two sound waves moves in the same direction .if the average power transmiitted across a cross - section by them are equal while their wavelengths are in the ratio of 1:2 . Their pressure amplitudes would be in the ratio of

Options

A. 1
B. 2
C. 4
D. `1/2`

Answer

To solve the problem, we need to find the ratio of the pressure amplitudes of two sound waves given that their average powers are equal and their wavelengths are in the ratio of 1:2.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- Let the wavelengths of the two sound waves be \( \lambda_1 \) and \( \lambda_2 \).
- According to the problem, \( \lambda_1 : \lambda_2 = 1 : 2 \). Therefore, we can write:
\[ \lambda_1 = \lambda \quad \text{and} \quad \lambda_2 = 2\lambda \]

2. **Relating Power and Intensity:**
- The average power transmitted by a wave is related to its intensity. The intensity \( I \) of a wave can be expressed as:
\[ I = \frac{P}{A} \]
where \( P \) is the power and \( A \) is the area.
- For sound waves, the intensity is also related to the pressure amplitude \( P_{\text{max}} \) by the formula:
\[ I = \frac{(P_{\text{max}})^2}{2 \rho v} \]
where \( \rho \) is the density of the medium and \( v \) is the speed of sound in that medium.

3. **Setting Up the Equations:**
- Since the average powers are equal, we have:
\[ P_1 = P_2 \]
- Therefore, the intensities must also be equal:
\[ I_1 = I_2 \]

4. **Expressing Intensities in Terms of Pressure Amplitudes:**
- For the first wave:
\[ I_1 = \frac{(P_{\text{max1}})^2}{2 \rho v} \]
- For the second wave:
\[ I_2 = \frac{(P_{\text{max2}})^2}{2 \rho v} \]
- Since \( I_1 = I_2 \), we can equate the two expressions:
\[ \frac{(P_{\text{max1}})^2}{2 \rho v} = \frac{(P_{\text{max2}})^2}{2 \rho v} \]

5. **Solving for the Ratio of Pressure Amplitudes:**
- Canceling out the common terms \( 2 \rho v \) from both sides gives:
\[ (P_{\text{max1}})^2 = (P_{\text{max2}})^2 \]
- Taking the square root of both sides, we find:
\[ P_{\text{max1}} = P_{\text{max2}} \]
- Thus, the ratio of the pressure amplitudes is:
\[ \frac{P_{\text{max1}}}{P_{\text{max2}}} = 1 \]

### Final Answer:
The ratio of the pressure amplitudes \( P_{\text{max1}} : P_{\text{max2}} = 1 : 1 \).