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

A transverse progressive wave traveling along a stretched string is represented by the displacement equation $y = 0.05 \sin\left(20\pi t - \frac{\pi}{4} x\right)$ , where $x$ and $y$ are in meters and $t$ is in seconds. What is the phase difference between two points on the string separated by a distance of $2.0\text{ m}$ ?

A
$\frac{\pi}{4}\text{ rad}$
$\frac{\pi}{2}\text{ rad}$ Answer
C
$\pi\text{ rad}$
D
$2\pi\text{ rad}$

Answer

The phase difference between the two points is

\frac{\pi}{2}\text{ rad}

Comparing the given equation

y = 0.05 \sin\left(20\pi t - \frac{\pi}{4} x\right)

to the standard wave equation

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

, the wavenumber is

k = \frac{\pi}{4}\text{ rad m}^{-1}

. The phase difference between two points separated by

\Delta x = 2.0\text{ m}

\Delta \phi = k \Delta x = \frac{\pi}{4} \times 2.0 = \frac{\pi}{2}\text{ rad}

. Thus, the option stating

\frac{\pi}{2}\text{ rad}

is correct.

Step-by-Step Solution

Identify the wavenumber

k

from the standard wave equation

Comparing

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

with

y = 0.05 \sin\left(20\pi t - \frac{\pi}{4} x\right)

gives

k = \frac{\pi}{4}\text{ rad m}^{-1}

The coefficient of

x

in the wave equation represents the wavenumber

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

Calculate the phase difference using

\Delta \phi = k \Delta x

\Delta \phi = \left(\frac{\pi}{4}\text{ rad m}^{-1}\right) \times 2.0\text{ m} = \frac{\pi}{2}\text{ rad}.

Phase difference is directly proportional to the spatial separation between two points along the path of propagation.

Key Concept

Phase difference in a progressive wave

Estimated Time:1m 0s

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