A wave is represented by the equation `y= A sin ( 10 pi x + 15 pi l + (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 analyze the wave represented by the equation \( y = A \sin(10\pi x + 15\pi t + \frac{\pi}{6}) \), we will break down the equation step by step.

### Step 1: Identify the wave equation format
The standard wave equation is given by:
\[ y = A \sin(kx - \omega t + \phi) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

### Step 2: Compare the given equation with the standard form
The given equation is:
\[ y = A \sin(10\pi x + 15\pi t + \frac{\pi}{6}) \]
We can rewrite it in the standard form by recognizing that:
- The term \( 10\pi x \) corresponds to \( kx \),
- The term \( 15\pi t \) corresponds to \( -\omega t \).

This means that the wave is moving in the negative x direction because of the positive sign in front of \( \omega t \).

### Step 3: Determine the wave number \( k \) and angular frequency \( \omega \)
From the equation:
- \( k = 10\pi \)
- \( \omega = 15\pi \)

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

### Step 5: Calculate the wavelength \( \lambda \)
The wavelength can be found using the relation:
\[ k = \frac{2\pi}{\lambda} \]
Substituting for \( k \):
\[ 10\pi = \frac{2\pi}{\lambda} \]
Solving for \( \lambda \):
\[ \lambda = \frac{2\pi}{10\pi} = \frac{2}{10} = 0.2 \text{ m} \]

### Conclusion
The expression represents a wave traveling in the negative x direction with a velocity of \( 1.5 \text{ m/s} \) and a wavelength of \( 0.2 \text{ m} \).

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