two particle of medium disturbed by the wave propagation are at `x_(1)=0 and x_(2)=1 cm`. The respective displacement (in cm) of the particles can be given by the equation:
`y_(1)=2 sin 3pi t, y_(2) sin (3pi t-pi//8)` the wave velocity is

Options

A. `16 cm//s`
B. `24 cm//s`
C. `12 cm//s`
D. `8 cm//s`

Explanation

To find the wave velocity given the displacements of two particles in a medium, we can follow these steps:

### Step 1: Identify the equations of the displacements
The displacements of the particles at positions \( x_1 = 0 \) cm and \( x_2 = 1 \) cm are given by:
- \( y_1 = 2 \sin(3\pi t) \)
- \( y_2 = \sin(3\pi t - \frac{\pi}{8}) \)

### Step 2: Compare with the standard wave equation
The standard equation for a traveling wave can be expressed as:
\[ y = a \sin(\omega t - kx) \]
where:
- \( a \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 3: Determine the angular frequency \( \omega \)
From the equation for \( y_1 \):
\[ y_1 = 2 \sin(3\pi t) \]
we can see that:
- \( \omega = 3\pi \)

### Step 4: Determine the wave number \( k \)
From the equation for \( y_2 \):
\[ y_2 = \sin(3\pi t - \frac{\pi}{8}) \]
we can rewrite it in the form \( y = a \sin(\omega t - kx) \) by noting that the term \( -\frac{\pi}{8} \) corresponds to the spatial part. Since \( y_2 \) is given at \( x_2 = 1 \) cm, we can express it as:
\[ y_2 = \sin(3\pi t - k \cdot 1) \]
This implies:
\[ k = 3\pi - \frac{\pi}{8} \]

### Step 5: Calculate the wave number \( k \)
To find \( k \), we need to express it in terms of the wave number:
From the equation:
\[ k = \frac{\omega}{v} \]
We can also find \( k \) directly from the wave number relationship. However, since we already have \( \omega \), we can find \( k \) from the given displacement equations.

### Step 6: Calculate the wave velocity \( v \)
The wave velocity \( v \) is given by the formula:
\[ v = \frac{\omega}{k} \]

Substituting the values we found:
- \( \omega = 3\pi \)
- \( k = \frac{\pi}{8} \)

Thus:
\[ v = \frac{3\pi}{\frac{\pi}{8}} = 3\pi \cdot \frac{8}{\pi} = 24 \, \text{cm/s} \]

### Conclusion
The wave velocity is \( 24 \, \text{cm/s} \).