The energy per unit area associated with a progressive sound wave will be doubled if

Options

A. the amplitude of the wave is doubled
B. the amplitude of the wave is increased by 50%
C. the amplitude of the wave is increased by 41%
D. the frequency of the wave is increased by 41%

Answer

To solve the problem, we need to analyze how the energy per unit area associated with a progressive sound wave changes with variations in amplitude and frequency.

### Step-by-Step Solution:

1. **Understanding Energy in Sound Waves**:
The energy \( E \) associated with a progressive sound wave is given by the formula:
\[ E \propto A^2 \cdot f^2 \]
where \( A \) is the amplitude and \( f \) is the frequency of the sound wave.

2. **Effect of Amplitude on Energy**:
If the amplitude of the wave is increased, the energy changes according to the square of the amplitude:
\[ \frac{E_1}{E_2} = \frac{A_1^2}{A_2^2} \]
If the amplitude is doubled (\( A_2 = 2A_1 \)):
\[ \frac{E_1}{E_2} = \frac{A_1^2}{(2A_1)^2} = \frac{A_1^2}{4A_1^2} = \frac{1}{4} \]
Thus, \( E_2 = 4E_1 \) (the energy is quadrupled).

3. **Increasing Amplitude by 50%**:
If the amplitude is increased by 50%, then:
\[ A_2 = 1.5 A_1 \]
Therefore:
\[ \frac{E_1}{E_2} = \frac{A_1^2}{(1.5A_1)^2} = \frac{A_1^2}{2.25A_1^2} = \frac{1}{2.25} \]
This means \( E_2 = 2.25E_1 \) (the energy increases by 125%).

4. **Increasing Amplitude by 41%**:
If the amplitude is increased by 41%, then:
\[ A_2 = 1.41 A_1 \]
Thus:
\[ \frac{E_1}{E_2} = \frac{A_1^2}{(1.41A_1)^2} = \frac{A_1^2}{1.9881A_1^2} = \frac{1}{1.9881} \]
This gives \( E_2 \approx 2E_1 \) (the energy doubles).

5. **Effect of Frequency on Energy**:
The energy is also proportional to the square of the frequency:
\[ \frac{E_1}{E_2} = \frac{f_1^2}{f_2^2} \]
If the frequency is increased by 41%, then:
\[ f_2 = 1.41 f_1 \]
Therefore:
\[ \frac{E_1}{E_2} = \frac{f_1^2}{(1.41f_1)^2} = \frac{f_1^2}{1.9881f_1^2} = \frac{1}{1.9881} \]
This also results in \( E_2 \approx 2E_1 \) (the energy doubles).

### Conclusion:
The energy per unit area associated with a progressive sound wave will be doubled if:
- The amplitude of the wave is increased by 41%.
- The frequency of the wave is increased by 41%.