A progressive wave travels in a medium `M_(1)` and enters into another mediun `M_(2)` in which its speed decreases to `60%`. Then the ratio of the amplitude of the transmitted and the incident waves is

Options

A. `3//5`
B. `3//4`
C. `3//10`
D. `1`

Explanation

To solve the problem of finding the ratio of the amplitude of the transmitted wave to the amplitude of the incident wave when a progressive wave travels from medium \( M_1 \) to medium \( M_2 \) and its speed decreases to 60%, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables:**
Let:
- \( A_i \) = Amplitude of the incident wave
- \( A_t \) = Amplitude of the transmitted wave
- \( V_1 \) = Speed of the wave in medium \( M_1 \)
- \( V_2 \) = Speed of the wave in medium \( M_2 \)

2. **Determine the Speed in Medium \( M_2 \):**
Since the speed decreases to 60% in medium \( M_2 \), we can express \( V_2 \) as:
\[ V_2 = 0.6 \times V_1 = \frac{3}{5} V_1 \]

3. **Use the Amplitude Ratio Formula:**
The ratio of the amplitudes of the transmitted wave to the incident wave is given by the formula:
\[ \frac{A_t}{A_i} = \frac{2 V_2}{V_1 + V_2} \]

4. **Substitute \( V_2 \) into the Formula:**
Substitute \( V_2 = \frac{3}{5} V_1 \) into the amplitude ratio formula:
\[ \frac{A_t}{A_i} = \frac{2 \left(\frac{3}{5} V_1\right)}{V_1 + \frac{3}{5} V_1} \]

5. **Simplify the Denominator:**
The denominator becomes:
\[ V_1 + \frac{3}{5} V_1 = \frac{5}{5} V_1 + \frac{3}{5} V_1 = \frac{8}{5} V_1 \]

6. **Complete the Substitution:**
Now, substituting back into the ratio:
\[ \frac{A_t}{A_i} = \frac{2 \left(\frac{3}{5} V_1\right)}{\frac{8}{5} V_1} \]

7. **Cancel Out \( V_1 \):**
The \( V_1 \) terms cancel out:
\[ \frac{A_t}{A_i} = \frac{2 \times \frac{3}{5}}{\frac{8}{5}} = \frac{6/5}{8/5} = \frac{6}{8} = \frac{3}{4} \]

8. **Final Result:**
Thus, the ratio of the amplitude of the transmitted wave to the amplitude of the incident wave is:
\[ \frac{A_t}{A_i} = \frac{3}{4} \]

### Conclusion:
The ratio of the amplitude of the transmitted wave to the amplitude of the incident wave is \( \frac{3}{4} \).