A progressive wave traveling along a stretched string is represented by the equation y = 0… — JAMB UTME

A progressive wave traveling along a stretched string is represented by the equation $y = 0.02 \sin(120\pi t - 3\pi x)$ , where $x$ and $y$ are in meters and $t$ is in seconds. What is the velocity of the wave in $\text{m s}^{-1}$ ?

Cevap: 40 m s^{-1}

Cevap

The velocity of the wave is

40 \text{ m s}^{-1}

The standard progressive wave equation is

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

, where

\omega

is the angular frequency and

k

is the wave number (wave vector). Comparing the given equation

y = 0.02 \sin(120\pi t - 3\pi x)

with the standard form yields

\omega = 120\pi \text{ rad s}^{-1}

and

k = 3\pi \text{ m}^{-1}

. The velocity of propagation of the wave is given by

v = \frac{\omega}{k} = \frac{120\pi}{3\pi} = 40 \text{ m s}^{-1}

Adım Adım Çözüm

Compare given equation with the standard progressive wave equation

Matching

y = 0.02 \sin(120\pi t - 3\pi x)

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

gives

\omega = 120\pi \text{ rad s}^{-1}

and

k = 3\pi \text{ rad m}^{-1}

Direct parameter identification from the wave function gives the angular frequency and wave number.

Calculate wave velocity

Wave velocity

v = \frac{\omega}{k} = \frac{120\pi}{3\pi} = 40 \text{ m s}^{-1}

The ratio of angular frequency to wave number equals the phase velocity of the wave.

Anahtar Kavram

Extracting wave parameters (angular frequency and wave number) from the mathematical wave equation to determine wave velocity.

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