A plance sound waves passes from medium `1` into medium`2`. The speed of sound in medium `1` is `200 m//s` and in medium `2` is `100 m//s`. The ratio of amplitude of the transmitted waves to that of incident waves is

Options

A. `(3)/(4)`
B. `(4)/(5)`
C. `(5)/(6)`
D. `(2)/(3)`

Explanation

To solve the problem of finding the ratio of the amplitude of transmitted waves to that of incident waves when a plane sound wave passes from medium 1 to medium 2, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given data:**
- Speed of sound in medium 1, \( v_1 = 200 \, \text{m/s} \)
- Speed of sound in medium 2, \( v_2 = 100 \, \text{m/s} \)

2. **Understand the relationship between refractive indices and speeds:**
- The refractive index \( n \) of a medium is related to the speed of sound in that medium. The relationship is given by:
\[ \frac{n_1}{n_2} = \frac{v_2}{v_1} \]
- Here, \( n_1 \) is the refractive index of medium 1, and \( n_2 \) is the refractive index of medium 2.

3. **Calculate the ratio of refractive indices:**
- Substitute the values of \( v_1 \) and \( v_2 \):
\[ \frac{n_1}{n_2} = \frac{100}{200} = \frac{1}{2} \]
- From this, we can express \( n_2 \) in terms of \( n_1 \):
\[ n_2 = 2n_1 \]

4. **Use the formula for the transmission coefficient \( T \):**
- The transmission coefficient \( T \) is defined as the ratio of the amplitude of the transmitted wave \( A_t \) to the amplitude of the incident wave \( A_i \):
\[ T = \frac{A_t}{A_i} = \frac{2n_1}{n_1 + n_2} \]

5. **Substitute \( n_2 \) into the transmission coefficient formula:**
- Replace \( n_2 \) with \( 2n_1 \):
\[ T = \frac{2n_1}{n_1 + 2n_1} = \frac{2n_1}{3n_1} \]

6. **Simplify the expression:**
- The \( n_1 \) terms cancel out:
\[ T = \frac{2}{3} \]

7. **State the final answer:**
- The ratio of the amplitude of the transmitted waves to that of the incident waves is:
\[ \frac{A_t}{A_i} = \frac{2}{3} \]

### Final Answer:
The ratio of the amplitude of the transmitted waves to that of the incident waves is \( \frac{2}{3} \).