If `lambda_(1), lamda_(2)and lamda_(3)` are the wavelengths of the wave giving resonance with the fundamental, first and second overtones respectively of a closed organ pipe Then the ratio of wavelength `lambda_(1), lamda_(2)and lamda_(3)` is

Options

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

Answer

To solve the problem of finding the ratio of wavelengths \( \lambda_1, \lambda_2, \) and \( \lambda_3 \) for a closed organ pipe, we can follow these steps:

### Step 1: Understand the Resonance Conditions
For a closed organ pipe, the fundamental frequency (first harmonic) has one node at the closed end and one antinode at the open end. The wavelengths for the fundamental and overtones can be described as follows:
- **Fundamental (1st harmonic)**: \( \lambda_1 = 4L \)
- **First overtone (2nd harmonic)**: \( \lambda_2 = \frac{4L}{3} \)
- **Second overtone (3rd harmonic)**: \( \lambda_3 = \frac{4L}{5} \)

### Step 2: Set Up the Equations
From the resonance conditions:
1. For the fundamental mode:
\[ L = \frac{\lambda_1}{4} \quad \Rightarrow \quad \lambda_1 = 4L \]

2. For the first overtone:
\[ L = \frac{3\lambda_2}{4} \quad \Rightarrow \quad \lambda_2 = \frac{4L}{3} \]

3. For the second overtone:
\[ L = \frac{5\lambda_3}{4} \quad \Rightarrow \quad \lambda_3 = \frac{4L}{5} \]

### Step 3: Find the Ratios
Now we can find the ratios of the wavelengths:
- From the equations derived:
- \( \lambda_1 = 4L \)
- \( \lambda_2 = \frac{4L}{3} \)
- \( \lambda_3 = \frac{4L}{5} \)

We can express the ratios \( \lambda_1 : \lambda_2 : \lambda_3 \) as follows:
\[ \frac{\lambda_1}{\lambda_2} = \frac{4L}{\frac{4L}{3}} = 3 \]
\[ \frac{\lambda_2}{\lambda_3} = \frac{\frac{4L}{3}}{\frac{4L}{5}} = \frac{5}{3} \]

### Step 4: Combine the Ratios
We can combine these ratios:
- Let \( \lambda_1 : \lambda_2 = 3 : 1 \)
- Let \( \lambda_2 : \lambda_3 = 5 : 3 \)

Now, we can express all three wavelengths in terms of a common variable:
- Let \( \lambda_2 = 1k \)
- Then \( \lambda_1 = 3k \)
- And \( \lambda_3 = \frac{3}{5}k \)

To express them in whole numbers, we can multiply through by 15 (the least common multiple of the denominators):
- \( \lambda_1 = 9 \)
- \( \lambda_2 = 5 \)
- \( \lambda_3 = 3 \)

Thus, the final ratio is:
\[ \lambda_1 : \lambda_2 : \lambda_3 = 15 : 9 : 5 \]

### Final Answer
The ratio of wavelengths \( \lambda_1 : \lambda_2 : \lambda_3 \) is \( 15 : 9 : 5 \).

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