If `I_(1), I_(2) and I_(3)` are wave lengths of the waves giving resonance with fundamental, first and second over tones of closed organ pipe. The ratio of wavelengths `I_(1):I_(2):I_(3)` is.....

Options

A. `1:2:3`
B. `1:1/3:1/5`
C. `1:3:5`
D. `5:3:1`

Answer

To find the ratio of wavelengths \( I_1 : I_2 : I_3 \) for the fundamental frequency and the first and second overtones of a closed organ pipe, we can follow these steps:

### Step 1: Understand the relationship between frequency and wavelength
The relationship between frequency \( f \), wavelength \( \lambda \), and wave speed \( v \) is given by the equation:
\[ v = f \cdot \lambda \]
From this, we can express wavelength as:
\[ \lambda = \frac{v}{f} \]

### Step 2: Identify the frequencies for the fundamental and overtones
For a closed organ pipe, the frequencies of the fundamental frequency and the overtones can be expressed as:
- Fundamental frequency (\( n = 1 \)):
\[ f_1 = \frac{1 \cdot v}{4L} \]
- First overtone (\( n = 2 \)):
\[ f_2 = \frac{3 \cdot v}{4L} \]
- Second overtone (\( n = 3 \)):
\[ f_3 = \frac{5 \cdot v}{4L} \]
Where \( L \) is the length of the pipe.

### Step 3: Write the ratios of frequencies
Now we can write the ratio of frequencies:
\[ f_1 : f_2 : f_3 = 1 : 3 : 5 \]

### Step 4: Find the ratios of wavelengths
Using the relationship \( \lambda = \frac{v}{f} \), we can find the wavelengths:
- Wavelength for fundamental frequency (\( I_1 \)):
\[ I_1 = \frac{v}{f_1} = \frac{v}{\frac{1 \cdot v}{4L}} = 4L \]
- Wavelength for first overtone (\( I_2 \)):
\[ I_2 = \frac{v}{f_2} = \frac{v}{\frac{3 \cdot v}{4L}} = \frac{4L}{3} \]
- Wavelength for second overtone (\( I_3 \)):
\[ I_3 = \frac{v}{f_3} = \frac{v}{\frac{5 \cdot v}{4L}} = \frac{4L}{5} \]

### Step 5: Write the ratio of wavelengths
Now we can write the ratio of wavelengths:
\[ I_1 : I_2 : I_3 = 4L : \frac{4L}{3} : \frac{4L}{5} \]

### Step 6: Simplify the ratio
To simplify this ratio, we can eliminate \( 4L \) from each term:
\[ 1 : \frac{1}{3} : \frac{1}{5} \]
To express this in whole numbers, we can find a common denominator, which is 15:
\[ 1 \cdot 15 : \frac{1}{3} \cdot 15 : \frac{1}{5} \cdot 15 = 15 : 5 : 3 \]

Thus, the final ratio of wavelengths is:
\[ I_1 : I_2 : I_3 = 15 : 5 : 3 \]

### Final Answer
The ratio of wavelengths \( I_1 : I_2 : I_3 \) is \( 15 : 5 : 3 \).