A progressive wave propagating through an initial elastic medium is represented by the equ… — JAMB UTME

A progressive wave propagating through an initial elastic medium is represented by the equation $y = 0.08 \sin(120\pi t - 4\pi x)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. Upon crossing a boundary into a second medium, the wave speed decreases by $25\%$ . What is the wavelength of the wave in the second medium?

$0.375\text{ m}$ Cevap
B
$0.667\text{ m}$
C
$0.125\text{ m}$
D
$0.625\text{ m}$

Cevap

0.375\text{ m}

Comparing

y = 0.08 \sin(120\pi t - 4\pi x)

with the standard wave equation

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

yields an angular frequency

\omega = 120\pi\text{ rad/s}

and wave number

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

. The frequency is

f = \frac{120\pi}{2\pi} = 60\text{ Hz}

, and the initial speed is

v_1 = \frac{120\pi}{4\pi} = 30\text{ m/s}

. When passing into a new medium, the frequency remains constant at

60\text{ Hz}

. A

25\%

reduction in speed gives

v_2 = 0.75 \times 30\text{ m/s} = 22.5\text{ m/s}

. The new wavelength is therefore

\lambda_2 = \frac{22.5\text{ m/s}}{60\text{ Hz}} = 0.375\text{ m}

Adım Adım Çözüm

Extract angular frequency and wave number from the wave equation

\omega = 120\pi\text{ rad/s}

and

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

Comparing the given equation

y = 0.08 \sin(120\pi t - 4\pi x)

with standard form

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

gives the values for

\omega

and

k

Calculate the frequency and initial wave speed in medium 1

f = \frac{\omega}{2\pi} = 60\text{ Hz}

and

v_1 = \frac{\omega}{k} = \frac{120\pi}{4\pi} = 30\text{ m/s}

Frequency is related to angular frequency by

f = \frac{\omega}{2\pi}

, and wave velocity is

v = \frac{\omega}{k}

Determine the speed in medium 2 and apply the principle of constant frequency

v_2 = 30 \times 0.75 = 22.5\text{ m/s}

and

f_2 = f_1 = 60\text{ Hz}

Wave frequency depends solely on the source and remains unchanged across boundary refraction, whereas speed decreases by

25\%

Calculate the new wavelength in medium 2

\lambda_2 = \frac{v_2}{f} = \frac{22.5}{60} = 0.375\text{ m}

Using the wave equation relationship

\lambda = \frac{v}{f}

Anahtar Kavram

Wave refraction across media boundaries and mathematical wave equation parameter matching

Tahmini Süre:2m 0s

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