Question

A wave travels along the positive x-direction with a speed of 20 m s−1. The amplitude of the wave is 0⋅20 cm and the wavelength 2⋅0 cm. (a) Write the 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

Explanation:

A wave goes in the positive x-direction.

Amplitude wave $(A)=0.20 \mathrm{cm}=2 \times 10^{-3}$

Wavelength $(\lambda)=20 \mathrm{cm}$

Wave speed $(v)=20 \mathrm{m} / \mathrm{s}$

(a) Specific equation of wave along the x-axis:

$y=A \sin (k x-w)$

$k=\frac{2 \pi}{\lambda}$

$k=\frac{2 \pi}{2}$

$k=\pi \mathrm{cm}^{-1}$

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

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

$T=\frac{1}{1000}$

$T=10^{-3} s$

$w=\frac{2 \pi}{\mathrm{T}}$

$w=\frac{2 \pi}{10^{-3}}$

$w=2 \pi \times 10^{-3} s^{-1}$

Wave equation  

$y=0.2 \sin \left(\pi x-2 \pi \times 10^{-3}\right)$

(b) The displacement and velocity at x=2 cm and t=0 must be found according to the problem

$y=0.2 \sin (2 \pi)=0$

$v=A w \cos (\pi x)$

$v=0.2 \times 2000 \cos (2 \pi)$

$v=400 \pi \mathrm{cm} / \mathrm{s}$

$v=40 \pi \mathrm{m} / \mathrm{s}$