Consider a system of three connected strings, $ S_1, S_2 $ and $ S_3 $ with uniform linear mass densities $ \mu \, \text{kg/m}, 4\mu \, \text{kg/m} $ and $ 16\mu \, \text{kg/m} $, respectively, as shown in the figure. $ S_1 $ and $ S_2 $ are connected at 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:
Consider a system of three connected strings, $ S_1, S_2 $ and $ S_3 $ with uniform linear mass densities $ \mu \, \text{kg/m}, 4\mu \, \text{kg/m} $ and $ 16\mu \, \text{kg/m} $, respectively, as shown in the figure. $ S_1 $ and $ S_2 $ are connected at 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:
Show Hint
- 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.
- 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.
- When the wave reflects from \( Q \) for the first time, the reflected wave is represented by \( y = \alpha_3 y_0 \cos(\omega t - kx + \pi) \) cm, where \( \alpha_3 \) is a positive constant.
- When the wave transmits through \( Q \) for the first time, the transmitted wave is represented by \( y = \alpha_4 y_0 \cos(\omega t - 4kx) \) cm, where \( \alpha_4 \) is a positive constant. \bigskip
The Correct Option is A, D
Solution and Explanation
Understanding the Wave Motion: The wave originates from the leftmost string \( S_1 \), which has a linear mass density \( \mu \), and moves toward the junction \( P \) where it meets a string of higher mass density \( 4\mu \). Then it travels through \( S_2 \) and hits junction \( Q \), again entering a denser string \( S_3 \) with \( 16\mu \), which is fixed at its far right end (a rigid wall).
Let’s analyze the reflection and transmission at each junction.
Step 1: Reflection at Junction \( P \) - A wave moves from a lighter string (\( \mu \)) to a denser string (\( 4\mu \)). - When a wave travels from a lighter medium to a denser medium, partial reflection occurs with a phase change of \( \pi \). - The reflected wave travels in the opposite direction, and the phase shift \( \pi \) changes the cosine to: \[ \cos(\omega t - kx + \pi) = -\cos(\omega t - kx) \] - But since it's moving in the opposite direction, the wave becomes: \[ \cos(\omega t + kx + \pi) \] - Therefore, the reflected wave is: \[ y = \alpha_1 y_0 \cos(\omega t + kx + \pi) \] ✓ So, (A) is correct.
Step 2: Transmission through Junction \( P \) - The wave that transmits into \( S_2 \) continues to travel forward (positive \( x \)-direction), retaining its original phase (no phase change in transmission). - Thus, the transmitted wave would still be of the form: \[ y = \alpha_2 y_0 \cos(\omega t - kx) \] However, the wave number \( k \) could change due to different medium properties. Since \( k \propto 1/v \propto \sqrt{\mu} \), technically it should be adjusted. But the given option doesn't reflect this properly. (B) is partially correct in form but lacks proper adjustment of \( k \). We'll consider it not fully correct.
Step 3: Reflection at Junction \( Q \) - The wave in \( S_2 \) travels toward \( Q \), where \( S_2 \) (with \( 4\mu \)) meets \( S_3 \) (with \( 16\mu \)). Again, this is a reflection at a denser medium, which causes a \( \pi \) phase shift. - The reflected wave moves back to the left, i.e., in the negative \( x \)-direction. - So the correct form of the reflected wave should be: \[ y = \alpha_3 y_0 \cos(\omega t + kx + \pi) \] But in Option (C), it is wrongly written as \( \cos(\omega t - kx + \pi) \), which corresponds to a forward-moving wave. Hence, (C) is incorrect.
Step 4: Transmission through Junction \( Q \) - The wave continues into string \( S_3 \), which has mass density \( 16\mu \). - Since the wave number \( k \propto \sqrt{\mu} \), the new wave number \( k' \) in \( S_3 \) becomes: \[ k' = \sqrt{\frac{16\mu}{\mu}} \cdot k = 4k \] - So the transmitted wave becomes: \[ y = \alpha_4 y_0 \cos(\omega t - 4kx) \] (D) is correct.
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