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

A progressive transverse wave propagating along a taut string is governed by the equation $y = 0.03 \sin(80\pi t - 5\pi x)$ , where $x$ and $y$ are in meters and $t$ is in seconds. What is the phase difference in radians between two points on the string separated by a distance of $0.1\text{ m}$ ?

$\frac{\pi}{2}\text{ rad}$ Cevap
B
$\pi\text{ rad}$
C
$\frac{\pi}{4}\text{ rad}$
D
$8\pi\text{ rad}$

Cevap

The phase difference between the two points is

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

Comparing the given equation

y = 0.03 \sin(80\pi t - 5\pi x)

with the standard form

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

shows that

k = 5\pi\text{ rad/m}

. Substituting this wave number and the distance separation

\Delta x = 0.1\text{ m}

into

\Delta \phi = k \Delta x

yields

\Delta \phi = 5\pi \times 0.1 = 0.5\pi\text{ rad}

, which is equal to

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

Adım Adım Çözüm

Identify the wave number

k

from the given wave equation.

k = 5\pi\text{ rad/m}

The standard progressive wave equation is

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

, where the coefficient of

x

k

Calculate the phase difference

\Delta \phi

for a spatial separation

\Delta x = 0.1\text{ m}

\Delta \phi = k \Delta x = 5\pi \times 0.1 = 0.5\pi\text{ rad} = \frac{\pi}{2}\text{ rad}

Phase difference is directly proportional to spatial separation via

\Delta \phi = k \Delta x = \frac{2\pi}{\lambda}\Delta x

Anahtar Kavram

Phase Difference in Progressive Waves

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Cevap

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