(A): Wave velocity and particle velocity for transverse wave are mutually perpendi cular to each other.
(R): The wave velocity and particle velocity have a constant ratio of their magnitudes

Options

A. Both 'A' and 'R' are true and 'R' is the correct, explanation of 'A'.
B. Both 'A' and 'R' are true and 'R' is not the correct explanation of 'A'
C. A' is true and 'R' is false
D. Both 'A' and 'R' are false

Answer

To analyze the assertion and reason provided in the question, we will break down the concepts of wave velocity and particle velocity in transverse waves step by step.

### Step-by-Step Solution:

1. **Understanding Wave Velocity and Particle Velocity**:
- In a transverse wave, the wave travels in a direction (let's say along the y-axis), while the particles of the medium oscillate perpendicular to this direction (along the x-axis).
- Therefore, the wave velocity (v) is directed along the y-axis, and the particle velocity (u) is directed along the x-axis.

2. **Assertion Analysis**:
- The assertion states that the wave velocity and particle velocity for transverse waves are mutually perpendicular to each other.
- Since wave velocity is in the direction of wave propagation and particle velocity is in the direction of particle oscillation, they indeed are perpendicular.
- **Conclusion**: The assertion (A) is true.

3. **Reason Analysis**:
- The reason states that the wave velocity and particle velocity have a constant ratio of their magnitudes.
- The wave velocity (v) can be expressed as \( v = \lambda f \), where \( \lambda \) is the wavelength and \( f \) is the frequency.
- The particle velocity (u) can be derived from the wave equation. For a simple harmonic wave, it can be expressed as \( u = \frac{dy}{dt} = A \omega \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( t \) is time.
- The ratio of wave velocity to particle velocity is given by \( \frac{v}{u} = \frac{\lambda f}{A \omega \cos(\omega t + \phi)} \).
- Since \( \lambda \), \( f \), and \( A \) are constants, but \( \cos(\omega t + \phi) \) varies with time, the ratio \( \frac{v}{u} \) is not constant.
- **Conclusion**: The reason (R) is false.

4. **Final Conclusion**:
- Since the assertion is true and the reason is false, the correct option is that A is true and R is false.