The path difference between the two waves
`y_(1)=a_(1) sin(omega t-(2pi x)/(lambda)) and y(2)=a_(2) cos(omega t-(2pi x)/(lambda)+phi)` is

Options

A. `(lambda)/(2pi)/phi`
B. `(lambda)/(2pi)(phi+(pi)/(2))`
C. `(2pi)/(lambda)(phi-(pi)/(2))`
D. `(2pi)/(lambda)(phi)`

Explanation

To find the path difference between the two waves given by the equations:

1. \( y_1 = A_1 \sin\left(\omega t - \frac{2\pi x}{\lambda}\right) \)
2. \( y_2 = A_2 \cos\left(\omega t - \frac{2\pi x}{\lambda} + \phi\right) \)

we can follow these steps:

### Step 1: Convert the cosine function to sine
We know from trigonometric identities that:
\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]
Thus, we can rewrite \( y_2 \):
\[ y_2 = A_2 \cos\left(\omega t - \frac{2\pi x}{\lambda} + \phi\right) = A_2 \sin\left(\omega t - \frac{2\pi x}{\lambda} + \phi + \frac{\pi}{2}\right) \]

### Step 2: Identify the phase difference
The phase of \( y_1 \) is:
\[ \phi_1 = \omega t - \frac{2\pi x}{\lambda} \]
The phase of \( y_2 \) is:
\[ \phi_2 = \omega t - \frac{2\pi x}{\lambda} + \phi + \frac{\pi}{2} \]
The phase difference \( \Delta \phi \) is given by:
\[ \Delta \phi = \phi_2 - \phi_1 = \left(\omega t - \frac{2\pi x}{\lambda} + \phi + \frac{\pi}{2}\right) - \left(\omega t - \frac{2\pi x}{\lambda}\right) \]
This simplifies to:
\[ \Delta \phi = \phi + \frac{\pi}{2} \]

### Step 3: Calculate the path difference
The path difference \( \Delta x \) can be calculated using the formula:
\[ \Delta x = \frac{\lambda}{2\pi} \Delta \phi \]
Substituting the expression for \( \Delta \phi \):
\[ \Delta x = \frac{\lambda}{2\pi} \left(\phi + \frac{\pi}{2}\right) \]

### Step 4: Final expression for path difference
Thus, we can express the path difference as:
\[ \Delta x = \frac{\lambda}{2\pi} \phi + \frac{\lambda}{2\pi} \cdot \frac{\pi}{2} \]
This simplifies to:
\[ \Delta x = \frac{\lambda}{2\pi} \phi + \frac{\lambda}{4} \]
Since \( \frac{\lambda}{4} \) is equivalent to \( \frac{\lambda}{2\pi} \cdot \frac{\pi}{2} \).

### Conclusion
The final expression for the path difference is:
\[ \Delta x = \frac{\lambda}{2\pi} \left(\phi + \frac{\pi}{2}\right) \]