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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?
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Solution
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.
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