Question

The equation of a plane progressive wave is $y = 0.02\sin 8\pi \left[ {t - \dfrac{x}{{20}}} \right]$. When it is reflected at the rare medium its amplitude is 75% of the previous value. The equation of the reflected wave is,
$A.y = 0.02\sin 8\pi \left[ {t - \dfrac{x}{{20}}} \right] \\\ B.y = 0.02\sin 8\pi \left[ {t + \dfrac{x}{{20}}} \right] \\\ C.y = 0.15\sin 8\pi \left[ {t + \dfrac{x}{{20}}} \right] \\\ D.y = 0.015\sin 8\pi \left[ {t - \dfrac{x}{{20}}} \right] \\\$

Answer 1

In order to solve this problem you need to know that there is no change in the equation of the wave if there is no change in phase difference but when the wave is reflected and it is said that the amplitude of the wave will be 75 percent of the previous value then the amplitude will change without changing the rest of the things. Use the fact that the phase does not change when reflected through a rare medium.

Complete step-by-step answer :
The given equation is $y = 0.02\sin 8\pi \left[ {t - \dfrac{x}{{20}}} \right]$ and it is said that when the wave is reflected at the rare medium its amplitude is 75% of the previous value then we need to find the equation of the reflected wave.
Amplitude earlier is 0.02. Now after reflecting it is said that the amplitude is 75 % of the previous one then the new amplitude will be $0.02 \times \dfrac{{75}}{{100}} = 0.015$.
Hence the new amplitude will be 0.015.
No phase difference is created between incident & reflected waves at boundary of rare medium, only direction of motion reversed. So, the wave equation will not change at all except amplitude.
The new equation is $y = 0.015\sin 8\pi \left[ {t - \dfrac{x}{{20}}} \right]$
So, the correct option is D.

Note :When you get to solve such problems of wave equation varying sinusoidally then you need to know that there will be no change in the equation of wave when it is reflected except the change in amplitude if the wave is realistic its intensity or amplitude decreases when any barrier comes on its way. The rest of the things will remain as usual provided no phase difference is created. Reflection of waves is the change in the direction of a wave upon striking the interface between two materials. When a wave strikes any interface between any two mediums the bouncing back of the wave is termed as reflection of waves. Knowing this will help you further to get the right answer.