A wave equation is given by `y=4 sin[pi((t)/(5)-(x)/(9)+(1)/(6))]` where x is in cm and t is in second. Which of the following is true ?

Options

A. `lambda=18cm`
B. v = 4 m/s
C. a = 0.4 m
D. f = 50 Hz

Explanation

To solve the given wave equation \( y = 4 \sin\left[\pi\left(\frac{t}{5} - \frac{x}{9} + \frac{1}{6}\right)\right] \), we will compare it with the standard wave equation:

\[ y = A \sin(\omega t - kx + \phi) \]

where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( \phi \) is the phase constant.

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

From the given equation, we can identify:
- Amplitude \( A = 4 \) cm
- The term inside the sine function can be rewritten as:
\[ \omega t - kx + \phi = \pi\left(\frac{t}{5}\right) - \pi\left(\frac{x}{9}\right) + \pi\left(\frac{1}{6}\right) \]

### Step 2: Calculate \( \omega \) and \( k \)

From the equation, we can extract:
- \( \omega = \frac{\pi}{5} \)
- \( k = \frac{\pi}{9} \)

### Step 3: Calculate the wavelength \( \lambda \)

The wavelength \( \lambda \) is given by the formula:
\[ \lambda = \frac{2\pi}{k} \]
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{\frac{\pi}{9}} = 2 \times 9 = 18 \text{ cm} \]

### Step 4: Calculate the velocity \( v \)

The velocity \( v \) of the wave can be calculated using:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{\frac{\pi}{5}}{\frac{\pi}{9}} = \frac{9}{5} = 1.8 \text{ cm/s} \]

### Step 5: Calculate the frequency \( f \)

The frequency \( f \) is related to the angular frequency \( \omega \) by:
\[ f = \frac{\omega}{2\pi} \]
Substituting the value of \( \omega \):
\[ f = \frac{\frac{\pi}{5}}{2\pi} = \frac{1}{10} = 0.1 \text{ Hz} \]

### Step 6: Summary of results

- Amplitude \( A = 4 \) cm
- Wavelength \( \lambda = 18 \) cm
- Velocity \( v = 1.8 \) cm/s
- Frequency \( f = 0.1 \) Hz

### Conclusion

From the calculations, we find that the only correct statement is regarding the wavelength \( \lambda = 18 \) cm.