A sine wave has an amplitude A and wavelength `lamda`. The ratio of particle velocity and the wave velocity is equal to `(2piA=lamda)`

Options

A. `le1`
B. `=1`
C. `ge1`
D. data insufficient

Answer

To solve the problem, we need to find the ratio of particle velocity to wave velocity for a sine wave with amplitude \( A \) and wavelength \( \lambda \).

### Step-by-Step Solution:

1. **Understanding Particle Velocity and Wave Velocity**:
- The **particle velocity** in a wave is the velocity of the individual particles of the medium through which the wave is traveling. For a sine wave, the maximum particle velocity can be expressed as:
\[ v_p = 2 \pi f A \]
where \( f \) is the frequency of the wave and \( A \) is the amplitude.

2. **Wave Velocity**:
- The **wave velocity** (or phase velocity) is the speed at which the wave propagates through the medium. It can be calculated using the formula:
\[ v_w = f \lambda \]
where \( \lambda \) is the wavelength.

3. **Finding the Ratio**:
- We need to find the ratio of particle velocity to wave velocity:
\[ \frac{v_p}{v_w} = \frac{2 \pi f A}{f \lambda} \]
- Here, the \( f \) in the numerator and denominator cancels out:
\[ \frac{v_p}{v_w} = \frac{2 \pi A}{\lambda} \]

4. **Conclusion**:
- The ratio of particle velocity to wave velocity is given by:
\[ \frac{v_p}{v_w} = \frac{2 \pi A}{\lambda} \]
- This indicates that the particle velocity is proportional to the amplitude and inversely proportional to the wavelength.