Two strings with linear mass densities `mu_(1)=0.1kg//m` and `mu_(2)=0.3kg//m` are joined seamlesly. They are under tension of `20N`. A travelling wave of triangular shape is moving from lighter to heavier string.

Options

A. The reflection coefficient at interface is zero
B. The reflection coefficient at interface is `2-sqrt(3)`
C. The transmission coefficient at interface is 1
D. The transmission coefficient at inerface is `sqrt(3)-1`

Explanation

To solve the problem, we need to analyze the wave propagation through two strings with different linear mass densities. We will calculate the velocities of the waves in both strings, determine the reflection and transmission coefficients, and finally identify the correct options based on these calculations.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Linear mass density of string 1, \( \mu_1 = 0.1 \, \text{kg/m} \)
- Linear mass density of string 2, \( \mu_2 = 0.3 \, \text{kg/m} \)
- Tension in the strings, \( T = 20 \, \text{N} \)

2. **Calculate the Velocity in String 1:**
The velocity of a wave in a string is given by the formula:
\[ v_1 = \sqrt{\frac{T}{\mu_1}} \]
Substituting the known values:
\[ v_1 = \sqrt{\frac{20}{0.1}} = \sqrt{200} = 10\sqrt{2} \, \text{m/s} \]

3. **Calculate the Velocity in String 2:**
Similarly, for the second string:
\[ v_2 = \sqrt{\frac{T}{\mu_2}} \]
Substituting the known values:
\[ v_2 = \sqrt{\frac{20}{0.3}} = \sqrt{\frac{200}{3}} = \frac{10\sqrt{2}}{\sqrt{3}} \, \text{m/s} \]

4. **Calculate the Reflection Coefficient (R):**
The reflection coefficient is given by:
\[ R = \frac{v_2 - v_1}{v_2 + v_1} \]
Substituting the values of \( v_1 \) and \( v_2 \):
\[ R = \frac{\frac{10\sqrt{2}}{\sqrt{3}} - 10\sqrt{2}}{\frac{10\sqrt{2}}{\sqrt{3}} + 10\sqrt{2}} \]
Simplifying:
\[ R = \frac{10\sqrt{2} \left( \frac{1}{\sqrt{3}} - 1 \right)}{10\sqrt{2} \left( \frac{1}{\sqrt{3}} + 1 \right)} = \frac{\frac{1}{\sqrt{3}} - 1}{\frac{1}{\sqrt{3}} + 1} \]
This can be further simplified to:
\[ R = 2 - \sqrt{3} \]

5. **Calculate the Transmission Coefficient (T):**
The transmission coefficient is given by:
\[ T = \frac{2v_1}{v_1 + v_2} \]
Substituting the values:
\[ T = \frac{2 \cdot 10\sqrt{2}}{10\sqrt{2} + \frac{10\sqrt{2}}{\sqrt{3}}} \]
Simplifying:
\[ T = \frac{20\sqrt{2}}{10\sqrt{2} \left( 1 + \frac{1}{\sqrt{3}} \right)} = \frac{2}{1 + \frac{1}{\sqrt{3}}} \]
This can be simplified to:
\[ T = \sqrt{3} - 1 \]

6. **Final Results:**
- Reflection Coefficient \( R = 2 - \sqrt{3} \)
- Transmission Coefficient \( T = \sqrt{3} - 1 \)

### Conclusion:
Based on the calculations, the reflection coefficient is \( 2 - \sqrt{3} \) and the transmission coefficient is \( \sqrt{3} - 1 \). Therefore, the correct options from the provided choices are B and D.