Figure shows a snapshot of a sinusoidal travelling wave taken at t = 0.3 s. The wavelength is 7.5 cm and the…
Figure shows a snapshot of a sinusoidal travelling wave taken at t = 0.3 s. The wavelength is 7.5 cm and the amplitude is 2 cm. If the crest P was at x = 0 at t = 0, write the equation of travelling wave.
Solution
Sol. Given, $A = 2\;\text{cm},\; \lambda = 7.5\;\text{cm}$
$\therefore\; k = \frac{2\pi}{\lambda} = 0.84\;\text{cm}^{-1}$
The wave has travelled a distance of 1.2 cm in 0.3 s. Hence, speed of the wave,
$v = \frac{1.2}{0.3} = 4\;\text{cm s}^{-1}$
∴ Angular frequency, $\omega = (v) (k) = 3.36 \text{ rad s}^{-1}$
Since, the wave is travelling along positive x-direction and crest (maximum displacement) is at $x = 0$ at $t = 0$, we can write the wave equation as
$y (x, t) = A \cos (kx - \omega t)$
or
$y (x, t) = A \cos (\omega t - kx)$ [∵ $\cos(-\theta) = \cos\theta$]
Therefore, the desired equation is
$y (x, t) = 2\cos (0.84x - 3.36 t)$ cm
Answer: $v = 4$ cm s$^{-1}$