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 need to analyze the given wave equation and extract relevant parameters such as amplitude, wave number, angular frequency, velocity, frequency, and wavelength.

### Step-by-Step Solution:

1. **Identify the Wave Equation**:
The given wave equation is:
\[ y = 2\sin^2(3x + 5t) + 6 \]
Here, \(y\) is the displacement in meters, \(x\) is the position in centimeters, and \(t\) is the time in seconds.

2. **Rearranging the Equation**:
We can rearrange the equation to isolate the sine term:
\[ y - 6 = 2\sin^2(3x + 5t) \]
Dividing both sides by 2 gives:
\[ \frac{y - 6}{2} = \sin^2(3x + 5t) \]

3. **Using the Identity for Sine Squared**:
We can use the identity \(\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}\):
\[ \sin^2(3x + 5t) = \frac{1 - \cos(2(3x + 5t))}{2} \]
Substituting this into our equation:
\[ \frac{y - 6}{2} = \frac{1 - \cos(6x + 10t)}{2} \]
Multiplying through by 2:
\[ y - 6 = 1 - \cos(6x + 10t) \]
Rearranging gives:
\[ \cos(6x + 10t) = 7 - y \]

4. **Identifying Wave Parameters**:
From the expression \(\cos(6x + 10t)\), we can identify:
- Wave number \(k = 6 \, \text{cm}^{-1}\)
- Angular frequency \(\omega = 10 \, \text{s}^{-1}\)

5. **Calculating Velocity**:
The velocity \(v\) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} = \frac{10}{6} = \frac{5}{3} \, \text{cm/s} \]

6. **Calculating Frequency**:
The frequency \(f\) can be found using the relation:
\[ f = \frac{\omega}{2\pi} = \frac{10}{2\pi} = \frac{5}{\pi} \, \text{Hz} \]

7. **Calculating Wavelength**:
The wavelength \(\lambda\) can be calculated using:
\[ k = \frac{2\pi}{\lambda} \Rightarrow \lambda = \frac{2\pi}{k} = \frac{2\pi}{6} = \frac{\pi}{3} \, \text{cm} \]

8. **Summary of Results**:
- Amplitude \(A = 1 \, \text{m}\)
- Velocity \(v = \frac{5}{3} \, \text{cm/s}\)
- Frequency \(f = \frac{5}{\pi} \, \text{Hz}\)
- Wavelength \(\lambda = \frac{\pi}{3} \, \text{cm}\)

### Conclusion:
All calculated parameters are consistent with the wave equation provided. Therefore, the correct options for the parameters of the wave are:
- Amplitude: 1 m
- Velocity: \( \frac{5}{3} \, \text{cm/s} \)
- Frequency: \( \frac{5}{\pi} \, \text{Hz} \)
- Wavelength: \( \frac{\pi}{3} \, \text{cm} \)

**Final Answer**: All options A, B, C, and D are correct.