The effects are produced at a given point in space by two wave decribed by the equations `y_(1) = y_(m) sin omegat` and `y_(2) = y_(m) sin (omegat + phi)` where `y_(m)` is the same for both the waves and `phi` is a phase angle. Tick the incorrect statement among the following.

Options

A. The maximum intensity that can be achieved at a point is twice the intensity of either wave and occurs if `phi = 0`
B. The maximum intensity that can be achieved at a point is four times the intensity of either wave and occurs if `phi = 0`
C. The maximum intensity that can be achieved at a point its twice the amplitude either wave and occurs at `phi = 0`
D. When the intensity is zero, the net amplitude is zero, and at this point `phi = pi`

Answer

To solve the problem step by step, we need to analyze the given wave equations and the statements about their intensities.

### Step 1: Understand the Wave Equations
The two waves are given by:
- \( y_1 = y_m \sin(\omega t) \)
- \( y_2 = y_m \sin(\omega t + \phi) \)

Here, \( y_m \) is the same for both waves, and \( \phi \) is the phase difference between them.

### Step 2: Calculate the Resultant Intensity
Using the principle of superposition, the resultant displacement \( y_R \) at a point due to both waves can be expressed as:
\[ y_R = y_1 + y_2 \]

The intensity \( I \) of a wave is proportional to the square of its amplitude. Therefore, the intensity of each wave can be expressed as:
- \( I_1 \propto y_m^2 \)
- \( I_2 \propto y_m^2 \)

The resultant intensity \( I_R \) can be calculated using the formula:
\[ I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]

### Step 3: Analyze Each Statement
Now let's analyze the statements one by one.

#### Statement A: "The maximum intensity that can be achieved at a point is twice the intensity of either wave for \( \phi = 0 \)."
- For \( \phi = 0 \):
- \( I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \cdot 1 \)
- Since \( I_1 = I_2 = I \), we have:
\[ I_R = I + I + 2\sqrt{I \cdot I} = 2I + 2I = 4I \]
- This statement is incorrect because the maximum intensity is four times the intensity of either wave, not twice.

#### Statement B: "The maximum intensity that can be achieved is four times the intensity of either of the wave."
- From the calculation above, we found that the maximum intensity is indeed \( 4I \).
- This statement is correct.

#### Statement C: "The maximum intensity that can be achieved at a point is twice the amplitude of either wave and occurs at \( \phi = 0 \)."
- The maximum amplitude \( A_R \) when \( \phi = 0 \) is:
\[ A_R = 2y_m \]
- The intensity is proportional to the square of the amplitude:
\[ I_R \propto A_R^2 = (2y_m)^2 = 4y_m^2 \]
- Thus, this statement is also correct.

#### Statement D: "When the intensity is 0, the net amplitude is 0 at the point \( \phi = \pi \)."
- For \( \phi = \pi \):
- \( I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \cdot (-1) \)
- Since \( I_1 = I_2 = I \):
\[ I_R = I + I - 2\sqrt{I \cdot I} = 2I - 2I = 0 \]
- This means the resultant amplitude is also zero. This statement is correct.

### Conclusion
The incorrect statement among the options is **Statement A**.