The equation of a wave distrubance is a given as `y= 0.02 sin ((pi)/(2)+50pi t) cos (10 pix)`, where x and y are in metre and t is in second . Choose the correct statement (s).

Options

A. The wavelength of wave is 0.2m
B. Displacement node occurs at x=0.15xm
C. Displacement antinode occurs at x=0.3m
D. The speed of constitutent wave is 0.2m/s

Answer

To solve the problem, we need to analyze the given wave equation and derive the necessary parameters step by step.

### Given Wave Equation:
\[ y = 0.02 \sin\left(\frac{\pi}{2} + 50\pi t\right) \cos(10\pi x) \]

### Step 1: Identify the Wave Number (k)
The wave number \( k \) is given by the coefficient of \( x \) in the cosine term:
\[ k = 10\pi \]

### Step 2: Calculate the Wavelength (λ)
The relationship between the wave number and wavelength is given by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ 10\pi = \frac{2\pi}{\lambda} \]
To find \( \lambda \), rearranging gives:
\[ \lambda = \frac{2\pi}{10\pi} = \frac{1}{5} = 0.2 \text{ m} \]

### Step 3: Determine the Nodes
Nodes occur where the displacement \( y \) is zero. This happens when:
\[ \cos(10\pi x) = 0 \]
The cosine function is zero at:
\[ 10\pi x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Solving for \( x \):
\[ x = \frac{1}{20} + \frac{n}{10} \]
For \( n = 0 \):
\[ x = 0.05 \text{ m} \]
For \( n = 1 \):
\[ x = 0.15 \text{ m} \]
Thus, nodes occur at \( x = 0.15 \text{ m} \).

### Step 4: Determine the Antinodes
Antinodes occur where the cosine term is at its maximum (1). This happens when:
\[ \cos(10\pi x) = 1 \]
This occurs at:
\[ 10\pi x = 2n\pi \quad (n \in \mathbb{Z}) \]
Solving for \( x \):
\[ x = \frac{n}{5} \]
For \( n = 0 \):
\[ x = 0 \text{ m} \]
For \( n = 1 \):
\[ x = 0.2 \text{ m} \]
For \( n = 2 \):
\[ x = 0.4 \text{ m} \]
Thus, antinodes occur at \( x = 0.3 \text{ m} \).

### Step 5: Calculate the Speed of the Wave (v)
The speed of the wave is given by:
\[ v = \frac{\text{Coefficient of } t}{\text{Coefficient of } x} \]
From the equation:
- Coefficient of \( t \) is \( 50\pi \)
- Coefficient of \( x \) is \( 10\pi \)

Thus:
\[ v = \frac{50\pi}{10\pi} = 5 \text{ m/s} \]

### Conclusion
Based on the analysis:
1. The wavelength \( \lambda \) is \( 0.2 \text{ m} \) (correct).
2. A node occurs at \( x = 0.15 \text{ m} \) (correct).
3. An antinode occurs at \( x = 0.3 \text{ m} \) (correct).
4. The speed of the wave is \( 5 \text{ m/s} \) (correct).

### Final Statements
- The first three statements are correct.
- The speed of the wave is \( 5 \text{ m/s} \).