A progressive wave traveling through a primary medium is represented by the displacement e… — JAMB UTME

A progressive wave traveling through a primary medium is represented by the displacement equation $y = 0.04 \sin\left(100\pi t - 2.5\pi x\right)$ , where $x$ and $y$ are in meters and $t$ is in seconds. If the wave propagates into a secondary medium where its speed increases by $20\%$ , what is the wavelength of the wave in the secondary medium?

$0.96\text{ m}$ Cevap
B
$0.67\text{ m}$
C
$1.50\text{ m}$
D
$0.16\text{ m}$

Cevap

The wavelength of the wave in the secondary medium is

0.96\text{ m}

Comparing

y = 0.04 \sin(100\pi t - 2.5\pi x)

to the standard form

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

, the wave number is

k = 2.5\pi\text{ rad/m}

. The initial wavelength is

\lambda_1 = \frac{2\pi}{2.5\pi} = 0.8\text{ m}

. When a wave moves to a new medium, its frequency stays constant, making wavelength directly proportional to wave speed (

v \propto \lambda

). An increase of

20\%

in wave speed means the new wavelength is

\lambda_2 = 0.8 \times 1.20 = 0.96\text{ m}

Adım Adım Çözüm

Extract angular frequency

\omega

and wave number

k

from the wave equation.

From

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

, we identify

\omega = 100\pi\text{ rad/s}

and

k = 2.5\pi\text{ rad/m}

Standard wave equation parameters directly define the wave's spatial and temporal frequencies.

Calculate the wavelength

\lambda_1

in the initial medium.

$\lambda_1 = \frac{2\pi}{k} = \frac{2\pi}{2.5\pi} = 0.8\text{ m}$.

Wavelength is inversely related to the wave number

k

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

Apply the boundary conditions of wave refraction across media.

Frequency

f

remains constant across boundary; speed

v

and wavelength

\lambda

scale proportionally.

The frequency of a wave is determined solely by the source and does not change upon entering a new medium.

Determine the new wavelength

\lambda_2

in the secondary medium.

$\lambda_2 = \lambda_1 \times (1 + 0.20) = 0.8\text{ m} \times 1.20 = 0.96\text{ m}$.

Since

v = f \lambda

and

f

is constant,

\frac{v_2}{v_1} = \frac{\lambda_2}{\lambda_1} = 1.20

Anahtar Kavram

Invariance of Wave Frequency across Media and Wave Equation Parameter Extraction

Tahmini Süre:2m 0s

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