A progressive wave is given by
`y=3 sin 2pi [(t//0.04)-(x//0.01)]`
where x, y are in cm and t in s. the frequency of wave and maximum accelration will be:

Options

A. `100 Hz,4.7xx10^(3) m//s^(2)`
B. `50 Hz, 7.5xx10^(3) m//s^(2)`
C. `25 Hz, 4.7xx10^(4) m//s^(2)`
D. `25 Hz, 7.5xx10^(4) m//s^(2)`

Explanation

To solve the problem, we need to find the frequency of the wave and the maximum acceleration from the given wave equation:

\[ y = 3 \sin \left( 2\pi \left( \frac{t}{0.04} - \frac{x}{0.01} \right) \right) \]

### Step 1: Identify the wave parameters
The standard form of a progressive wave is given by:

\[ y = A \sin(2\pi Ft - \frac{2\pi x}{\lambda}) \]

Where:
- \( A \) is the amplitude
- \( F \) is the frequency
- \( \lambda \) is the wavelength

From the given equation, we can identify:
- Amplitude \( A = 3 \) cm
- The term \( \frac{t}{0.04} \) indicates that \( T = 0.04 \) s (where \( T \) is the time period)
- The term \( \frac{x}{0.01} \) indicates that \( \lambda = 0.02 \) m (since \( \lambda = 2 \times 0.01 \))

### Step 2: Calculate the frequency
The frequency \( F \) is the reciprocal of the time period \( T \):

\[ F = \frac{1}{T} = \frac{1}{0.04} = 25 \text{ Hz} \]

### Step 3: Calculate the maximum acceleration
The maximum acceleration \( a_{max} \) can be found by taking the second derivative of the displacement \( y \) with respect to time \( t \).

1. First, differentiate \( y \) with respect to \( t \):

\[ \frac{dy}{dt} = 3 \cdot 2\pi \cdot \frac{1}{0.04} \cos\left(2\pi \left(\frac{t}{0.04} - \frac{x}{0.01}\right)\right) \]

2. Differentiate again to find acceleration:

\[ \frac{d^2y}{dt^2} = -3 \cdot (2\pi)^2 \cdot \frac{1}{0.04^2} \sin\left(2\pi \left(\frac{t}{0.04} - \frac{x}{0.01}\right)\right) \]

The maximum value of \( \sin \) is 1, so:

\[ a_{max} = 3 \cdot (2\pi)^2 \cdot \frac{1}{0.04^2} \]

3. Calculate \( a_{max} \):

\[ a_{max} = 3 \cdot 4\pi^2 \cdot \frac{1}{0.0016} = \frac{12\pi^2}{0.0016} \]

Calculating \( \pi^2 \approx 9.87 \):

\[ a_{max} \approx \frac{12 \cdot 9.87}{0.0016} \approx 7.5 \times 10^4 \text{ m/s}^2 \]

### Final Answers:
- Frequency \( F = 25 \text{ Hz} \)
- Maximum Acceleration \( a_{max} = 7.5 \times 10^4 \text{ m/s}^2 \)