The transverse displacement of a string clamped at its both ends is given by
`y(x,t)=2sin((2pi)/3x)cos(100pit)`
where x and y are in cm and t is in s.
Which of the following statements is correct?

Options

A. All the points on the string between two consecutive nodes vibrate with same frequency, phase and amplitude.
B. All the points on the string between two consecutive nodes vibrate with same frequency and phase but different amplitude.
C. All the points on the string between two consecutive nodes vibrate with different frequency and phase but same amplitude.
D. All the points on the string between two consecutive nodes vibrate with different frequency, phase and amplitude

Explanation

To solve the problem, we need to analyze the given wave equation for the transverse displacement of a string clamped at both ends:

\[ y(x,t) = 2 \sin\left(\frac{2\pi}{3} x\right) \cos(100 \pi t) \]

### Step 1: Identify the type of wave
The equation is in the form of a product of a sine function and a cosine function, which indicates that it represents a stationary (or standing) wave.

### Step 2: Determine the characteristics of the wave
1. **Amplitude**: The amplitude of the wave can be identified from the equation. The maximum displacement (amplitude) is given by the coefficient of the sine function, which is 2 cm.

2. **Wavelength**: The term \(\frac{2\pi}{3}\) in the sine function corresponds to the wave number \(k\). The wavelength \(\lambda\) can be calculated using the formula:
\[ k = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{\frac{2\pi}{3}} = 3 \text{ cm} \]

3. **Frequency**: The term \(100\pi\) in the cosine function corresponds to the angular frequency \(\omega\). The frequency \(f\) can be calculated using the formula:
\[ \omega = 2\pi f \implies f = \frac{100\pi}{2\pi} = 50 \text{ Hz} \]

### Step 3: Analyze the behavior of the wave
In a stationary wave, points between two consecutive nodes (points where the displacement is always zero) vibrate with the same frequency and are in the same phase, but they can have different amplitudes.

### Step 4: Evaluate the options
Given the characteristics of the wave, we can conclude that:
- All points between two consecutive nodes vibrate with the same frequency and phase but may have different amplitudes.

### Conclusion
Based on the analysis, the correct statement is:
- **Option B**: All points on the string between two consecutive nodes vibrate with the same frequency and phase but different amplitudes.