Consider a system of three connected strings, $S_1, S_2$ and $S_3$ with uniform linear mass densities $\mu \mathrm{kg} / \mathrm{m}$, $4 \mu \mathrm{~kg} / \mathrm{m}$ and $16 \mu \mathrm{~kg} / \mathrm{m}$, 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 $y=y_0$ $\cos (\omega t-k x) \mathrm{cm}$, where $y_0, \omega$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct:

• Correct answer
  A

  When the wave reflects from $P$ for the first time, the reflected wave is represented by $y=\alpha_1 \mathrm{y}_0 \cos (\omega t+k x+\pi) \mathrm{cm}$, where $\alpha_1$ is a positive constant.
• B

  When the wave transmits through $P$ for the first time, the transmitted wave is represented by $y=\alpha_2 \mathrm{y}_0 \cos (\omega t-k x) \mathrm{cm}$, where $\alpha_2$ is a positive constant.
• C

  When the wave reflects from $Q$ for the first time, the reflected wave is represented by $y=\alpha_3 y_0 \cos (\omega t-k x+\pi) \mathrm{cm}$, where $\alpha_3$ is a positive constant.
• Correct answer
  D

  When the wave transmits through $Q$ for the first time, the transmitted wave is represented by $y=\alpha_4 y_0 \cos (\omega t-4 k x) \mathrm{cm}$, where $\alpha_4$ is a positive constant.

Solution

(A)
$y_1=y_0 \cos (\omega t-k x)$ when wave going from Rarer to Denser, $\begin{aligned} & y_r=A_r \cos (\omega t+k x+\pi) \\ & y_r=a_1 y_0 \cos (\omega t+k x+\pi)\end{aligned}$ option (A) correct (B) For transmitted from point P $\begin{aligned} & \mathrm{y}_{\mathrm{t}}=\mathrm{A}_{\mathrm{t}} \cos \left[\omega \mathrm{t}-\mathrm{k}_1 \mathrm{x}\right] \\ & \frac{\mathrm{k}_1}{\mathrm{k}}=\sqrt{\frac{\mu_1}{\mu}}=\frac{\mathrm{k}_1}{\mathrm{k}}=\sqrt{\frac{4 \mu}{\mu}} \\ & \mathrm{k}_1=2 \mathrm{k} \\ & \mathrm{y}_{\mathrm{t}}=\mathrm{a}_2 \mathrm{y}_0 \cos [\omega \mathrm{t}-2 \mathrm{kx}]\end{aligned}$ option (B) incorrect (C) when reflected from Q $\begin{aligned} & y_i=a_2 y_0 \cos [\omega t-2 k x] \\ & y_r=a_3 y_0 \cos [\omega t+2 k x+\pi]\end{aligned}$ option (C) incorrect (D) when transmitted from Q $\begin{aligned} & y_{\mathrm{t}}=a_4 y_0 \cos \left[\omega \mathrm{t}=k_2 \mathrm{x}\right] \\ & \frac{k_2}{2 \mathrm{k}}=\sqrt{\frac{16 \mu}{4 \mu}} \Rightarrow k_2=4 \mathrm{k} \\ & y_{\mathrm{t}}=a_4 y_0 \cos [\omega \mathrm{t}-4 \mathrm{kx}]\end{aligned}$ option (D) correct

Hence, the correct answer is options: 'A', 'D'.