A standing wave pattern is formed on a string One of the waves if given by equation `y_1=acos(omegat-kx+(pi)/(3))` then the equation of the other wave such that at `x=0` a node is formed.

Options

A. `y_(2)=a sin (omegat+kx+(pi)/3)`
B. `y_(2)=a cos(omegat+kx+(pi)/3)`
C. `y_(2)=a cos(omegat+kx+(2pi)/3)`
D. `y_(2)=a cos (omegat+kx+(4pi)/3)`

Explanation

To find the equation of the second wave such that at \( x = 0 \) a node is formed, we can follow these steps:

### Step 1: Understand the given wave equation
The first wave is given by:
\[ y_1 = a \cos(\omega t - kx + \frac{\pi}{3}) \]
This represents a wave traveling in the positive x-direction.

### Step 2: Identify the condition for a node
A node is a point where the amplitude of the wave is zero. For a standing wave formed by the superposition of two waves, at \( x = 0 \), the total displacement must be zero.

### Step 3: Write the general form of the second wave
The second wave, which is reflected, can be represented as:
\[ y_2 = a \cos(\omega t + kx + \phi) \]
where \( \phi \) is the phase constant we need to determine.

### Step 4: Apply the condition for a node at \( x = 0 \)
At \( x = 0 \), the total displacement \( y \) of the standing wave is given by:
\[ y = y_1 + y_2 = a \cos(\omega t - k(0) + \frac{\pi}{3}) + a \cos(\omega t + k(0) + \phi) \]
This simplifies to:
\[ y = a \cos(\omega t + \frac{\pi}{3}) + a \cos(\omega t + \phi) \]
For a node to form at \( x = 0 \), we need:
\[ y = 0 \]
Thus, we have:
\[ a \cos(\omega t + \frac{\pi}{3}) + a \cos(\omega t + \phi) = 0 \]

### Step 5: Factor out the amplitude
Factoring out \( a \):
\[ a \left( \cos(\omega t + \frac{\pi}{3}) + \cos(\omega t + \phi) \right) = 0 \]
Since \( a \neq 0 \), we need:
\[ \cos(\omega t + \frac{\pi}{3}) + \cos(\omega t + \phi) = 0 \]

### Step 6: Use the cosine addition formula
Using the cosine addition formula:
\[ \cos A + \cos B = 0 \implies A + B = (2n + 1)\frac{\pi}{2} \text{ for } n \in \mathbb{Z} \]
Let \( A = \omega t + \frac{\pi}{3} \) and \( B = \omega t + \phi \):
\[ (\omega t + \frac{\pi}{3}) + (\omega t + \phi) = (2n + 1)\frac{\pi}{2} \]
This simplifies to:
\[ 2\omega t + \frac{\pi}{3} + \phi = (2n + 1)\frac{\pi}{2} \]

### Step 7: Solve for \( \phi \)
To find \( \phi \) such that this holds for all \( t \), we can set \( n = 0 \):
\[ \frac{\pi}{3} + \phi = \frac{\pi}{2} \]
Solving for \( \phi \):
\[ \phi = \frac{\pi}{2} - \frac{\pi}{3} = \frac{3\pi - 2\pi}{6} = \frac{\pi}{6} \]

### Step 8: Write the final equation for the second wave
Thus, the equation for the second wave is:
\[ y_2 = a \cos(\omega t + kx + \frac{\pi}{6}) \]

### Conclusion
The equation of the second wave such that at \( x = 0 \) a node is formed is:
\[ y_2 = a \cos(\omega t + kx + \frac{\pi}{6}) \]