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 `3A`
B. a stationary simple harmonic wave of amplitude `3A`
C. a stationary 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 the superposition of three simple harmonic waves with given phase angles, we can follow these steps:

### Step 1: Identify the Phase Angles
We have three waves with the following phase angles:
- Wave 1: \( \phi_1 = \phi \)
- Wave 2: \( \phi_2 = \phi + \frac{\pi}{2} \)
- Wave 3: \( \phi_3 = \phi + \pi \)

### 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 Wave Equations
Using the sine addition formulas, we can simplify:
- Wave 2 becomes:
\[ y_2 = A \sin(kx - \omega t + \phi) \cos\left(\frac{\pi}{2}\right) + A \cos(kx - \omega t + \phi) \sin\left(\frac{\pi}{2}\right) = A \cos(kx - \omega t + \phi) \]
- Wave 3 becomes:
\[ y_3 = A \sin(kx - \omega t + \phi + \pi) = -A \sin(kx - \omega t + \phi) \]

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

### Step 5: Analyze the Resultant Wave
From the combination, we see that the sine components from wave 1 and wave 3 cancel each other out. The only wave that contributes to the superposition is wave 2, which is a cosine wave.

### Conclusion
The resultant wave is a simple harmonic wave with amplitude \( A \) and the same frequency \( n \) as the original waves. The velocity of the resultant wave will also be the same as that of the individual waves.

### Final Result
The superposition results in a single harmonic wave with amplitude \( A \) and frequency \( n \).

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