For a certain organ pipe , three successive resonance observed are `425 , 595 and 765 Hz`. Taking the speed of sound to be `340 ms^(-1)` , find the length of the pipe , in meter .

Explanation

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

### Step 1: Identify the Frequencies
We are given three successive resonance frequencies:
- \( f_1 = 425 \, \text{Hz} \)
- \( f_2 = 595 \, \text{Hz} \)
- \( f_3 = 765 \, \text{Hz} \)

### Step 2: Determine the Ratio of Frequencies
The frequencies are in an odd number ratio. We can express them as:
- \( f_1 : f_2 : f_3 = 5 : 7 : 9 \)

### Step 3: Find the Fundamental Frequency
To find the fundamental frequency \( f \), we can use the first frequency \( f_1 \):
\[ f = \frac{425 \, \text{Hz}}{5} = 85 \, \text{Hz} \]

### Step 4: Use the Speed of Sound
We are given the speed of sound \( v = 340 \, \text{m/s} \).

### Step 5: Relate Frequency and Length of the Pipe
For a closed organ pipe, the fundamental frequency is given by:
\[ f = \frac{v}{4L} \]
where \( L \) is the length of the pipe.

### Step 6: Rearrange the Formula to Find Length
Rearranging the formula to solve for \( L \):
\[ L = \frac{v}{4f} \]

### Step 7: Substitute Values
Now, substitute the values of \( v \) and \( f \):
\[ L = \frac{340 \, \text{m/s}}{4 \times 85 \, \text{Hz}} = \frac{340}{340} = 1 \, \text{m} \]

### Final Answer
The length of the organ pipe is \( L = 1 \, \text{meter} \).

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