Travelling wave travels in medium '1' and enters into another medium '2' in which it's speed gets decreased to `25%`. Then magnitude of ratio of Amplitude of transmitted to rerflected wave is :

Options

A. `(6)/(5)`
B. `(2)/(3)`
C. `(1)/(7)`
D. `(5)/(9)`

Explanation

To solve the problem, we need to find the ratio of the amplitude of the transmitted wave to the amplitude of the reflected wave when a traveling wave enters from medium 1 to medium 2, where its speed decreases to 25% of its original speed.

### Step-by-Step Solution:

1. **Identify the velocities in both media:**
- Let the speed of the wave in medium 1 be \( V_1 \).
- The speed of the wave in medium 2 is given as \( V_2 = 0.25 V_1 \) (25% of \( V_1 \)).

2. **Use the formula for the ratio of amplitudes:**
- The formula for the ratio of the amplitude of the transmitted wave \( A_t \) to the amplitude of the reflected wave \( A_r \) is given by:
\[ \frac{A_t}{A_r} = \frac{2V_2}{V_2 + V_1} \quad \text{(for a wave moving from a medium of higher speed to lower speed)} \]

3. **Substitute the values into the formula:**
- Substitute \( V_2 = 0.25 V_1 \) into the formula:
\[ \frac{A_t}{A_r} = \frac{2(0.25 V_1)}{(0.25 V_1) + V_1} \]

4. **Simplify the expression:**
- The denominator becomes:
\[ 0.25 V_1 + V_1 = 0.25 V_1 + 1.00 V_1 = 1.25 V_1 \]
- Thus, the ratio becomes:
\[ \frac{A_t}{A_r} = \frac{0.5 V_1}{1.25 V_1} \]

5. **Cancel out \( V_1 \):**
- Since \( V_1 \) is common in both numerator and denominator, it cancels out:
\[ \frac{A_t}{A_r} = \frac{0.5}{1.25} \]

6. **Convert to a simpler fraction:**
- Simplifying \( \frac{0.5}{1.25} \):
\[ \frac{0.5}{1.25} = \frac{0.5 \times 4}{1.25 \times 4} = \frac{2}{5} \]

7. **Final Result:**
- Therefore, the magnitude of the ratio of the amplitude of the transmitted wave to the amplitude of the reflected wave is:
\[ \frac{A_t}{A_r} = \frac{2}{5} \]

### Answer:
The magnitude of the ratio of the amplitude of the transmitted wave to the amplitude of the reflected wave is \( \frac{2}{5} \).