Question

A wave travels along the positive x- direction with a speed of $20m{s}^{-1}$.The amplitude of the wave is $0.20cm$ and the wavelength $2.0cm$.
(a) write a suitable wave equation which describes this wave.
(b) What is the displacement and velocity of the particle at $x=2.0cm$ at time $t=0$ according to the wave equation written ?

Answer 1

Wave speed, $v=20m/s$

$A=0.20cm$

$\lambda =2cm$

(a) Equation of wave along the x-axis

$y=A\sin (kx-wt)$

$\therefore k=2\pi /\lambda =2\pi /2=\pi c{m}^{-1}$

$T=\lambda /v=2/2000=1/1000\sec ={10}^{-3}\sec$

$\Rightarrow \omega =2\pi /T=2\pi \times {10}^{-3}{\sec }^{-1}$

So, the wave equation is,

$\therefore y=\left(0.2cm\right)\sin \left[\right(\pi c{m}^{-1})x-(2\pi \times {10}^3{\sec }^{-1}\left)t\right]$

(b) At $x=2cm,$ and $t=0$

$y=\left(0.2cm\right)\sin \left(\pi /2\right)=0$

$\therefore v=r\omega \cos \pi x=0.2\times 2000\pi \times \cos 2\pi =400\pi$

$=400\times \left(3.14\right)=1256cm/s$

$=400\pi cm/s=4\pi m/s$