A wave is represented by the equation `y= A sin ( 10 pi x + 15 pi t + (pi)/(6))` wher x is in metre and t in second. The expression represents

Options

A. A wave travelling in negative x - direction with a velocity of 1.5 m/s
B. A wave travelling in positive x direction with a velocity of 1.5 m/s
C. A wave travelling in position x - direction with wavelength 0.2 m
D. A wave travelling in negative x-direction with a velocity of 150 m/s

Answer

To solve the problem, we need to analyze the wave equation given:

**Step 1: Identify the wave equation.**
The wave is represented by the equation:
\[ y = A \sin(10\pi x + 15\pi t + \frac{\pi}{6}) \]

**Step 2: Determine the wave parameters.**
In the general form of a wave equation, \( y = A \sin(kx \pm \omega t + \phi) \):
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

From the given equation:
- \( k = 10\pi \)
- \( \omega = 15\pi \)
- \( \phi = \frac{\pi}{6} \)

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

Since our equation has \( +\omega t \), it indicates that the wave is traveling in the **negative x-direction**.

**Step 4: Calculate the wave velocity.**
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]

Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{15\pi}{10\pi} \]

Now, simplifying this:
\[ v = \frac{15}{10} = 1.5 \, \text{m/s} \]

**Step 5: Summarize the findings.**
- The wave is traveling in the **negative x-direction**.
- The velocity of the wave is **1.5 m/s**.

Thus, the expression represents a wave traveling in the negative x-direction with a velocity of 1.5 m/s.