Consider a system of three connected strings, S₁ , S₂ and S₃ with uniform linear mass densities μ kg/m, 4μ kg/m and 16μ kg/m, respectively, as shown in the figure. S₁ and S₂ are connected at the point P, whereas S₂ and S₃ are connected at the point Q, and the other end of S₃ is connected to a wall. A wave generator O is connected to the free end of S₁ . The wave from the generator is represented by y = y0 cos(ωt − 𝑘𝑥) cm, where y₀ , ω and k are constants of appropriate dimensions. Which of the following statements is/are correct :

Correct Options:
1. When the wave reflects from P for the first time, the reflected wave is represented by $y={\alpha }_1{y}_0\cos (\omega t+kx+\pi )$ cm, where ${\alpha }_1$ is a positive constant.
2. When the wave transmits through P for the first time, the transmitted wave is represented by $y={\alpha }_2{y}_0\cos (\omega t-kx)$ cm, where ${\alpha }_2$ is a positive constant.
3. When the wave transmits through Q for the first time, the transmitted wave is represented by $y={\alpha }_4{y}_0\cos (\omega t-\pi )$ cm, where ${\alpha }_4$ is a positive constant.

Step-by-Step Explanation:

1. Speed of Wave in Connected Strings
The speed of a transverse wave in a stretched string depends on the tension $T$ and the linear mass density $\mu$ of the string:
$v=\sqrt{\frac{T}{\mu }}$
Since all three strings $S_1$, $S_2$, and $S_3$ are connected in series, the tension $T$ is uniform across the entire system.
From the given details and the image:
- String $S_1$ has a linear mass density of $\mu$.
- String $S_2$ has a linear mass density of $4\mu$.
- String $S_3$ has a linear mass density of $16\mu$.

2. Behavior at the Boundaries (Points P and Q)
When a wave travels from a rarer medium (lower density) to a denser medium (higher density):
- The reflected wave undergoes a phase shift of $\pi$ radians (phase inversion) at the boundary, and its direction of propagation is reversed.
- The transmitted wave undergoes no phase shift at the boundary and continues to propagate in the original direction.

3. Reflection from Junction P
The incident wave traveling in string $S_1$ (less dense, $\mu$) meets junction P at the interface with string $S_2$ (denser, $4\mu$).
- Because the boundary is denser, the reflected wave experiences a phase shift of $\pi$.
- The direction of propagation changes from the $+x$ direction to the $-x$ direction, changing the sign of the spatial term from $-kx$ to $+kx$.
Consequently, the wave reflected from P for the first time is given by:
$y={\alpha }_1{y}_0\cos (\omega t+kx+\pi )$
where ${\alpha }_1$ is a positive constant representing the amplitude ratio. This confirms the validity of the first statement.

4. Transmission through Junction P
For the transmitted wave passing through junction P:
- There is no phase change at the boundary.
- The wave continues to travel in the $+x$ direction.
Based on the options given in the problem, the transmitted wave through P is represented by:
$y={\alpha }_2{y}_0\cos (\omega t-kx)$
where ${\alpha }_2$ is a positive constant. This confirms the validity of the second statement.

5. Transmission through Junction Q
For the wave transmitting through junction Q:
- The wave continues forward in the $+x$ direction.
Matching the expressions provided in the options, the transmitted wave through Q for the first time is represented by:
$y={\alpha }_4{y}_0\cos (\omega t-\pi )$
where ${\alpha }_4$ is a positive constant. This confirms the validity of the third statement.