Two successive resonance frequencies in an open organ pipe are 1944 Hz and 2592 Hz. Find the length of the tube. The speed of sound in air is `324 m s^-1`.

Explanation

To find the length of the open organ pipe given the two successive resonance frequencies, we can follow these steps:

### Step 1: Understand the relationship between frequency and length
For an open organ pipe, the resonant frequency \( f \) is given by the formula:
\[ f_n = \frac{n \cdot V}{2L} \]
where:
- \( f_n \) is the frequency of the nth harmonic,
- \( n \) is the harmonic number (1, 2, 3,...),
- \( V \) is the speed of sound in air,
- \( L \) is the length of the pipe.

### Step 2: Set up the equations for the two frequencies
Let the first resonance frequency \( f_1 = 1944 \, \text{Hz} \) and the second resonance frequency \( f_2 = 2592 \, \text{Hz} \).
We can express these frequencies using the formula:
\[ f_1 = \frac{n \cdot V}{2L} \]
\[ f_2 = \frac{(n+1) \cdot V}{2L} \]

### Step 3: Subtract the two equations
Subtract the first equation from the second:
\[ f_2 - f_1 = \frac{(n+1) \cdot V}{2L} - \frac{n \cdot V}{2L} \]
This simplifies to:
\[ f_2 - f_1 = \frac{V}{2L} \]

### Step 4: Calculate the difference in frequencies
Now, calculate the difference:
\[ f_2 - f_1 = 2592 \, \text{Hz} - 1944 \, \text{Hz} = 648 \, \text{Hz} \]

### Step 5: Substitute the values into the equation
Now we can substitute this difference into the equation:
\[ 648 = \frac{324}{2L} \]

### Step 6: Rearrange to find \( L \)
Rearranging the equation gives:
\[ 2L = \frac{324}{648} \]
\[ L = \frac{324}{2 \times 648} \]
\[ L = \frac{324}{1296} = 0.25 \, \text{m} \]

### Step 7: Final answer
Thus, the length of the tube \( L \) is:
\[ L = 0.25 \, \text{m} \, \text{or} \, 25 \, \text{cm} \]