The transverse displacement of a string (clamped at its both ends ) is given by `y(x,t)=0.06sin(2pix//3)cos(120pit).`
All the points on the string between two consecutive nodes vibrate with

Options

A. same frequency
B. same phase
C. same energy
D. different amplitude

Explanation

To solve the problem, we need to analyze the given equation of the transverse displacement of a string, which is:

\[ y(x,t) = 0.06 \sin\left(\frac{2\pi x}{3}\right) \cos(120\pi t) \]

This equation represents a standing wave on a string that is clamped at both ends.

### Step 1: Identify the form of the wave equation
The equation can be compared to the general form of a standing wave:

\[ y(x,t) = A \sin(kx) \cos(\omega t) \]

Here, \( A = 0.06 \), \( k = \frac{2\pi}{3} \), and \( \omega = 120\pi \).

### Step 2: Determine the wave properties
1. **Wavelength (\( \lambda \))**: The wave number \( k \) is related to the wavelength by the formula \( k = \frac{2\pi}{\lambda} \). Therefore, we can find \( \lambda \):

\[ k = \frac{2\pi}{3} \implies \lambda = 3 \]

2. **Frequency (\( f \))**: The angular frequency \( \omega \) is related to the frequency by \( \omega = 2\pi f \). Thus, we can find \( f \):

\[ \omega = 120\pi \implies f = 60 \text{ Hz} \]

### Step 3: Analyze the points between two consecutive nodes
In a standing wave, nodes are points where the displacement is always zero. The distance between two consecutive nodes is half the wavelength:

\[ \text{Distance between nodes} = \frac{\lambda}{2} = \frac{3}{2} = 1.5 \]

The points between two consecutive nodes will vibrate with the same frequency, but they will have different amplitudes due to the nature of standing waves.

### Step 4: Conclusion on the vibration characteristics
- **Same Frequency**: All points between two consecutive nodes vibrate with the same frequency \( f = 60 \text{ Hz} \).
- **Different Amplitudes**: The amplitude varies with position \( x \) as given by the sine function, which means the amplitude is not constant.
- **Same Phase**: All points between two nodes are in phase at any given instant of time due to the nature of the cosine function.

### Final Answer
All the points on the string between two consecutive nodes vibrate with the **same frequency** and **same phase**, but they have **different amplitudes**.