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 :

Explanation

To solve the problem of finding the ratio of the amplitude of the transmitted wave to the reflected wave when a traveling wave enters a second medium where its speed decreases to 25%, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Speeds 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 25% of \( V_1 \), which can be expressed as:
\[ V_2 = 0.25 V_1 = \frac{V_1}{4} \]

2. **Use the Formula for Amplitude Ratio:**
- The ratio of the amplitude of the transmitted wave \( A_T \) to the reflected wave \( A_R \) is given by the formula:
\[ \frac{A_T}{A_R} = \frac{2V_2}{V_2 - V_1} \]

3. **Substitute \( V_2 \) into the Formula:**
- Substitute \( V_2 = \frac{V_1}{4} \) into the amplitude ratio formula:
\[ \frac{A_T}{A_R} = \frac{2 \left(\frac{V_1}{4}\right)}{\left(\frac{V_1}{4}\right) - V_1} \]

4. **Simplify the Denominator:**
- The denominator simplifies as follows:
\[ \frac{V_1}{4} - V_1 = \frac{V_1}{4} - \frac{4V_1}{4} = \frac{-3V_1}{4} \]

5. **Substitute Back into the Ratio:**
- Now substituting back into the ratio gives:
\[ \frac{A_T}{A_R} = \frac{2 \left(\frac{V_1}{4}\right)}{\left(\frac{-3V_1}{4}\right)} = \frac{2V_1/4}{-3V_1/4} \]

6. **Cancel Common Terms:**
- The \( V_1 \) and \( 4 \) terms cancel out:
\[ \frac{A_T}{A_R} = \frac{2}{-3} = -\frac{2}{3} \]

7. **Magnitude of the Ratio:**
- Since we are interested in the magnitude, we take the absolute value:
\[ \left|\frac{A_T}{A_R}\right| = \frac{2}{3} \]

### Final Answer:
The magnitude of the ratio of the amplitude of the transmitted wave to the reflected wave is:
\[ \frac{2}{3} \]