What is the fundamental frequency and second harmonic of a sound wave that propagates inside an open tube, if the air is at a temperature of 35 Celsius degrees? The length of the tube is 40 cm.

To determine the fundamental frequency and the second harmonic of a sound wave propagating inside an open tube, we need to consider the physical properties of the tube and the speed of sound in air at the given temperature.

### Step-by-Step Solution:

1. **Calculate the Speed of Sound in Air at 35°C:**

   The speed of sound in air can be approximated by the formula:
   \[
   v = 331.4 + 0.6 \cdot T
   \]
   where \( T \) is the temperature in Celsius.

   For \( T = 35^\circ \text{C} \):
   \[
   v = 331.4 + 0.6 \cdot 35 = 331.4 + 21 = 352.4 \text{ m/s}
   \]

2. **Determine the Wavelengths for the Fundamental and Second Harmonic Frequencies:**

   In an open tube, both ends are antinodes, and the tube supports standing waves at specific wavelengths.

   - The fundamental frequency (first harmonic) has a wavelength \( \lambda_1 \) that is twice the length of the tube:
     \[
     \lambda_1 = 2L
     \]
   - The second harmonic has a wavelength \( \lambda_2 \) that is the length of the tube:
     \[
     \lambda_2 = L
     \]

   Given the length of the tube \( L = 40 \text{ cm} = 0.4 \text{ m} \):
   \[
   \lambda_1 = 2 \cdot 0.4 = 0.8 \text{ m}
   \]
   \[
   \lambda_2 = 0.4 \text{ m}
   \]

3. **Calculate the Fundamental Frequency (First Harmonic):**

   The frequency \( f \) is related to the speed of sound \( v \) and the wavelength \( \lambda \) by:
   \[
   f = \frac{v}{\lambda}
   \]

   For the fundamental frequency:
   \[
   f_1 = \frac{v}{\lambda_1} = \frac{352.4}{0.8} = 440.5 \text{ Hz}
   \]

4. **Calculate the Second Harmonic Frequency:**

   For the second harmonic:
   \[
   f_2 = \frac{v}{\lambda_2} = \frac{352.4}{0.4} = 880.9 \text{ Hz}
   \]

### Summary:

- **Fundamental frequency (first harmonic):** \( 440.5 \text{ Hz} \)
- **Second harmonic:** \( 880.9 \text{ Hz} \)

Would you like more details on any part of this solution, or do you have any questions?

### Further Exploration:

1. How does the speed of sound change with temperature?
2. What are the differences in harmonic frequencies for open and closed tubes?
3. How would the fundamental frequency change if the tube length were doubled?
4. What are the implications of using different gases in the tube on the speed of sound?
5. How do environmental factors like humidity affect the speed of sound?

**Tip:** Always remember to convert all units to a consistent system (e.g., meters, seconds) when performing calculations to avoid errors.

Learn how to calculate the fundamental frequency (440.5 Hz) and second harmonic (880.9 Hz) of a sound wave in an open tube with a length of 40 cm and air temperature at 35°C. This step-by-step solution explains key wave concepts and formulas for frequency, wavelength, and speed of sound.

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