A wave pulse, travelling on a two piece string, gets partically reflected and partically transmitted at the junction, The reflected wave is inverted in shape as compard to the incident one. If the incident wave has speed `v` and the transmitted wave `v'`

Options

A. `v' gt v`
B. `v' = v`
C. `v' lt v`
D. nothing can be said about the relation of `v` and `v'`.

Explanation

To solve the problem of comparing the speeds of the incident wave (v) and the transmitted wave (v'), we need to analyze the behavior of waves at the junction of two different media. Here is a step-by-step solution:

### Step 1: Understand the wave behavior at the junction
When a wave pulse travels from one medium to another, it can be partially reflected and partially transmitted. At the junction, if the reflected wave is inverted, it indicates that the wave is reflecting from a medium of higher linear mass density (μ).

**Hint:** Remember that an inverted reflection occurs when a wave hits a boundary where the second medium has a greater mass density.

### Step 2: Identify the relationship between wave speed and mass density
The speed of a wave on a string is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the string and \( \mu \) is the linear mass density.

**Hint:** The wave speed is inversely related to the square root of the mass density.

### Step 3: Analyze the two media
Let:
- Medium 1 have mass density \( \mu \) and wave speed \( v \)
- Medium 2 have mass density \( \mu' \) and wave speed \( v' \)

Since the reflected wave is inverted, we know that:
\[ \mu' > \mu \]

**Hint:** If the mass density of the second medium is greater, it affects the speed of the transmitted wave.

### Step 4: Write the relationship between the speeds
From the wave speed formula, we can express the speeds in terms of mass densities:
\[ v = \sqrt{\frac{T}{\mu}} \quad \text{and} \quad v' = \sqrt{\frac{T}{\mu'}} \]

Now, we can set up a ratio of the speeds:
\[ \frac{v}{v'} = \sqrt{\frac{\mu'}{\mu}} \]

**Hint:** This ratio helps us understand how the speeds relate based on the mass densities.

### Step 5: Analyze the ratio
Since \( \mu' > \mu \), it follows that:
\[ \frac{\mu'}{\mu} > 1 \]
This implies:
\[ \sqrt{\frac{\mu'}{\mu}} > 1 \]
Thus, we can conclude:
\[ \frac{v}{v'} > 1 \]
or
\[ v > v' \]

**Hint:** The conclusion here is that the speed of the incident wave is greater than the speed of the transmitted wave.

### Final Conclusion
The incident wave speed \( v \) is greater than the transmitted wave speed \( v' \):
\[ v > v' \]