Which of the following is true if , equation of progressive wave is `y = a sin[10pi x + 11pit + pi/3]` ?

Options

A. Its wavelength is 0.2 units
B. It is travelling in the positive x-direction
C. Wave velocity is 1.5 unit
D. Time period of SHM is 1 s

Answer

To analyze the given equation of the progressive wave \( y = a \sin(10\pi x + 11\pi t + \frac{\pi}{3}) \), we will identify the parameters involved, including the amplitude, angular frequency, wave number, wavelength, wave velocity, and time period.

### Step 1: Identify the parameters from the wave equation

The general form of a progressive wave 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.

From the given equation \( y = a \sin(10\pi x + 11\pi t + \frac{\pi}{3}) \), we can identify:
- \( k = 10\pi \)
- \( \omega = 11\pi \)
- \( \phi = \frac{\pi}{3} \)

### Step 2: Calculate the wavelength

The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:

\[ k = \frac{2\pi}{\lambda} \]

Substituting the value of \( k \):

\[ 10\pi = \frac{2\pi}{\lambda} \]

To find \( \lambda \), rearranging gives:

\[ \lambda = \frac{2\pi}{10\pi} = \frac{1}{5} \text{ units} \]

### Step 3: Calculate the time period

The angular frequency \( \omega \) is related to the time period \( T \) by the formula:

\[ \omega = \frac{2\pi}{T} \]

Substituting the value of \( \omega \):

\[ 11\pi = \frac{2\pi}{T} \]

Rearranging gives:

\[ T = \frac{2\pi}{11\pi} = \frac{2}{11} \text{ seconds} \]

### Step 4: Calculate the wave velocity

The wave velocity \( v \) can be calculated using the relationship:

\[ v = f \lambda \]

where \( f \) is the frequency. The frequency \( f \) can be calculated from the time period \( T \):

\[ f = \frac{1}{T} = \frac{11}{2} \text{ Hz} \]

Now substituting \( f \) and \( \lambda \):

\[ v = f \lambda = \left(\frac{11}{2}\right) \left(\frac{1}{5}\right) = \frac{11}{10} \text{ units/s} \]

### Conclusion

From the analysis, we have determined:
- Wavelength \( \lambda = \frac{1}{5} \) units
- Time period \( T = \frac{2}{11} \) seconds
- Wave velocity \( v = \frac{11}{10} \) units/s