The equation of a transverse mechanical wave propagating along the 'x' axis is given by the expression `y=2sin^(2)(3x+5t)+6` where y is displacement in 'm', x is position on the 'x' axis in 'cm', t is time in 'secs'
Choose the correct option

Options

A. The amplitude of the wave is 1 m
B. The velocity of propagation is `(5)/(3)cm//s`
C. The frequency of wave is `5/pi"sec"^(-1)`
D. The wavelength of the wave is `(pi)/(3)` cm

Answer

To solve the problem, we start with the given equation of the transverse mechanical wave:

\[ y = 2 \sin^2(3x + 5t) + 6 \]

where \( y \) is the displacement in meters, \( x \) is the position along the x-axis in centimeters, and \( t \) is the time in seconds.

### Step 1: Simplify the Equation
First, we can rewrite the equation by isolating the sine term:

\[ y - 6 = 2 \sin^2(3x + 5t) \]

### Step 2: Use the Identity for Sine Squared
We can use the trigonometric identity:

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

Applying this identity, we have:

\[ y - 6 = 2 \left( \frac{1 - \cos(2(3x + 5t))}{2} \right) \]

This simplifies to:

\[ y - 6 = 1 - \cos(6x + 10t) \]

### Step 3: Rearranging the Equation
Rearranging gives us:

\[ y = 7 - \cos(6x + 10t) \]

### Step 4: Identify Parameters
From the equation \( y = 7 - \cos(6x + 10t) \), we can identify the following parameters:

- Amplitude \( A = 1 \) meter (since the coefficient of cosine is 1)
- Wave number \( k = 6 \) (coefficient of \( x \))
- Angular frequency \( \omega = 10 \) (coefficient of \( t \))

### Step 5: Calculate Velocity of Propagation
The velocity \( v \) of the wave can be calculated using the relationship:

\[ \omega = v \cdot k \]

Rearranging gives:

\[ v = \frac{\omega}{k} = \frac{10}{6} = \frac{5}{3} \text{ cm/s} \]

### Step 6: Calculate Frequency
The frequency \( f \) can be calculated using the relationship:

\[ \omega = 2\pi f \]

Rearranging gives:

\[ f = \frac{\omega}{2\pi} = \frac{10}{2\pi} = \frac{5}{\pi} \text{ Hz} \]

### Step 7: Calculate Wavelength
The wavelength \( \lambda \) can be calculated using the relationship:

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

Rearranging gives:

\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{6} = \frac{\pi}{3} \text{ cm} \]

### Summary of Results
- Amplitude: \( 1 \) meter
- Velocity of propagation: \( \frac{5}{3} \) cm/s
- Frequency: \( \frac{5}{\pi} \) Hz
- Wavelength: \( \frac{\pi}{3} \) cm

### Conclusion
All calculated values match the options provided in the question, confirming the correctness of the solution.

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