Equation of a plane progressive wave is given by $y = 0.6 \sin 2\pi \left( t - \frac{x}{2} \right)$. On…

Equation of a plane progressive wave is given by $y = 0.6 \sin 2\pi \left( t - \frac{x}{2} \right)$. On reflection from a denser medium, its amplitude becomes $(2/3)$ of the amplitude of the incident wave. The equation of the reflected wave is [NCERT Exemplar]
  1. $y = 0.6 \sin 2\pi \left( t + \frac{x}{2} \right)$
  2. $y = - 0.4 \sin 2\pi \left( t + \frac{x}{2} \right)$
  3. $y = 0.4 \sin 2\pi \left( t + \frac{x}{2} \right)$
  4. $y = - 0.4 \sin 2\pi \left( t - \frac{x}{2} \right)$

Solution

Amplitude of reflected wave, $A_r = \frac{2}{3} \times A_i = \frac{2}{3} \times 0.6 = 0.4\text{ units}$ Given equation of incident wave, $y_i = 0.6 \sin 2\pi \left(t - \frac{x}{2}\right)$ Equation of reflected wave, $y_r = A_r \sin 2\pi \left(t + \frac{x}{2} + \pi\right)$ ($\because$ At denser medium, phase changes by $\pi$) The positive sign is due to reversal of direction of propagation. So, $y_r = - 0.4 \sin 2\pi \left(t + \frac{x}{2}\right)$ [$\because \sin(\pi + \theta) = - \sin \theta$]

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