The first resonance length of a resonance tube is `40 cm` and the second resonance length is `122 cm` . The third resonance length of the tube will be

Options

A. `200 cm`
B. `202 cm`
C. `203 cm`
D. `204 cm`

Explanation

To find the third resonance length of the resonance tube, we can follow these steps:

### Step 1: Understand the concept of resonance lengths
In a resonance tube, the resonance lengths correspond to the positions where standing waves are formed. The first resonance length (L1) corresponds to the first harmonic, the second resonance length (L2) corresponds to the third harmonic, and so on. The difference between consecutive resonance lengths is related to the wavelength (λ) of the sound wave.

### Step 2: Identify the given values
From the question, we have:
- First resonance length (L1) = 40 cm
- Second resonance length (L2) = 122 cm

### Step 3: Calculate the difference between the resonance lengths
The difference between the second and first resonance lengths gives us the value of half the wavelength (λ/2):
\[ \Delta L = L2 - L1 = 122 \, \text{cm} - 40 \, \text{cm} = 82 \, \text{cm} \]
This means:
\[ \frac{\lambda}{2} = 82 \, \text{cm} \]
Thus, the wavelength (λ) can be calculated as:
\[ \lambda = 2 \times 82 \, \text{cm} = 164 \, \text{cm} \]

### Step 4: Find the third resonance length
The third resonance length (L3) can be found by adding the wavelength (λ/2) to the second resonance length (L2):
\[ L3 = L2 + \frac{\lambda}{2} \]
Substituting the values:
\[ L3 = 122 \, \text{cm} + 82 \, \text{cm} = 204 \, \text{cm} \]

### Step 5: Conclusion
The third resonance length of the tube is:
\[ \boxed{204 \, \text{cm}} \]

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