Consider a wave rpresented by `y= a cos^(2) (omega t-kx)` where symbols have their usual meanings. This wave has

Options

A. an amplitude `a`, frequency `omega`, and wavelength `lambda`.
B. an amplitude `a`, frequency `2omega`, and wabvelength `2lambda`.
C. an amplitude `a//2` frequency `2omega` and wavelength `lambda//2`.
D. am amplitude `a//2` frequency `2omega`, and wavelength `lambda`.

Explanation

To solve the problem step by step, we need to analyze the given wave equation \( y = a \cos^2(\omega t - kx) \) and extract the amplitude, frequency, and wavelength.

### Step 1: Rewrite the Wave Equation
We start with the wave equation:
\[ y = a \cos^2(\omega t - kx) \]

Using the trigonometric identity:
\[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \]
we can rewrite the equation as:
\[ y = a \left(\frac{1 + \cos(2(\omega t - kx))}{2}\right) \]
This simplifies to:
\[ y = \frac{a}{2} + \frac{a}{2} \cos(2\omega t - 2kx) \]

### Step 2: Identify the Amplitude
From the rewritten equation:
\[ y = \frac{a}{2} + \frac{a}{2} \cos(2\omega t - 2kx) \]
we can see that the amplitude of the wave is the coefficient of the cosine term. Thus, the amplitude \( A \) is:
\[ A = \frac{a}{2} \]

### Step 3: Determine the Angular Frequency
In the term \( \cos(2\omega t - 2kx) \), the angular frequency \( \omega' \) is the coefficient of \( t \):
\[ \omega' = 2\omega \]

### Step 4: Determine the Wavenumber
The wavenumber \( k' \) is the coefficient of \( x \) in the cosine term:
\[ k' = 2k \]

### Step 5: Calculate the Wavelength
The relationship between wavenumber and wavelength is given by:
\[ k = \frac{2\pi}{\lambda} \]
Thus, for \( k' \):
\[ k' = \frac{2\pi}{\lambda'} \]
Setting these equal gives:
\[ 2k = \frac{2\pi}{\lambda'} \]
Substituting \( k = \frac{2\pi}{\lambda} \):
\[ 2 \left(\frac{2\pi}{\lambda}\right) = \frac{2\pi}{\lambda'} \]
This simplifies to:
\[ \frac{4\pi}{\lambda} = \frac{2\pi}{\lambda'} \]
Cross-multiplying gives:
\[ 4\lambda' = \lambda \]
Thus, the new wavelength \( \lambda' \) is:
\[ \lambda' = \frac{\lambda}{2} \]

### Final Results
- Amplitude: \( \frac{a}{2} \)
- Frequency: \( 2\omega \) (since frequency \( f = \frac{\omega}{2\pi} \))
- Wavelength: \( \frac{\lambda}{2} \)

### Summary
The wave has:
- Amplitude: \( \frac{a}{2} \)
- Frequency: \( 2\omega \)
- Wavelength: \( \frac{\lambda}{2} \)