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

A transverse progressive wave traveling along a stretched string is represented by the mathematical wave equation $y = 0.05 \sin(160\pi t - 8\pi x)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. What is the speed of propagation of the wave in $\text{m/s}$ ?

Cevap: 20 m/s

Cevap

The speed of propagation of the wave is

20\text{ m/s}

Comparing the given equation

y = 0.05 \sin(160\pi t - 8\pi x)

with the standard wave equation

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

gives the angular frequency

\omega = 160\pi\text{ rad/s}

and wave number

k = 8\pi\text{ rad/m}

. Substituting these into

v = \frac{\omega}{k}

yields

v = \frac{160\pi}{8\pi} = 20\text{ m/s}

Adım Adım Çözüm

Compare the given wave equation with the standard progressive wave equation.

The standard form is

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

. Comparing parameters yields

\omega = 160\pi\text{ rad/s}

and

k = 8\pi\text{ rad/m}

Matching coefficients allows direct extraction of angular frequency and wave number.

Calculate the wave speed using the relation between angular frequency and wave number.

v = \frac{\omega}{k} = \frac{160\pi\text{ rad/s}}{8\pi\text{ rad/m}} = 20\text{ m/s}

Wave speed is defined as the ratio of angular frequency to wave number (

v = \lambda f = \frac{\omega}{k}

Anahtar Kavram

Wave Equation Parameter Extraction and Wave Speed Calculation

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