The maximum particle velocity is 3 times the wave velocity of a progressive wave. If the amplitude of the particle is "a". The phase difference between the two particles seperated by a distance of 'x' is

Options

A. `x/a`
B. `(3x)/(a)`
C. `(3a)/(x)`
D. `(3pi x)/(a)`

Answer

To solve the problem step by step, we will analyze the given information and use relevant formulas from wave motion.

### Step 1: Understand the relationship between maximum particle velocity and wave velocity
We know that the maximum particle velocity \( v_{max} \) of a progressive wave is given by the formula:
\[ v_{max} = \omega A \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude of the wave.

The wave velocity \( v \) is given by:
\[ v = \frac{\omega}{k} \]
where \( k \) is the wave number.

### Step 2: Set up the equation based on the given information
According to the problem, the maximum particle velocity is 3 times the wave velocity:
\[ \omega A = 3 \left(\frac{\omega}{k}\right) \]

### Step 3: Simplify the equation
We can cancel \( \omega \) from both sides (assuming \( \omega \neq 0 \)):
\[ A = \frac{3}{k} \]
From this, we can express the wave number \( k \):
\[ k = \frac{3}{A} \]

### Step 4: Relate wave number to wavelength
The wave number \( k \) is also related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Setting the two expressions for \( k \) equal gives:
\[ \frac{2\pi}{\lambda} = \frac{3}{A} \]

### Step 5: Solve for the wavelength \( \lambda \)
Rearranging the equation to find \( \lambda \):
\[ \lambda = \frac{2\pi A}{3} \]

### Step 6: Calculate the phase difference
The phase difference \( \Delta \phi \) between two particles separated by a distance \( x \) is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot x \]
Substituting the expression for \( \lambda \):
\[ \Delta \phi = \frac{2\pi}{\frac{2\pi A}{3}} \cdot x = \frac{3x}{A} \]

### Final Answer
Thus, the phase difference between the two particles separated by a distance \( x \) is:
\[ \Delta \phi = \frac{3x}{A} \]

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