Wave Reflection and Transmission in Three Connected Strings

To determine the correct statements, we analyze the propagation, reflection, and transmission of the wave across the three connected strings.

1. Wave Velocities and Wave Numbers in the Three Strings

Let the tension in the system of strings be $T$, which is uniform throughout. The linear mass densities of strings $S_1, S_2,$ and $S_3$ are given as: $\mu_1 = \mu, \quad \mu_2 = 4\mu, \quad \mu_3 = 16\mu$

The wave speed in a string of linear mass density $\mu_i$ is given by $v_i = \sqrt{\frac{T}{\mu_i}}$. Therefore: $v_1 = \sqrt{\frac{T}{\mu}} = v$ $v_2 = \sqrt{\frac{T}{4\mu}} = \frac{v}{2}$ $v_3 = \sqrt{\frac{T}{16\mu}} = \frac{v}{4}$

Since the angular frequency $\omega$ remains constant across all interfaces, the wave numbers $k_i = \frac{\omega}{v_i}$ in the respective strings are: $k_1 = \frac{\omega}{v} = k$ $k_2 = \frac{\omega}{v/2} = 2k$ $k_3 = \frac{\omega}{v/4} = 4k$

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2. Analysis of Reflection and Transmission at Junction $P$

The incident wave traveling in string $S_1$ towards junction $P$ (along $+x$ direction) is represented as: $y_i = y_0 \cos(\omega t - kx)$

Reflected Wave at $P$:

The amplitude reflection coefficient at the junction $P$ between string $S_1$ and string $S_2$ is: $A_r = \left(\frac{v_2 - v_1}{v_1 + v_2}\right) y_0 = \left(\frac{v/2 - v}{v + v/2}\right) y_0 = -\frac{1}{3} y_0$

The negative sign indicates a phase inversion of $\pi$ upon reflection from a denser medium. Since the reflected wave travels in the $-x$ direction in string $S_1$ (where $k_1 = k$), its equation is: $y_{r, P} = \frac{1}{3} y_0 \cos(\omega t + kx + \pi)$

Comparing this with $y = \alpha_1 y_0 \cos(\omega t + kx + \pi)$, we get $\alpha_1 = \frac{1}{3} > 0$.
Hence, Option (A) is CORRECT.

Transmitted Wave through $P$:

The wave transmitted into string $S_2$ travels in the $+x$ direction with a wave number $k_2 = 2k$. Thus, its mathematical form must be of the type: $y_{t, P} \propto \cos(\omega t - 2kx)$

Option (B) incorrectly represents the wave number as $k$ instead of $2k$.
Hence, Option (B) is INCORRECT.

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3. Analysis of Reflection and Transmission at Junction $Q$

The wave transmitted through $P$ travels in string $S_2$ towards junction $Q$.

Reflected Wave at $Q$:

The wave reflected at junction $Q$ travels back in string $S_2$ in the $-x$ direction with wave number $k_2 = 2k$. Its spatial dependence must be $+2kx$.
Option (C) incorrectly represents the phase as $(\omega t - kx + \pi)$.
Hence, Option (C) is INCORRECT.

Transmitted Wave through $Q$:

The wave transmitted through $Q$ travels into string $S_3$ in the $+x$ direction with wave number $k_3 = 4k$. Its equation is represented as: $y_{t, Q} = \alpha_4 y_0 \cos(\omega t - 4kx)$ where $\alpha_4$ is a positive constant representing the net transmission amplitude factor.
Hence, Option (D) is CORRECT.

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Conclusion

The correct options are A and D.