An equation ` y = a cos ^( 2 ) (2 pi nt - (2pi x ) / ( lamda ) ) ` represents a wave with : -

Options

A. amplitude a, frequency n and wavelength ` lamda `
B. amplitude a, frequency 2n and wavelength 2`lamda `
C. amplitude ` (a ) /( 2 ) ` , frequency 2n and wavelength `lamda`
D. amplitude ` ( a ) / ( 2 ) `, frequency 2n and wavelength ` ( lamda ) / ( 2 ) `

Explanation

To solve the problem, we need to analyze the given wave equation:

\[ y = a \cos^2(2\pi nt - \frac{2\pi x}{\lambda}) \]

### Step 1: Rewrite the Cosine Squared Function
We can use the trigonometric identity for cosine squared:

\[ \cos^2(\theta) = \frac{1}{2}(1 + \cos(2\theta)) \]

Applying this identity to our equation:

\[ y = a \cdot \frac{1}{2}(1 + \cos(2(2\pi nt - \frac{2\pi x}{\lambda}))) \]

This simplifies to:

\[ y = \frac{a}{2} + \frac{a}{2} \cos(4\pi nt - \frac{4\pi x}{\lambda}) \]

### Step 2: Identify the Amplitude
From the rewritten equation, we can see that the term multiplying the cosine function represents the amplitude of the wave. Thus, the amplitude \( A \) is:

\[ A = \frac{a}{2} \]

### Step 3: Identify the Frequency
The general form of a wave equation is given by:

\[ y = A \cos(2\pi ft - kx) \]

where \( f \) is the frequency and \( k \) is the wave number. In our case, we have:

- The frequency \( f \) can be identified from the term \( 4\pi nt \):

\[ f = 2n \]

### Step 4: Identify the Wavelength
The wave number \( k \) is defined as:

\[ k = \frac{2\pi}{\lambda} \]

From our equation, we have:

- The term \( \frac{4\pi}{\lambda} \) corresponds to the wave number \( k \):

Thus, we can find the wavelength \( \lambda' \) as follows:

\[ k = \frac{2\pi}{\lambda'} \implies \frac{4\pi}{\lambda} = \frac{2\pi}{\lambda'} \]

Solving for \( \lambda' \):

\[ \lambda' = \frac{\lambda}{2} \]

### Final Results
Putting it all together, we have:

- Amplitude \( A = \frac{a}{2} \)
- Frequency \( f = 2n \)
- Wavelength \( \lambda' = \frac{\lambda}{2} \)

### Conclusion
The equation represents a wave with:
- Amplitude: \( \frac{a}{2} \)
- Frequency: \( 2n \)
- Wavelength: \( \frac{\lambda}{2} \)