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 can follow these steps:

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

### Step 2: Identify Given Values
From the problem, we have:
- Amplitude \( A = 0.2 \, \text{m} \)
- Wave speed \( v = 360 \, \text{m/s} \)
- Wavelength \( \lambda = 60 \, \text{m} \)

### 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 \( \lambda \):
\[ k = \frac{2\pi}{60} = \frac{\pi}{30} \, \text{rad/m} \]

### Step 4: Calculate the Angular Frequency \( \omega \)
The angular frequency \( \omega \) can be calculated using the relationship between wave speed, wavelength, and frequency:
\[ v = f \lambda \]
where \( f \) is the frequency. Rearranging gives:
\[ f = \frac{v}{\lambda} = \frac{360}{60} = 6 \, \text{Hz} \]
Now, we can find \( \omega \):
\[ \omega = 2\pi f = 2\pi \cdot 6 = 12\pi \, \text{rad/s} \]

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

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

### Final Expression
Thus, the correct expression for the wave is:
\[ y(x, t) = 0.2 \sin\left(\frac{\pi}{30} x - 12\pi t\right) \]