Consider an incident wave pulse which is travelling along the negative $x$-axis in the string $\left(1\right)$ as shown in figure. One end of the string is joined with another heavy string $\left(2\right)$ whose other ends is fixed at $O$ on a rigid vertical wall as shown in figure. Mass per unit length of string $\left(1\right)$ and $\left(2\right)$ are $\mu$ and $2\mu$ respectively. If equation of incident wave is
$y_i=A_i\sin \left(kx+\omega t+\frac{\pi }{4}\right)$, then equation of reflected wave which is reflected from rigid wall will be ($A_R$ is the amplitude of reflected wave)