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 \( y_1 = a_1 \sin\left(\omega t - \frac{2\pi x}{\lambda}\right) \) and \( y_2 = a_2 \cos\left(\omega t - \frac{2\pi x}{\lambda} + \phi\right) \), we can follow these steps:

### Step 1: Rewrite the cosine function
We can express the cosine function in terms of sine using the identity:
\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]
Thus, we can rewrite \( y_2 \) as:
\[ y_2 = a_2 \sin\left(\omega t - \frac{2\pi x}{\lambda} + \phi + \frac{\pi}{2}\right) \]

### Step 2: Identify the phase of each wave
Now we can identify the phase of each wave:
- For \( y_1 \): The phase is \( \phi_1 = \omega t - \frac{2\pi x}{\lambda} \).
- For \( y_2 \): The phase is \( \phi_2 = \omega t - \frac{2\pi x}{\lambda} + \phi + \frac{\pi}{2} \).

### Step 3: Calculate the phase difference
The phase difference \( \Delta \phi \) between the two waves is given by:
\[ \Delta \phi = \phi_2 - \phi_1 \]
Substituting the values we found:
\[ \Delta \phi = \left(\omega t - \frac{2\pi x}{\lambda} + \phi + \frac{\pi}{2}\right) - \left(\omega t - \frac{2\pi x}{\lambda}\right) \]
This simplifies to:
\[ \Delta \phi = \phi + \frac{\pi}{2} \]

### Step 4: Relate phase difference to path difference
The path difference \( \Delta x \) can be related to the phase difference using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
So, we can rearrange this to find \( \Delta x \):
\[ \Delta x = \frac{\Delta \phi \cdot \lambda}{2\pi} \]
Substituting \( \Delta \phi = \phi + \frac{\pi}{2} \):
\[ \Delta x = \frac{\left(\phi + \frac{\pi}{2}\right) \cdot \lambda}{2\pi} \]

### Final Result
Thus, the path difference between the two waves is:
\[ \Delta x = \frac{\lambda}{2\pi} \left(\phi + \frac{\pi}{2}\right) \]