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:

\[y = A\sin\left( kx - \omega \right)\] 

\[ \therefore k = \frac{2\pi}{\lambda} = \frac{2\pi}{2} = \pi   {cm}^{- 1} \] 

\[T = \frac{\lambda}{\nu} = \frac{2}{2000}\] 

\[         = \frac{1}{1000} =  {10}^{- 3}   s\] 

\[\omega = \frac{2\pi}{T} = 2\pi \times  {10}^3    s^{- 1}\]

Wave equation:
\[y = \left( 0 . 2  cm \right)  \sin\left[ \left( \pi  {cm}^{- 1} \right)  x - \left( 2\pi \times {10}^{- 3} s^{- 1} \right) \right]\]
(b) As per the question
For the wave equation ,we need to find the displacement and velocity at x = 2 cm and t = 0.

\[y = \left( 0 . 2 \right)  cm  \sin2\pi = 0\] 

\[ \therefore \nu = A\omega cos\pi x\] 

\[             = 0 . 2 \times 2000\pi \times \cos2\pi\] 

\[             = 400\pi\] 

\[             = 400\pi  cm/s = 4\pi  m/s\]

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