If `lambda_(1), lambda_(2), lambda_(3)` are the wavelengths of the waves giving resonance in the fundamental, first and second overtone modes respecively in a open organ pipe, then the ratio of the wavelengths `lambda_(1) : lambda_(2) : lambda_(3)`, is :

Options

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

Explanation

To find the ratio of the wavelengths \( \lambda_1 : \lambda_2 : \lambda_3 \) for the fundamental frequency and the first and second overtone modes in an open organ pipe, we can follow these steps:

### Step 1: Understand the relationship between frequency and wavelength
In an open organ pipe, the fundamental frequency (first harmonic) corresponds to one wavelength fitting in the length of the pipe, while the first overtone (second harmonic) corresponds to two wavelengths fitting in the length of the pipe, and the second overtone (third harmonic) corresponds to three wavelengths fitting in the length of the pipe.

### Step 2: Write the equations for the wavelengths
For an open organ pipe:
- Fundamental frequency (first harmonic):
\[ L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L \]
- First overtone (second harmonic):
\[ L = \frac{2\lambda_2}{2} \implies \lambda_2 = L \]
- Second overtone (third harmonic):
\[ L = \frac{3\lambda_3}{2} \implies \lambda_3 = \frac{2L}{3} \]

### Step 3: Write the ratio of the wavelengths
Now we can express the wavelengths in terms of \( L \):
- \( \lambda_1 = 2L \)
- \( \lambda_2 = L \)
- \( \lambda_3 = \frac{2L}{3} \)

Now, we can write the ratio:
\[ \lambda_1 : \lambda_2 : \lambda_3 = 2L : L : \frac{2L}{3} \]

### Step 4: Simplify the ratio
To simplify the ratio, we can eliminate \( L \) from all terms:
\[ = 2 : 1 : \frac{2}{3} \]

To express all terms with a common denominator, we can multiply through by 3:
\[ = 2 \times 3 : 1 \times 3 : \frac{2}{3} \times 3 = 6 : 3 : 2 \]

### Step 5: Final ratio
Thus, the final ratio of the wavelengths is:
\[ \lambda_1 : \lambda_2 : \lambda_3 = 6 : 3 : 2 \]