Equation of a progressive wave is given by, `y=4sin[pi((t)/(5)-(x)/(9))+(pi)/(6)]` where x and y are in metre. Then :

Options

A. `(a)v=1m//s`
B. (b)`lamda=18m`
C. (c)`A=0.04m`
D. (d)`f=50Hz`

Explanation

To solve the given problem, we need to analyze the equation of the progressive wave provided:

**Given Equation:**
\[ y = 4 \sin \left[ \pi \left( \frac{t}{5} - \frac{x}{9} \right) + \frac{\pi}{6} \right] \]

### Step 1: Identify the General Form
The general form of a progressive wave can be expressed as:
\[ y = A \sin(\omega t - kx + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( \phi \) is the phase constant.

### Step 2: Compare the Given Equation with the General Form
From the given equation, we can rewrite it as:
\[ y = 4 \sin \left( \pi \cdot \frac{t}{5} - \pi \cdot \frac{x}{9} + \frac{\pi}{6} \right) \]

Now, we can identify:
- Amplitude \( A = 4 \)
- Angular frequency \( \omega = \pi \cdot \frac{1}{5} = \frac{\pi}{5} \)
- Wave number \( k = \pi \cdot \frac{1}{9} = \frac{\pi}{9} \)
- Phase constant \( \phi = \frac{\pi}{6} \)

### Step 3: Calculate the Wavelength
The wave number \( k \) is related to the wavelength \( \lambda \) by the equation:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ \frac{\pi}{9} = \frac{2\pi}{\lambda} \]

To find \( \lambda \), rearranging gives:
\[ \lambda = \frac{2\pi}{\frac{\pi}{9}} = 2 \cdot 9 = 18 \text{ meters} \]

### Step 4: Calculate the Velocity of the Wave
The velocity \( v \) of the wave is given by:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{\frac{\pi}{5}}{\frac{\pi}{9}} = \frac{9}{5} = 1.8 \text{ meters/second} \]

### Step 5: Calculate the Frequency of the Wave
The frequency \( f \) can be calculated using:
\[ f = \frac{\omega}{2\pi} \]
Substituting the value of \( \omega \):
\[ f = \frac{\frac{\pi}{5}}{2\pi} = \frac{1}{10} = 0.1 \text{ Hz} \]

### Summary of Results
- Amplitude \( A = 4 \) meters
- Wavelength \( \lambda = 18 \) meters
- Velocity \( v = 1.8 \) meters/second
- Frequency \( f = 0.1 \) Hz

### Conclusion
After comparing the calculated values with the options provided in the question, we can determine which option is correct.