The amplitude of two waves are in ratio 5 : 2. If all other conditions for the two waves are same, then what is the ratio of their energy densities

Options

A. `5:2`
B. `5:4`
C. `4:5`
D. `25:4`

Answer

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

### Step-by-Step Solution:

1. **Understand the Relationship Between Energy and Amplitude**:
The energy \(E\) of a wave is proportional to the square of its amplitude \(A\). This can be expressed mathematically as:
\[ E \propto A^2 \]

2. **Define Energy Density**:
Energy density \(u\) is defined as the energy stored per unit volume. Therefore, the energy density is also proportional to the square of the amplitude:
\[ u \propto A^2 \]

3. **Set Up the Ratio of Energy Densities**:
Let the amplitudes of the two waves be \(A_1\) and \(A_2\). The ratio of their energy densities can be expressed as:
\[ \frac{u_1}{u_2} = \frac{A_1^2}{A_2^2} \]

4. **Substitute the Given Amplitude Ratio**:
We are given that the ratio of the amplitudes is:
\[ \frac{A_1}{A_2} = \frac{5}{2} \]
Squaring both sides gives:
\[ \left(\frac{A_1}{A_2}\right)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]

5. **Final Ratio of Energy Densities**:
Thus, the ratio of the energy densities is:
\[ \frac{u_1}{u_2} = \frac{25}{4} \]

### Conclusion:
The ratio of the energy densities of the two waves is \( \frac{25}{4} \).