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:

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

### Step 1: Identify the angular frequency (ω) and wave number (k)

From the wave equation, we can identify:
- The angular frequency \( \omega \) is the coefficient of \( t \) in the sine function. Here, \( \omega = 50\pi \) rad/s.
- The wave number \( k \) is the coefficient of \( x \) in the cosine function. Here, \( k = 10\pi \) rad/m.

### Step 2: Calculate the speed of the wave (v)

The speed of the wave can be calculated using the formula:

\[ v = \frac{\omega}{k} \]

Substituting the values we found:

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

### Step 3: Calculate the wavelength (λ)

The wavelength \( \lambda \) can be calculated using the formula:

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

Substituting the value of \( k \):

\[ \lambda = \frac{2\pi}{10\pi} = \frac{1}{5} = 0.2 \text{ m} \]

### Step 4: Determine the positions of nodes and antinodes

Nodes occur where the cosine term is zero, and antinodes occur where the cosine term is ±1.

1. **Finding Nodes:**
- The cosine function \( \cos(10\pi x) = 0 \) when \( 10\pi x = \frac{\pi}{2} + n\pi \) for \( n = 0, 1, 2, \ldots \)
- Solving for \( x \):
\[ x = \frac{1}{20} + \frac{n}{10} \]
- For \( n = 0 \): \( x = 0.05 \) m
- For \( n = 1 \): \( x = 0.15 \) m
- For \( n = 2 \): \( x = 0.25 \) m
- For \( n = 3 \): \( x = 0.35 \) m

2. **Finding Antinodes:**
- The cosine function \( \cos(10\pi x) = ±1 \) when \( 10\pi x = n\pi \) for \( n = 0, 1, 2, \ldots \)
- Solving for \( x \):
\[ x = \frac{n}{10} \]
- For \( n = 0 \): \( x = 0 \) m
- For \( n = 1 \): \( x = 0.1 \) m
- For \( n = 2 \): \( x = 0.2 \) m
- For \( n = 3 \): \( x = 0.3 \) m

### Conclusion

- Speed of the wave: **5 m/s**
- Wavelength: **0.2 m**
- Nodes at: **0.15 m** (and other calculated positions)
- Antinodes at: **0.3 m** (and other calculated positions)