A wave travelling in positive X-direction with `A=0.2m` has a velocity of `360 m//sec` if `lamda=60m`, then correct exression for the wave is

Options

A. `y=0.2 sin [2pi(6t+x/60)]`
B. `y=0.2 sin [pi(6t+x/60)]`
C. `y=0.2 sin [2pi(6t-x/60)]`
D. `y=0.2 sin [pi(6t-x/60)]`

Answer

To find the correct expression for the wave traveling in the positive x-direction, we will follow these steps:

### Step 1: Identify the wave equation format
The general form of a wave traveling in the positive x-direction is given by:
\[ y = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Given values
From the question, we have:
- Amplitude \( A = 0.2 \, m \)
- Velocity \( V = 360 \, m/s \)
- Wavelength \( \lambda = 60 \, m \)

### Step 3: Calculate the wave number \( k \)
The wave number \( k \) is calculated using the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{60} = \frac{\pi}{30} \, \text{m}^{-1} \]

### Step 4: Calculate the angular frequency \( \omega \)
Using the relationship between velocity, angular frequency, and wave number:
\[ V = \frac{\omega}{k} \]
Rearranging gives:
\[ \omega = V \cdot k \]
Substituting the values of \( V \) and \( k \):
\[ \omega = 360 \cdot \frac{\pi}{30} \]
Calculating this:
\[ \omega = 360 \cdot \frac{\pi}{30} = 12\pi \, \text{rad/s} \]

### Step 5: Substitute values into the wave equation
Now we can substitute \( A \), \( \omega \), and \( k \) into the wave equation:
\[ y = 0.2 \sin(12\pi t - \frac{\pi}{30} x) \]

### Step 6: Simplify the expression
To express it in a more standard form, we can factor out \( 2\pi \):
\[ y = 0.2 \sin\left(12\pi t - \frac{\pi}{30} x\right) \]
This can be rewritten as:
\[ y = 0.2 \sin\left(2\pi (6t - \frac{x}{60})\right) \]

### Final Expression
Thus, the correct expression for the wave is:
\[ y = 0.2 \sin\left(2\pi \left(6t - \frac{x}{60}\right)\right) \]