A progressive transverse wave traveling along a medium is described by the displacement eq… — JAMB UTME

A progressive transverse wave traveling along a medium is described by the displacement equation $y = 0.05 \sin\left(160\pi t - 4\pi x\right)$ , where $x$ and $y$ are in meters and $t$ is in seconds. What is the ratio of the maximum particle velocity to the wave propagation velocity? (Take $\pi = 3.142$ .)

Answer: 0.628

Answer

The ratio of the maximum particle velocity to the wave propagation velocity is 0.628.

For a progressive wave

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

, the wave advances at speed

v = \frac{\omega}{k} = \frac{160\pi}{4\pi} = 40\text{ m/s}

. Meanwhile, individual particles vibrate transversely with simple harmonic motion where maximum velocity is

v_{p,\text{max}} = A\omega = 0.05 \times 160\pi = 8\pi\text{ m/s}

. Taking the ratio gives

\frac{v_{p,\text{max}}}{v} = \frac{8\pi}{40} = 0.2\pi = 0.2 \times 3.142 = 0.6284

, which rounds to 0.628.

Step-by-Step Solution

Extract parameters from the standard wave equation

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

A = 0.05\text{ m}

\omega = 160\pi\text{ rad/s}

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

Matching the given equation with standard progressive wave form yields the required wave constants.

Calculate the wave propagation velocity

v

v = \frac{\omega}{k} = \frac{160\pi}{4\pi} = 40\text{ m/s}

The speed at which the wave energy advances through the medium depends on angular frequency and wave number.

Calculate the maximum transverse particle velocity

v_{p,\text{max}}

v_{p,\text{max}} = A\omega = 0.05 \times 160\pi = 8\pi\text{ m/s} \approx 25.136\text{ m/s}

Particles perform simple harmonic motion, whose maximum speed is given by the product of amplitude and angular frequency.

Compute the ratio of maximum particle velocity to wave velocity

Ratio

= \frac{v_{p,\text{max}}}{v} = \frac{8\pi}{40} = 0.2\pi = 0.2 \times 3.142 = 0.6284

Dividing the maximum particle speed by the wave speed gives the desired dimensionless ratio.

Key Concept

Distinction between particle oscillation velocity and wave propagation velocity

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