Question

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

Answer 1

Wave speed, $v=20m/s$

$A=0.20cm$

$\lambda =20cm$

(a) Equation of wave along the x-axis

$y=Asin(kx-wt)$

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

$T=\frac{\lambda }{v}=\frac{2}{2000}$

$=\frac{1}{1000}sec={10}^{-8}sec$

$\omega =\frac{2\pi }{T}$

$=2\pi \times {10}^{8}se{c}^{-1}$

So, the wave equation is,

$\therefore y=\left(0.2cm\right)sin\left[\right(\pi c{m}^{-1})x-(2\pi \times {10}^{-8}\left)\right]$

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

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

$\therefore v=r\omega cos\pi x$

$=0.2\times 2000\pi \times cos2\pi$

$=400\pi$

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