The equation of stationary wave along a stretched string is given by `y = 5 sin(pi/3 x) cos 40pi t` where x and y are in centimetre and t in second. The separation between two adjacent nodes is :

Options

A. 1.5
B. 3
C. 6
D. 4

Answer

To find the separation between two adjacent nodes in the given stationary wave equation, we can follow these steps:

### Step-by-Step Solution:

1. **Write the Given Wave Equation**:
The equation of the stationary wave is given as:
\[ y = 5 \sin\left(\frac{\pi}{3} x\right) \cos(40 \pi t) \]

2. **Identify the General Form of the Wave Equation**:
The general form of a stationary wave can be expressed as:
\[ y = 2A \sin(kx) \cos(\omega t) \]
where \( k \) is the wave number and \( \omega \) is the angular frequency.

3. **Compare the Given Equation with the General Form**:
From the given equation, we can identify:
\[ k = \frac{\pi}{3} \]

4. **Relate Wave Number to Wavelength**:
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ \frac{\pi}{3} = \frac{2\pi}{\lambda} \]

5. **Solve for Wavelength \( \lambda \)**:
Cross-multiplying gives:
\[ \pi \lambda = 6\pi \]
Dividing both sides by \( \pi \):
\[ \lambda = 6 \text{ cm} \]

6. **Calculate the Separation Between Adjacent Nodes**:
The separation between two adjacent nodes is given by:
\[ \text{Separation} = \frac{\lambda}{2} \]
Substituting the value of \( \lambda \):
\[ \text{Separation} = \frac{6}{2} = 3 \text{ cm} \]

7. **Final Answer**:
The separation between two adjacent nodes is:
\[ \boxed{3 \text{ cm}} \]