Three simple harmonic waves , identical in frequency `n` and amplitude `A` moving in the same direction are superimposed in air in such a way , that the first , second and the third wave have the phase angles `phi , phi + ( pi//2) and (phi + pi)`, respectively at a given point `P` in the superposition
Then as the waves progress , the superposition will result in

Options

A. a periodic , non - simple harmonic wave of amplitude `3 A`
B. a stationary simple harmonic wave of amplitude ` 3 A`
C. a simple harmonic progressive wave of amplitude `A`
D. the velocity of the superposed resultant wave will be the same as the velocity of each wave

Explanation

To solve the problem of superimposing three simple harmonic waves with given phase angles, we can follow these steps:

### Step 1: Identify the Waves
We have three waves with the following phase angles:
- Wave 1: \( \phi \)
- Wave 2: \( \phi + \frac{\pi}{2} \)
- Wave 3: \( \phi + \pi \)

### Step 2: Write the Wave Equations
The general form of a simple harmonic wave can be expressed as:
\[ y = A \sin(kx - \omega t + \phi) \]
For our three waves, we can write:
- Wave 1: \( y_1 = A \sin(kx - \omega t + \phi) \)
- Wave 2: \( y_2 = A \sin(kx - \omega t + \phi + \frac{\pi}{2}) \)
- Wave 3: \( y_3 = A \sin(kx - \omega t + \phi + \pi) \)

### Step 3: Simplify the Waves
Using the sine addition formula:
- \( \sin(\phi + \frac{\pi}{2}) = \cos(\phi) \)
- \( \sin(\phi + \pi) = -\sin(\phi) \)

Thus, we can rewrite the waves as:
- Wave 1: \( y_1 = A \sin(kx - \omega t + \phi) \)
- Wave 2: \( y_2 = A \cos(kx - \omega t + \phi) \)
- Wave 3: \( y_3 = -A \sin(kx - \omega t + \phi) \)

### Step 4: Combine the Waves
Now we can combine the waves:
\[ y = y_1 + y_2 + y_3 \]
Substituting the expressions:
\[ y = A \sin(kx - \omega t + \phi) + A \cos(kx - \omega t + \phi) - A \sin(kx - \omega t + \phi) \]

### Step 5: Simplify the Result
Notice that the \( A \sin(kx - \omega t + \phi) \) terms cancel out:
\[ y = A \cos(kx - \omega t + \phi) \]

### Step 6: Determine the Resultant Amplitude
The resultant wave is thus:
\[ y = A \cos(kx - \omega t + \phi) \]
This indicates that the amplitude of the resultant wave is \( A \).

### Step 7: Conclusion
The superposition of the three waves results in a single wave with amplitude \( A \).

### Final Answer
The amplitude of the resultant wave is \( A \).

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