The equation of the progressive wave, where t is the time in second , x is the distance in metre is `y= A cos 240 (t-(x)/(12))` . The phase difference ( in SI units ) between two position 0.5 m apart is

Options

A. 40
B. 20
C. 10
D. 5

Answer

To find the phase difference between two positions 0.5 m apart for the given wave equation \( y = A \cos(240(t - \frac{x}{12})) \), we can follow these steps:

### Step 1: Identify the wave parameters
The wave equation is given in the form:
\[ y = A \cos(\omega t - kx) \]
From the equation \( y = A \cos(240(t - \frac{x}{12})) \), we can identify:
- Angular frequency \( \omega = 240 \, \text{rad/s} \)
- Wave number \( k = \frac{2\pi}{\lambda} \)

### Step 2: Determine the wavelength \( \lambda \)
We can find the wave number \( k \) from the equation:
\[ k = \frac{2\pi}{\lambda} \]
From the equation, we can see that:
\[ k = \frac{240}{12} = 20 \, \text{rad/m} \]
Now, we can find the wavelength \( \lambda \):
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{20} = \frac{\pi}{10} \, \text{m} \]

### Step 3: Calculate the phase difference \( \Delta \phi \)
The phase difference \( \Delta \phi \) between two points separated by a distance \( \Delta x \) is given by:
\[ \Delta \phi = k \Delta x \]
Here, \( \Delta x = 0.5 \, \text{m} \):
\[ \Delta \phi = 20 \times 0.5 = 10 \, \text{radians} \]

### Final Answer
The phase difference between two positions 0.5 m apart is:
\[ \Delta \phi = 10 \, \text{radians} \]

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