A wave is represented by the equation `y=[Asin{10pix+15pit+(pi//4)}]` where x is in meters and t is in seconds. The expression represents

Options

A. A wave travelling in positive x-direction with a velocity 1.5 m/s
B. A wave travelling in negative x-direction with a velocity 1.5 m/s
C. A wave travelling in the negative x-direction having a wavelength 2 m
D. A wave travelling in positive x-direction having a wavelength 2m

Answer

To solve the problem, we need to analyze the given wave equation and extract the relevant parameters to determine the direction of the wave and its velocity.

### Step-by-Step Solution:

1. **Identify the Wave Equation**:
The given wave equation is:
\[ y = A \sin(10\pi x + 15\pi t + \frac{\pi}{4}) \]
Here, \(A\) is the amplitude, \(10\pi\) is the coefficient of \(x\), \(15\pi\) is the coefficient of \(t\), and \(\frac{\pi}{4}\) is the phase constant.

2. **Compare with the Standard Wave Equation**:
The standard form of a wave equation can be written as:
\[ y = A \sin(kx + \omega t + \phi) \]
where:
- \(k\) is the wave number,
- \(\omega\) is the angular frequency,
- \(\phi\) is the phase constant.

By comparing, we identify:
- \(k = 10\pi\)
- \(\omega = 15\pi\)
- \(\phi = \frac{\pi}{4}\)

3. **Determine the Direction of the Wave**:
The sign in front of the \(kx\) term indicates the direction of wave propagation:
- If the equation is of the form \(kx + \omega t\), the wave travels in the negative x-direction.
- If the equation is of the form \(kx - \omega t\), the wave travels in the positive x-direction.

In our case, since we have \(10\pi x + 15\pi t\), the wave is traveling in the **negative x-direction**.

4. **Calculate the Velocity of the Wave**:
The velocity \(v\) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \(\omega\) and \(k\):
\[ v = \frac{15\pi}{10\pi} = \frac{15}{10} = 1.5 \text{ m/s} \]

5. **Final Conclusion**:
The wave travels in the negative x-direction with a velocity of \(1.5 \text{ m/s}\).

### Summary of Results:
- **Direction of Wave**: Negative x-direction
- **Velocity of Wave**: \(1.5 \text{ m/s}\)