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

Answer

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

### Step 1: Understand the given wave equation
The first wave is given by:
\[ y_1 = a \cos(\omega t - kx + \frac{\pi}{3}) \]
We need to analyze this wave at \( x = 0 \).

### Step 2: Evaluate the first wave at \( x = 0 \)
Substituting \( x = 0 \) into the equation:
\[ y_1(0, t) = a \cos(\omega t + \frac{\pi}{3}) \]
This expression shows how the wave behaves at \( x = 0 \).

### Step 3: Determine the condition for a node
A node is formed when the total displacement is zero. For two waves to create a node, they must interfere destructively at that point. This means that the second wave \( y_2 \) must be the negative of the first wave at \( x = 0 \):
\[ y_2(0, t) = -y_1(0, t) = -a \cos(\omega t + \frac{\pi}{3}) \]

### Step 4: Write the equation for the second wave
To express \( y_2 \) in a similar form to \( y_1 \), we can use the property of cosine:
\[ \cos(\theta) = -\cos(\theta + \pi) \]
Thus, we can rewrite \( y_2 \) as:
\[ y_2 = a \cos(\omega t + kx + \frac{4\pi}{3}) \]
Here, we have added \( \pi \) to the phase angle to achieve the negative amplitude, and we have adjusted the angle to maintain the cosine function's periodicity.

### Step 5: Final equation of the second wave
The complete equation for the second wave is:
\[ y_2 = a \cos(\omega t + kx + \frac{4\pi}{3}) \]

### Conclusion
Thus, the equation of the other wave such that at \( x = 0 \) a node is formed is:
\[ y_2 = a \cos(\omega t + kx + \frac{4\pi}{3}) \]

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