Equation of a plane progressive wave is given by . On reflection from a denser medium, its amplitude becomes  $\frac{2}{3}$rd of the amplitude of the incident wave. The equation of the reflected wave is:

1.

2.

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(2) Hint: There would be a phase difference in the incident wave due to reflection.
Step 1: Find the amplitude of the reflected wave.
The amplitude of the reflected wave,

Given the equation of the incident wave,
${y}_{\mathit{i}}=0.6\mathrm{sin}2\mathrm{\pi }\left(\mathrm{t}-\frac{\mathrm{x}}{2}\right)$
Step 2: Find the equation of the reflected wave.
Equation of the reflected wave is,
${\mathrm{y}}_{\mathrm{r}}={\mathrm{A}}_{\mathrm{r}}\mathrm{sin}2\mathrm{\pi }\left(\mathrm{t}+\frac{\mathrm{x}}{2}+\mathrm{\pi }\right)$
[$\because$At a denser medium, the phase changes by $\mathrm{\pi }$]
The positive sign is due to the reversal of direction of propagation.
So,                                   ${\mathrm{y}}_{\mathrm{r}}=-0.4\mathrm{sin}2\mathrm{\pi }\left(\mathrm{t}+\frac{\mathrm{x}}{2}\right)$                       $\left[\because \mathrm{sin}\left(\mathrm{\pi }+\mathrm{\theta }\right)=-\mathrm{sin\theta }\right]$