Final answer:
To calculate the first three standing wave frequencies for a tube open at both ends, one must use the fundamental relationship f = v/λ, where v is the speed of sound and λ is the wavelength. The fundamental wavelength is twice the length of the tube, and the first three harmonics have wavelengths of 2L, L, and 2L/3 respectively. Using the speed of sound in air, the frequencies can be determined.
Step-by-step explanation:
The student is asking about calculating the first three standing wave frequencies for a tube that is open at both ends, considering the tube's length and diameter. Based on the information provided, the fundamental wavelength for a tube open at both ends is twice the length of the tube (2L). Since the frequencies of the standing waves are integer multiples of the fundamental frequency, the first three frequencies can be calculated using the formula f = v/λ, where v is the speed of sound in the tube and λ is the wavelength of the standing wave.
To find the first three standing wave frequencies, we need to calculate the wavelengths for the fundamental and the next two higher harmonics. The fundamental frequency (first harmonic) corresponds to a wavelength that is twice the length of the tube (λ = 2L). The second harmonic corresponds to a wavelength equal to the length of the tube (λ = L), and the third harmonic is two-thirds the length of the tube (λ = 2L/3).
The relationships between the lengths of the tube and the wavelengths for the harmonics are:
- Fundamental (first harmonic): λ₁ = 2L
- First overtone (second harmonic): λ₂ = L
- Second overtone (third harmonic): λ₃ = 2L/3
Once the wavelengths are determined, we can use the speed of sound in air (which depends on temperature) to calculate the frequencies for each harmonic by using the fundamental relationship f = v/λ.