Consider a wave propagating in the negative x-direction whose frequency is 100Hz at t = 5 s, the…
Consider a wave propagating in the negative x-direction whose frequency is 100Hz at t = 5 s, the displacement associated with the wave is given by
$y = 0.5\cos(0.1\ x)$
Here, x and y are measured in centimetres and t in seconds.
Obtain the displacement (as a function of x) at t = 10 s. What is the wavelength and velocity associated with the wave?
Solution
Sol. A wave travelling in negative x-direction can be represented as, $y(x,t) = A\cos(kx + \omega t + \phi)$
At $t = 5\ \mathrm{s}$, $y(x,t = 5) = A\cos(kx + 5\omega + \phi)$
Comparing this with the given equation, we get
$A = 0.5\ \mathrm{cm},\ k = 0.1\ \mathrm{cm}^{-1}\ \text{and}\ 5\omega + \phi = 0\ \ldots(i)$
Now,
$\lambda = \frac{2\pi}{k} = \frac{2\pi}{0.1} = 20\pi\ \mathrm{cm}$
$\omega = 2\pi f = 200\pi\ \mathrm{rad\ s}^{-1}\ \ (\because f = 100\ \mathrm{Hz})$
$\therefore\ v = \frac{\omega}{k} = \frac{200\pi}{0.1} = 2000\pi\ \mathrm{cm\ s}^{-1}$
From Eq. (i), we get
$\phi = -5\omega$
At $t = 10\ \mathrm{s},$
$y(x,t = 10) = 0.5\cos (0.1 x + 10\omega - 5\omega)$
$= 0.5\cos (0.1 x + 5\omega)$
Substituting $\omega = 200\pi\ \mathrm{rad\ s}^{-1}$, we get
$y(x, t = 10) = 0.5\cos (0.1 x + 1000\pi)\ \mathrm{cm}$
Answer: $y(x, t = 10) = 0.5\cos (0.1 x + 1000\pi)\ \mathrm{cm}$