The equation of the progressive wave is `y = 3 sin[pi(t/3-x/5)+pi/4]` , where x and y are in metre and time in second. Which of the following is correct.

Options

A. Velocity `V = 1.5 m//s`
B. Amplitude `A = 3 cm`
C. Frequency `F = 0.2 Hz`
D. Wavelength `lambda = 10m`

Answer

To solve the problem, we need to analyze the given wave equation and extract relevant parameters such as amplitude, angular frequency, wave number, wavelength, wave velocity, and frequency.

### Step-by-Step Solution:

1. **Identify the wave equation**:
The given wave equation is:
\[ y = 3 \sin\left[\pi\left(\frac{t}{3} - \frac{x}{5}\right) + \frac{\pi}{4}\right] \]

2. **Rewrite the wave equation in standard form**:
The standard form of a progressive wave is:
\[ y = A \sin(\omega t - kx + \phi) \]
Here, \(A\) is the amplitude, \(\omega\) is the angular frequency, \(k\) is the wave number, and \(\phi\) is the phase constant.

3. **Extract parameters from the equation**:
From the equation, we can identify:
- Amplitude \(A = 3\) m
- The term inside the sine function can be rearranged to match the standard form:
\[ \omega t - kx + \phi = \pi\left(\frac{t}{3} - \frac{x}{5}\right) + \frac{\pi}{4} \]
- This gives:
- \(\omega = \frac{\pi}{3}\)
- \(k = \frac{\pi}{5}\)
- \(\phi = \frac{\pi}{4}\)

4. **Calculate the wavelength**:
The wavelength \(\lambda\) is given by:
\[ \lambda = \frac{2\pi}{k} \]
Substituting \(k = \frac{\pi}{5}\):
\[ \lambda = \frac{2\pi}{\frac{\pi}{5}} = 10 \text{ m} \]

5. **Calculate the wave velocity**:
The wave velocity \(v\) is given by:
\[ v = \frac{\omega}{k} \]
Substituting the values of \(\omega\) and \(k\):
\[ v = \frac{\frac{\pi}{3}}{\frac{\pi}{5}} = \frac{5}{3} \approx 1.67 \text{ m/s} \]

6. **Calculate the frequency**:
The frequency \(f\) is given by:
\[ f = \frac{\omega}{2\pi} \]
Substituting \(\omega = \frac{\pi}{3}\):
\[ f = \frac{\frac{\pi}{3}}{2\pi} = \frac{1}{6} \approx 0.167 \text{ Hz} \]

7. **Summarize the results**:
- Amplitude \(A = 3\) m
- Wavelength \(\lambda = 10\) m
- Wave velocity \(v \approx 1.67\) m/s
- Frequency \(f \approx 0.167\) Hz

8. **Identify the correct option**:
Based on the calculated values, we can check the options provided in the question. The correct option is the one that matches the wavelength of 10 m.