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

A wave travels along the positive x-direction.
Wave amplitude (A) = 0.20 cm
Wavelength (λ) = 20 cm
Wave speed (v) = 20 m/s

(a) General wave equation along the x-axis:
$$\begin{gathered} y=A\sin \left(kx-\omega t\right) \\ \therefore k=\frac{2\pi }{\lambda }=\frac{2\pi }{2}=\pi {\mathrm{cm}}^{-1} \\ T=\frac{\lambda }{\nu }=\frac{2}{2000} \\ =\frac{1}{1000}={10}^{-3}\mathrm{s} \\ \omega =\frac{2\pi }{T}=2\pi \times {10}^3{\mathrm{s}}^{-1} \end{gathered}$$
Wave equation:
$y=\left(0.2\mathrm{cm}\right)\sin \left[\left(\pi {\mathrm{cm}}^{-1}\right)x-\left(2\pi \times {10}^{-3}{\mathrm{s}}^{-1}\right)\right]$

(b) As per the question
For the wave equation ,we need to find the displacement and velocity at x = 2 cm and t = 0.
$$\begin{gathered} y=\left(0.2\right)\mathrm{cm}\sin 2\pi =0 \\ \therefore \nu =A\omega \mathrm{cos\pi }x \\ =0.2\times 2000\pi \times \cos 2\pi \\ =400\pi \\ =400\pi \mathrm{cm}/\mathrm{s}=4\pi \mathrm{m}/\mathrm{s} \end{gathered}$$

If the wave equation is written in a different fashion, then also we will get the same values for these quantities.