two particle of medium disturbed by the wave propagation are at `x_(1)=0cm 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"^(-1)`
B. `"24 cm s"^(-1)`
C. `"12 cm s"^(-1)`
D. `"8 cm s"^(-1)`

Answer

To find the wave velocity for the given problem, we will follow these steps:

### Step 1: Write the general wave equation
The general equation for wave displacement 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, \( t \) is time, and \( x \) is the position.

### Step 2: Analyze the first particle's displacement
For the first particle located at \( x_1 = 0 \) cm, the displacement is given by:
\[ y_1 = 2 \sin(3\pi t) \]
This can be compared with the general wave equation:
\[ y_1 = A \sin(\omega t - kx) \]
Here, since \( x = 0 \):
\[ y_1 = A \sin(\omega t) \]
From the equation, we can see that:
- Amplitude \( A = 2 \)
- Angular frequency \( \omega = 3\pi \) rad/s

### Step 3: Analyze the second particle's displacement
For the second particle located at \( x_2 = 1 \) cm, the displacement is given by:
\[ y_2 = \sin(3\pi t - \frac{\pi}{8}) \]
This can also be compared with the general wave equation:
\[ y_2 = A \sin(\omega t - kx) \]
Here, we can identify:
- Amplitude \( A = 1 \)
- Angular frequency \( \omega = 3\pi \)
- The phase shift indicates that \( kx = \frac{\pi}{8} \) when \( x = 1 \) cm.

### Step 4: Determine the wave number \( k \)
From the equation \( kx = \frac{\pi}{8} \) at \( x = 1 \) cm, we can find \( k \):
\[ k \cdot 1 = \frac{\pi}{8} \]
Thus,
\[ k = \frac{\pi}{8} \text{ rad/cm} \]

### Step 5: Calculate the wave velocity \( v \)
The wave velocity \( v \) is given by the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{3\pi}{\frac{\pi}{8}} \]
This simplifies to:
\[ v = 3\pi \cdot \frac{8}{\pi} = 24 \text{ cm/s} \]

### Final Answer
The wave velocity is:
\[ \boxed{24 \text{ cm/s}} \]