The equation `y=A cos^(2) (2pi nt -2 pi (x)/(lambda))` represents a wave with

Options

A. Amplitude `A//2` , frequency 2n and wavelength `lambda//2`
B. Amplitude `A//2` , frequency 2n and wavelength `lambda`
C. Amplitude A , frequency 2n and wavelength `2lambda`
D. Amplitude A , frequency n and waves

Answer

To solve the problem given by the equation \( y = A \cos^2(2\pi nt - \frac{2\pi x}{\lambda}) \), we need to find the amplitude, frequency, and wavelength of the wave represented by this equation. Here’s a step-by-step breakdown of the solution:

### Step 1: Rewrite the Cosine Squared Function
We start with the equation:
\[ y = A \cos^2(2\pi nt - \frac{2\pi x}{\lambda}) \]
Using the trigonometric identity:
\[ \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \]
we can rewrite the equation as:
\[ y = A \cdot \frac{1 + \cos(2(2\pi nt - \frac{2\pi x}{\lambda}))}{2} \]
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:
\[ y = \frac{A}{2} + \frac{A}{2} \cos(4\pi nt - \frac{4\pi x}{\lambda}) \]
we can see that the amplitude of the wave is:
\[ \text{Amplitude} = \frac{A}{2} \]

### Step 3: Identify the Frequency
The term \( \cos(4\pi nt - \frac{4\pi x}{\lambda}) \) indicates the angular frequency \( \omega \). We know:
\[ \omega = 4\pi n \]
Since \( \omega = 2\pi f \), we can find the frequency \( f \):
\[ 2\pi f = 4\pi n \implies f = 2n \]

### Step 4: Identify the Wavelength
The wave number \( k \) is given by the coefficient of \( x \) in the cosine term:
\[ k = \frac{4\pi}{\lambda} \]
We know that:
\[ k = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{k} \]
From our equation, we have:
\[ k = 4\pi \implies \lambda = \frac{2\pi}{4\pi} = \frac{\lambda}{2} \]

### Summary of Results
- Amplitude: \( \frac{A}{2} \)
- Frequency: \( 2n \)
- Wavelength: \( \frac{\lambda}{2} \)

### Final Answer
The wave represented by the equation has:
- Amplitude = \( \frac{A}{2} \)
- Frequency = \( 2n \)
- Wavelength = \( \frac{\lambda}{2} \)