Two waves are passing through a region in the same direction at the same time . If the equation of these waves are
`y_(1) = a sin ( 2pi)/(lambda)( v t - x)`
and `y_(2) = b sin ( 2pi)/( lambda) [( vt - x) + x_(0) ]`
then the amplitude of the resulting wave for `x_(0) = (lambda//2)` is

Options

A. `| a - b|`
B. ` a + b`
C. `sqrt(a^(2) + b^(2))`
D. `sqrt( a^(2) + b^(2) + 2 ab cos x)`

Explanation

To find the amplitude of the resulting wave when two waves are superimposed, we can follow these steps:

### Step 1: Identify the wave equations
The equations of the two waves are given as:
1. \( y_1 = a \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \)
2. \( y_2 = b \sin\left(\frac{2\pi}{\lambda}\left[(vt - x) + x_0\right]\right) \)

### Step 2: Substitute the value of \( x_0 \)
We are given \( x_0 = \frac{\lambda}{2} \). Substituting this into the equation for \( y_2 \):
\[ y_2 = b \sin\left(\frac{2\pi}{\lambda}\left[(vt - x) + \frac{\lambda}{2}\right]\right) \]
This simplifies to:
\[ y_2 = b \sin\left(\frac{2\pi}{\lambda}(vt - x) + \pi\right) \]

### Step 3: Use the sine addition formula
Using the sine addition formula, we know that:
\[ \sin(A + \pi) = -\sin(A) \]
Thus, we can rewrite \( y_2 \):
\[ y_2 = -b \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \]

### Step 4: Write the resulting wave equation
Now we can write the total wave \( y \) as:
\[ y = y_1 + y_2 = a \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) - b \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \]
This simplifies to:
\[ y = (a - b) \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \]

### Step 5: Determine the amplitude of the resulting wave
The amplitude of the resulting wave is simply the coefficient of the sine function:
\[ R = |a - b| \]

### Conclusion
Thus, the amplitude of the resulting wave when \( x_0 = \frac{\lambda}{2} \) is:
\[ R = |a - b| \]