Consider a system of three connected strings, $$S_1$$, $$S_2$$ and $$S_3$$ with uniform linear mass densities $$\mu$$ kg/m, $$4\mu$$ kg/m and $$16\mu$$ kg/m, respectively, as shown in the figure. $$S_1$$ and $$S_2$$ are connected at the point $$P$$, whereas $$S_2$$ and $$S_3$$ are connected at the point $$Q$$, and the other end of $$S_3$$ is connected to a wall. A wave generator O is connected to the free end of $$S_1$$. The wave from the generator is represented by $$y = y_0 \cos(\omega t - kx)$$ cm, where $$y_0$$, $$\omega$$ and $$k$$ are constants of appropriate dimensions. Which of the following statements is/are correct:

Given, wave eqn, $$y = y_0 \cos(\omega t - kx)\text{ cm}$$ ($$y_0, \omega, k$$ are constants)

We know, wave speed, $$v = \sqrt{\frac{T}{\mu}}$$, where $$T = \text{tension in string}$$

So speeds : $$v_1 = \sqrt{\frac{T}{\mu}}, v_2 = \sqrt{\frac{T}{4\mu}} = \frac{1}{2}\sqrt{\frac{T}{\mu}}$$

$$\Rightarrow v_2 = \frac{v_1}{2}, v_3 = \sqrt{\frac{T}{16\mu}} = \frac{1}{4}\sqrt{\frac{T}{\mu}}$$

$$\Rightarrow v_3 = \frac{v_1}{4}$$

wave numbers: $$k = \frac{\omega}{v}$$

so $$k_1 = k, k_2 = \frac{\omega}{v_2} = 2k$$ and $$k_3 = \frac{\omega}{v_3} = 4k$$

Reflection of wave in string from fixed end, reflected wave phase change by $$\pi_0$$ and reflected wave from free (rarer) end has no phase change.

$$\therefore$$ For medium $$PQ$$ ($$s_2$$): $$k_{s_2} = k_{PQ} = 2k$$

For medium, $$S_3, k_{s_3} = 4k$$

Option (A), reflection from $$P$$, phase change by ' $$\pi$$ ', No Change in ' $$K$$ '. Hence, ($$A$$) is the correct option.

Option (B), $$k_{s_2} = 2k$$ but in option given $$k_s = k$$. Hence, wrong option.

Option (C), the direction of the reflected wave is in the negative direction. Hence, wrong.

Option (D), no phase change in transmitted wave and given $$k_{s_3} = 4k$$

So (A), (D) are correct.