The standing wave in a medium is expressed as `y=0.2 sin (0.8x) cos (3000 t ) m`. The distance between any two consecutive points of minimum of maximum displacement is

Options

A. `(pi)/(2)m`
B. `(pi)/(4) m`
C. `(pi)/(6) m`
D. None of these

Answer

To solve the problem of finding the distance between any two consecutive points of minimum (nodes) and maximum (antinodes) displacement in the standing wave described by the equation \( y = 0.2 \sin(0.8x) \cos(3000t) \), we can follow these steps:

### Step 1: Identify the wave equation components
The given wave equation is in the form of a standing wave:
\[ y = A \sin(kx) \cos(\omega t) \]
where:
- \( A = 0.2 \) m (amplitude),
- \( k = 0.8 \) (wave number),
- \( \omega = 3000 \) (angular frequency).

### Step 2: Determine the wave number \( k \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
From the equation, we have \( k = 0.8 \). We can rearrange the formula to solve for \( \lambda \):
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{0.8} \]

### Step 3: Calculate the wavelength \( \lambda \)
Calculating \( \lambda \):
\[ \lambda = \frac{2\pi}{0.8} = \frac{2\pi}{\frac{8}{10}} = \frac{2\pi \cdot 10}{8} = \frac{20\pi}{8} = \frac{5\pi}{2} \text{ m} \]

### Step 4: Find the distance between a node and an antinode
The distance between a node (minimum displacement) and an antinode (maximum displacement) is given by:
\[ \text{Distance} = \frac{\lambda}{4} \]
Now substituting the value of \( \lambda \):
\[ \text{Distance} = \frac{1}{4} \cdot \frac{5\pi}{2} = \frac{5\pi}{8} \text{ m} \]

### Step 5: Conclusion
The distance between any two consecutive points of minimum and maximum displacement is:
\[ \frac{5\pi}{8} \text{ m} \]