Two sinusoidal waves are superposed. Their equations are
`y_(1)=Asin(kx-omegat+(pi)/(6))and y_(2)=Asin(kx-omegat-(pi)/(6))`
the equation of their resultant is

Options

A. `(A)/(sqrt(3))sin(kx-omegat)`
B. `Asqrt(3)sin(kx-omegat)`
C. `Asqrt(3)sin(kx-omegat-(pi)/(3))`
D. `(A)/(sqrt(3))sin(kx-omegat-(pi)/(2))`

Answer

To find the equation of the resultant wave when two sinusoidal waves are superposed, we can follow these steps:

### Step 1: Write Down the Given Equations
The equations of the two waves are:
1. \( y_1 = A \sin(kx - \omega t + \frac{\pi}{6}) \)
2. \( y_2 = A \sin(kx - \omega t - \frac{\pi}{6}) \)

### Step 2: Add the Two Waves
The resultant wave \( y_r \) is given by the sum of the two waves:
\[ y_r = y_1 + y_2 = A \sin(kx - \omega t + \frac{\pi}{6}) + A \sin(kx - \omega t - \frac{\pi}{6}) \]

### Step 3: Use the Sine Addition Formula
To simplify the addition of the two sine functions, we can use the sine addition formula:
\[ \sin(a + b) + \sin(a - b) = 2 \sin(a) \cos(b) \]
Here, let:
- \( a = kx - \omega t \)
- \( b = \frac{\pi}{6} \)

Thus, we can rewrite the resultant wave as:
\[ y_r = 2A \sin(kx - \omega t) \cos\left(\frac{\pi}{6}\right) \]

### Step 4: Calculate \( \cos\left(\frac{\pi}{6}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

### Step 5: Substitute Back into the Resultant Equation
Now, substituting \( \cos\left(\frac{\pi}{6}\right) \) into the equation gives:
\[ y_r = 2A \sin(kx - \omega t) \cdot \frac{\sqrt{3}}{2} \]
This simplifies to:
\[ y_r = A\sqrt{3} \sin(kx - \omega t) \]

### Final Result
The equation of the resultant wave is:
\[ y_r = \sqrt{3}A \sin(kx - \omega t) \]