For a resonance tube, the air columns for the first and the second resonance differ in length by `31.5 cm`. The wavelength of the wave is

Options

A. `15.75 cm`
B. `31.5 cm`
C. `63.0 cm`
D. `126.0 cm`

Explanation

To find the wavelength of the wave in the resonance tube, we can follow these steps:

### Step 1: Understand the relationship between the lengths of the air columns and the wavelength.
In a resonance tube, the lengths of the air columns at the first and second resonance are related to the wavelength of the sound wave. The first resonance occurs at a length of \( \frac{\lambda}{4} \) and the second resonance occurs at a length of \( \frac{3\lambda}{4} \).

### Step 2: Set up the equation for the difference in lengths.
Given that the difference in lengths of the air columns for the first and second resonance is \( 31.5 \, \text{cm} \), we can express this mathematically:
\[ \frac{3\lambda}{4} - \frac{\lambda}{4} = 31.5 \, \text{cm} \]

### Step 3: Simplify the equation.
Now, simplify the left side of the equation:
\[ \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2} \]
So, we have:
\[ \frac{\lambda}{2} = 31.5 \, \text{cm} \]

### Step 4: Solve for the wavelength \( \lambda \).
To find \( \lambda \), multiply both sides of the equation by 2:
\[ \lambda = 2 \times 31.5 \, \text{cm} = 63 \, \text{cm} \]

### Conclusion:
The wavelength of the wave is \( \lambda = 63 \, \text{cm} \).

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