When a wave travels from one medium to another, the displacement of the particle is given as `y = a sin 2pi (mt - nx)` m and n are constants.

Options

A. The velocity of wave is `(m)/(n)`
B. The frequency of wave is m.
C. The maximum particle velocity is thrice the wave velocity if `n = (1)/(5 pi a^(2))`
D. The wavelength of the wave is 1/n

Explanation

To solve the problem, we need to analyze the wave equation given and derive the necessary quantities step by step.

### Given:
The displacement of the particle is given by the equation:
\[ y = a \sin(2\pi (mt - nx)) \]
where \( m \) and \( n \) are constants.

### Step 1: Identify Angular Frequency and Frequency
The standard form of a wave equation is:
\[ y = A \sin(\omega t - kx) \]
where:
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

From the given equation, we can identify:
- \( \omega = 2\pi m \)
- \( k = 2\pi n \)

The frequency \( f \) can be derived from the angular frequency:
\[ f = \frac{\omega}{2\pi} = m \]

### Step 2: Calculate Wave Velocity
The wave velocity \( v \) can be calculated using the relationship between angular frequency and wave number:
\[ v = \frac{\omega}{k} = \frac{2\pi m}{2\pi n} = \frac{m}{n} \]

### Step 3: Calculate Wavelength
The wavelength \( \lambda \) is related to the wave number \( k \):
\[ k = \frac{2\pi}{\lambda} \]
Thus,
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{2\pi n} = \frac{1}{n} \]

### Step 4: Calculate Particle Velocity
The particle velocity \( v_p \) can be derived from the wave velocity and the derivative of the displacement with respect to time:
\[ v_p = -v \frac{dy}{dx} \]

First, we need to find \( \frac{dy}{dx} \):
\[ y = a \sin(2\pi (mt - nx)) \]
Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = a \cdot \cos(2\pi (mt - nx)) \cdot (-2\pi n) = -2\pi n a \cos(2\pi (mt - nx)) \]

Now substituting into the particle velocity equation:
\[ v_p = -\left(\frac{m}{n}\right) \cdot (-2\pi n a \cos(2\pi (mt - nx))) = \frac{2\pi m a}{n} \cos(2\pi (mt - nx)) \]

### Step 5: Maximum Particle Velocity
The maximum particle velocity occurs when \( \cos(2\pi (mt - nx)) = 1 \):
\[ v_{p, \text{max}} = \frac{2\pi m a}{n} \]

### Step 6: Compare Particle Velocity and Wave Velocity
The question states that the maximum particle velocity is thrice the wave velocity:
\[ v_{p, \text{max}} = 3v \]
Substituting the values:
\[ \frac{2\pi m a}{n} = 3 \cdot \frac{m}{n} \]
This simplifies to:
\[ 2\pi a = 3 \]
Thus, \( a = \frac{3}{2\pi} \).

### Conclusion
From the analysis, we have derived the frequency, wave velocity, wavelength, and maximum particle velocity from the given wave equation. The relationship between the maximum particle velocity and the wave velocity shows that the statement about the maximum particle velocity being thrice the wave velocity is dependent on the value of \( a \).