Math Problem Statement

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.

Solution

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+0.6Tv = 331.4 + 0.6 \cdot T where TT is the temperature in Celsius.

    For T=35CT = 35^\circ \text{C}: v=331.4+0.635=331.4+21=352.4 m/sv = 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 λ1\lambda_1 that is twice the length of the tube: λ1=2L\lambda_1 = 2L
    • The second harmonic has a wavelength λ2\lambda_2 that is the length of the tube: λ2=L\lambda_2 = L

    Given the length of the tube L=40 cm=0.4 mL = 40 \text{ cm} = 0.4 \text{ m}: λ1=20.4=0.8 m\lambda_1 = 2 \cdot 0.4 = 0.8 \text{ m} λ2=0.4 m\lambda_2 = 0.4 \text{ m}

  3. Calculate the Fundamental Frequency (First Harmonic):

    The frequency ff is related to the speed of sound vv and the wavelength λ\lambda by: f=vλf = \frac{v}{\lambda}

    For the fundamental frequency: f1=vλ1=352.40.8=440.5 Hzf_1 = \frac{v}{\lambda_1} = \frac{352.4}{0.8} = 440.5 \text{ Hz}

  4. Calculate the Second Harmonic Frequency:

    For the second harmonic: f2=vλ2=352.40.4=880.9 Hzf_2 = \frac{v}{\lambda_2} = \frac{352.4}{0.4} = 880.9 \text{ Hz}

Summary:

  • Fundamental frequency (first harmonic): 440.5 Hz440.5 \text{ Hz}
  • Second harmonic: 880.9 Hz880.9 \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.

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Math Problem Analysis

Mathematical Concepts

Wave Propagation
Frequency
Wavelength
Speed of Sound

Formulas

v = 331.4 + 0.6 * T (Speed of sound in air at temperature T)
f = v / λ (Frequency of wave)
λ_1 = 2L (Wavelength for the fundamental frequency)
λ_2 = L (Wavelength for the second harmonic)

Theorems

Wave Harmonics in Open Tubes

Suitable Grade Level

Grades 10-12