A sonic source, located in a uniform medium, emits waves of frequency `n`. If intensity, energy density (energy per unit volume of the medium) and maximum speed of oscillations of medium particle are, respectively, `I`, `E` and `u_0` at a point, then which of the following graphs are correct?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationships between intensity (I), energy density (E), and maximum speed of oscillations of the medium particles (u₀) for a sonic source emitting waves of frequency n.

### Step-by-Step Solution:

1. **Understanding Intensity (I)**:
- Intensity (I) is defined as the energy per unit area per unit time. It can be expressed in terms of energy density (E) and the speed of sound in the medium.
- The relationship can be given as:
\[ I \propto E \]
- This implies that intensity is directly proportional to energy density.

2. **Graph of Intensity vs Energy Density**:
- Since intensity is proportional to energy density, if we plot intensity (I) on the y-axis and energy density (E) on the x-axis, we will get a straight line passing through the origin.
- **Graph**: A straight line graph indicating a linear relationship between I and E.

3. **Understanding Energy Density (E) and Frequency (n)**:
- The energy density (E) is related to the frequency (n) of the waves. The relationship can be derived from the formula for intensity:
\[ I = 2\pi^2 n^2 A^2 \rho v \]
- From this, we can express energy density as:
\[ E \propto n^2 \]
- This indicates that energy density is proportional to the square of the frequency.

4. **Graph of Energy Density vs Frequency**:
- If we plot energy density (E) on the y-axis and frequency (n) on the x-axis, we will get a parabolic graph that is symmetric about the E-axis.
- **Graph**: A parabola opening upwards, indicating that as frequency increases, energy density increases quadratically.

5. **Understanding Maximum Speed of Oscillation (u₀)**:
- The maximum speed of oscillation (u₀) is related to the amplitude (A) and frequency (n):
\[ u_0 = 2\pi n A \]
- Squaring both sides gives:
\[ E \propto u_0^2 \]
- This means energy density is also proportional to the square of the maximum speed of oscillation.

6. **Graph of Energy Density vs Maximum Speed of Oscillation**:
- If we plot energy density (E) on the y-axis and maximum speed of oscillation (u₀) on the x-axis, we will again get a parabolic graph that is symmetric about the E-axis.
- **Graph**: A parabola opening upwards, indicating that as the maximum speed of oscillation increases, energy density also increases quadratically.

### Summary of Graphs:
- **Graph 1**: Intensity (I) vs Energy Density (E) - Straight line through the origin.
- **Graph 2**: Energy Density (E) vs Frequency (n) - Parabola opening upwards.
- **Graph 3**: Energy Density (E) vs Maximum Speed of Oscillation (u₀) - Parabola opening upwards.