A transverse mechanical wave propagates along a stretched string. If the maximum speed of … — JAMB UTME

A transverse mechanical wave propagates along a stretched string. If the maximum speed of an oscillating particle of the string is equal to the speed of wave propagation, what is the ratio of the wave's amplitude to its wavelength?

A
$2\pi$
$\frac{1}{2\pi}$ Answer
C
$\frac{1}{\pi}$
D
$1$

Answer

The ratio of the wave's amplitude to its wavelength is

\frac{1}{2\pi}

For a transverse wave, the maximum speed of particle oscillation is

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

, while the propagation speed of the wave pattern through the medium is

v_w = \frac{\omega}{k}

. Setting these two speeds equal gives

A\omega = \frac{\omega}{k}

, which simplifies to

A k = 1

. Substituting

k = \frac{2\pi}{\lambda}

yields

A \left(\frac{2\pi}{\lambda}\right) = 1

, giving the ratio

\frac{A}{\lambda} = \frac{1}{2\pi}

Step-by-Step Solution

Express the maximum speed of a particle in simple harmonic oscillation along a transverse wave.

The maximum particle speed is given by

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

, where

A

is the amplitude and

\omega

is the angular frequency.

Particles in a transverse mechanical wave oscillate perpendicularly to wave propagation in simple harmonic motion.

Express the wave propagation speed in terms of angular frequency and wave number.

The wave propagation speed is

v_w = \frac{\omega}{k}

, where

k = \frac{2\pi}{\lambda}

is the angular wave number.

Wave propagation speed reflects the rate at which the wave energy and profile move through the medium.

Equate the maximum particle speed to the wave propagation speed and solve for

\frac{A}{\lambda}

A\omega = \frac{\omega}{k} \implies A k = 1 \implies A \left(\frac{2\pi}{\lambda}\right) = 1 \implies \frac{A}{\lambda} = \frac{1}{2\pi}

Setting

v_{p,\text{max}} = v_w

allows the angular frequency

\omega

to cancel out, directly linking amplitude and wavelength.

Key Concept

Transverse Wave Particle Oscillation vs. Wave Propagation Speed

Estimated Time:2m 0s

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