Ratio of amplitude for two wave is 1:7 .Find the ratio of intensity?

Answer

To solve the problem of finding the ratio of intensity given the ratio of amplitudes of two waves, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- The ratio of amplitudes of the two waves is given as \( A_1 : A_2 = 1 : 7 \). This can be expressed mathematically as:
\[ \frac{A_1}{A_2} = \frac{1}{7} \]

2. **Understand the Relationship Between Intensity and Amplitude**:
- The intensity \( I \) of a wave is directly proportional to the square of its amplitude. This can be expressed as:
\[ I \propto A^2 \]
- Therefore, the ratio of intensities \( I_1 \) and \( I_2 \) can be expressed in terms of the amplitudes:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 \]

3. **Substitute the Known Values**:
- We already have \( \frac{A_1}{A_2} = \frac{1}{7} \). Now, we can substitute this value into the intensity ratio formula:
\[ \frac{I_1}{I_2} = \left(\frac{1}{7}\right)^2 \]

4. **Calculate the Intensity Ratio**:
- Now, calculate \( \left(\frac{1}{7}\right)^2 \):
\[ \frac{I_1}{I_2} = \frac{1}{49} \]

5. **Conclusion**:
- The ratio of the intensities of the two waves is:
\[ I_1 : I_2 = 1 : 49 \]

### Final Answer:
The ratio of intensity is \( 1 : 49 \).

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