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 extract the frequency of the wave and the maximum acceleration from the given wave equation:

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

### Step 1: Identify the angular frequency (ω)
The general form of a wave equation is:
\[ y = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

From the given equation, we can see that:
\[ \omega = 2\pi \left( \frac{1}{0.04} \right) \]

### Step 2: Calculate the frequency (f)
The relationship between angular frequency (ω) and frequency (f) is given by:
\[ \omega = 2\pi f \]
Thus, we can rearrange this to find the frequency:
\[ f = \frac{\omega}{2\pi} \]

Substituting the value of ω:
\[ f = \frac{2\pi \left( \frac{1}{0.04} \right)}{2\pi} = \frac{1}{0.04} = 25 \text{ Hz} \]

### Step 3: Find the maximum acceleration (a_max)
The maximum acceleration for a wave can be calculated using the formula:
\[ a_{max} = A \omega^2 \]
Where:
- \( A = 3 \) cm (which we convert to meters: \( A = 0.03 \) m),
- \( \omega = 2\pi \left( \frac{1}{0.04} \right) \).

Now, calculate ω:
\[ \omega = \frac{2\pi}{0.04} = 50\pi \text{ rad/s} \]

Now, substituting these values into the acceleration formula:
\[ a_{max} = 0.03 \times (50\pi)^2 \]
\[ a_{max} = 0.03 \times 2500\pi^2 \]
\[ a_{max} = 75\pi^2 \text{ m/s}^2 \]

### Step 4: Approximate π²
Using \( \pi^2 \approx 10 \):
\[ a_{max} \approx 75 \times 10 = 750 \text{ m/s}^2 \]

### Final Results
- Frequency \( f = 25 \text{ Hz} \)
- Maximum Acceleration \( a_{max} \approx 750 \text{ m/s}^2 \)

### Summary
The frequency of the wave is \( 25 \text{ Hz} \) and the maximum acceleration is approximately \( 750 \text{ m/s}^2 \).

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