If the equation of progressive wave is given by `y=4 sin pi[(t)/(5)-(x)/(9)+(pi)/(6)]` then, which of the following is correct `?` (Assume SI units )

Options

A. v= 5cm
B. `lamda = 18 cm`
C. a = 0.04 cm
D. f =50 Hz

Answer

To solve the problem, we need to analyze the given wave equation and extract the necessary parameters such as amplitude, wavelength, velocity, and frequency. The equation provided is:

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

### Step 1: Identify the Amplitude
The amplitude \( A \) of the wave is the coefficient in front of the sine function. From the equation:

\[ A = 4 \, \text{meters} \]

### Step 2: Identify the Angular Frequency and Wavenumber
The general form of a wave equation is:

\[ y = A \sin \left( \omega t - kx + \phi_0 \right) \]

Where:
- \( \omega \) is the angular frequency,
- \( k \) is the wavenumber,
- \( \phi_0 \) is the phase constant.

From the equation, we can rewrite it as:

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

This gives us:

- \( \omega = \frac{\pi}{5} \)
- \( k = \frac{\pi}{9} \)

### Step 3: Calculate the Wavelength
The wavenumber \( k \) is related to the wavelength \( \lambda \) by the formula:

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

Rearranging gives:

\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{\frac{\pi}{9}} = 18 \, \text{meters} \]

### Step 4: Calculate the Velocity of the Wave
The velocity \( v \) of the wave can be calculated using the relationship:

\[ v = \frac{\omega}{k} \]

Substituting the values we found:

\[ v = \frac{\frac{\pi}{5}}{\frac{\pi}{9}} = \frac{9}{5} \, \text{m/s} \]

### Step 5: Calculate the Frequency
The frequency \( f \) is given by the formula:

\[ f = \frac{v}{\lambda} \]

Substituting the values of \( v \) and \( \lambda \):

\[ f = \frac{\frac{9}{5}}{18} = \frac{9}{90} = 0.1 \, \text{Hz} \]

### Summary of Results
- Amplitude \( A = 4 \, \text{m} \)
- Wavelength \( \lambda = 18 \, \text{m} \)
- Velocity \( v = \frac{9}{5} \, \text{m/s} \)
- Frequency \( f = 0.1 \, \text{Hz} \)

### Conclusion
Based on the calculations, we can determine the correct option from the given choices.

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