Three sinusoidal waves having same frequency have amplitudes in the ratio 6:3:2 and their phases are `0, pi/2 ,pi` . If 6a be the amplitude of the first wave , then which of the following is not correct

Options

A. the amplitude of the resultant wave of the three waves is 5a
B. phase of resultant wave is `tan^(-1) (3/4)`
C. the amplitude of resultant wave is 5a/6
D. the frequency of the resultant wave will be that of given wave

Answer

To solve the problem step by step, we will analyze the given information about the three sinusoidal waves and derive the resultant wave's amplitude and phase.

### Step 1: Define the Amplitudes and Phases
We are given three sinusoidal waves with amplitudes in the ratio 6:3:2. Let's denote the amplitudes as:
- Amplitude of the first wave, \( A_1 = 6k \)
- Amplitude of the second wave, \( A_2 = 3k \)
- Amplitude of the third wave, \( A_3 = 2k \)

Since it is given that \( 6a \) is the amplitude of the first wave, we can set \( 6k = 6a \), which gives us \( k = a \).

Thus, we have:
- \( A_1 = 6a \)
- \( A_2 = 3a \)
- \( A_3 = 2a \)

The phases of the waves are given as:
- Phase of the first wave, \( \phi_1 = 0 \)
- Phase of the second wave, \( \phi_2 = \frac{\pi}{2} \)
- Phase of the third wave, \( \phi_3 = \pi \)

### Step 2: Write the Equations of the Waves
The general equation for a sinusoidal wave is given by:
\[ y = A \sin(\omega t + \phi) \]

Using this, we can write the equations for the three waves:
1. \( y_1 = 6a \sin(\omega t + 0) = 6a \sin(\omega t) \)
2. \( y_2 = 3a \sin(\omega t + \frac{\pi}{2}) = 3a \cos(\omega t) \) (since \( \sin(x + \frac{\pi}{2}) = \cos(x) \))
3. \( y_3 = 2a \sin(\omega t + \pi) = -2a \sin(\omega t) \) (since \( \sin(x + \pi) = -\sin(x) \))

### Step 3: Combine the Waves
Now, we can combine these waves to find the resultant wave:
\[ y_{\text{resultant}} = y_1 + y_2 + y_3 \]
\[ y_{\text{resultant}} = 6a \sin(\omega t) + 3a \cos(\omega t) - 2a \sin(\omega t) \]
\[ y_{\text{resultant}} = (6a - 2a) \sin(\omega t) + 3a \cos(\omega t) \]
\[ y_{\text{resultant}} = 4a \sin(\omega t) + 3a \cos(\omega t) \]

### Step 4: Find the Amplitude of the Resultant Wave
To find the amplitude of the resultant wave, we can use the formula:
\[ A_{\text{resultant}} = \sqrt{(4a)^2 + (3a)^2} \]
\[ A_{\text{resultant}} = \sqrt{16a^2 + 9a^2} \]
\[ A_{\text{resultant}} = \sqrt{25a^2} = 5a \]

### Step 5: Find the Phase of the Resultant Wave
The phase \( \phi \) of the resultant wave can be found using:
\[ \tan(\phi) = \frac{\text{Coefficient of } \cos(\omega t)}{\text{Coefficient of } \sin(\omega t)} \]
\[ \tan(\phi) = \frac{3a}{4a} = \frac{3}{4} \]
Thus, \( \phi = \tan^{-1}\left(\frac{3}{4}\right) \).

### Step 6: Analyze the Statements
Now we need to determine which statement is not correct based on our findings:
1. Amplitude of resultant wave is \( 5a \) - **Correct**
2. Phase of resultant wave is \( \tan^{-1}\left(\frac{3}{4}\right) \) - **Correct**
3. Amplitude of resultant wave is \( \frac{5a}{6} \) - **Not Correct**
4. Frequency of the resultant wave will be that of the given waves - **Correct**

### Conclusion
The statement that is not correct is:
**Amplitude of resultant wave is \( \frac{5a}{6} \)**.