equation with the standard progressive wave form.

Standard form:

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

. Matching coefficients gives

A = 0.04\text{ m}

\omega = 200\text{ rad/s}

, and

k = 8\text{ rad/m}

Extracting amplitude and angular frequency is essential for evaluating particle motion.

Differentiate displacement with respect to time to find the expression for particle velocity.

v_p(x, t) = \frac{\partial y}{\partial t} = A \omega \cos(\omega t - kx)

Particle velocity represents the time rate of change of transverse displacement.

Determine maximum particle speed by setting the magnitude of the cosine factor to 1.

v_{p,\text{max}} = A \omega = 0.04 \text{ m} \times 200 \text{ rad/s} = 8.0 \text{ m/s}

The maximum absolute value of the cosine function is 1.

Key Concept

Maximum particle velocity in a progressive wave (

v_{p,\text{max}} = A\omega

)

Estimated Time:1m 30s

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