The equation of a longitudinal stationary wave produced in a closed organ pipe is
`y=6 sin (2pix)/(6) cos 160 pi t`
where x, y are in cm and t in second.Find (i)the frequency , amplitude and wavelength of the original progressive wave (ii)separation between two successive nodes and (iii)equation of the original progressive wave.

Answer

To solve the problem step by step, we will analyze the given equation of the stationary wave and extract the required information.

### Given:
The equation of the longitudinal stationary wave is:
\[ y = 6 \sin\left(\frac{2\pi x}{6}\right) \cos(160\pi t) \]
where \( x \) and \( y \) are in cm, and \( t \) is in seconds.

### Step 1: Identify the parameters of the wave equation
The general form of a stationary wave is:
\[ y = A \sin(kx) \cos(\omega t) \]
From the given equation, we can identify:
- Amplitude \( A = 6 \) cm
- Wave number \( k = \frac{2\pi}{6} = \frac{\pi}{3} \)
- Angular frequency \( \omega = 160\pi \)

### Step 2: Calculate the frequency
The frequency \( f \) can be calculated using the relationship:
\[ f = \frac{\omega}{2\pi} \]
Substituting the value of \( \omega \):
\[ f = \frac{160\pi}{2\pi} = 80 \text{ Hz} \]

### Step 3: Calculate the wavelength
The wavelength \( \lambda \) is related to the wave number \( k \) by the formula:
\[ \lambda = \frac{2\pi}{k} \]
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{\frac{\pi}{3}} = 6 \text{ cm} \]

### Step 4: Separation between two successive nodes
The separation between two successive nodes in a stationary wave is given by:
\[ \text{Separation} = \frac{\lambda}{2} \]
Substituting the value of \( \lambda \):
\[ \text{Separation} = \frac{6}{2} = 3 \text{ cm} \]

### Step 5: Write the equation of the original progressive wave
The original progressive wave can be expressed as:
\[ y = A \sin(kx - \omega t) \]
Using the values of \( A \), \( k \), and \( \omega \):
\[ y = 6 \sin\left(\frac{\pi}{3} x - 160\pi t\right) \]

### Summary of Results:
1. Frequency \( f = 80 \) Hz
2. Amplitude \( A = 6 \) cm
3. Wavelength \( \lambda = 6 \) cm
4. Separation between two successive nodes = 3 cm
5. Equation of the original progressive wave:
\[ y = 6 \sin\left(\frac{\pi}{3} x - 160\pi t\right) \]