Final answer:
The first two resonant positions in a resonance tube closed at one end are one half of the wavelength apart. The tube length accommodates a quarter of a wavelength at the fundamental frequency and three-quarters of a wavelength at the first overtone, indicating the addition of half a wavelength between these two resonant positions. Option c is the correct answer.
Step-by-step explanation:
When discussing resonance in a tube closed at one end, we must consider how standing waves are formed and what portion of the wavelength fits within the tube. At the first resonant position, the tube supports a standing wave with the length of the tube equal to one quarter of the wavelength of the sound wave. This is because there is an antinode at the open end where air displacement is maximum, and a node at the closed end where there is no air displacement.
Thus, the first harmonic or fundamental frequency occurs when the length of the tube, L, is equal to one quarter of the wavelength (λ), that is L = λ/4. For the first overtone (second harmonic), three-fourths of the wavelength fits within the tube, making the length of the tube one-third larger than the wavelength present for the fundamental frequency, or L = (3/4)λ, which rearranges to λ = 4L/3.
The resonant positions are essentially the points at which a new antinode or node is added to the system, thereby changing the number of quarter-wavelengths that fit within the length of the tube. When considering the distance between the first two resonant positions, we must understand that from the fundamental frequency to the first overtone, the length of the tube accommodates an additional half of a wavelength. This means that from the fundamental frequency (one quarter of a wavelength) to the first overtone (three-quarters of a wavelength) there is an increase by half a wavelength within the tube.
Therefore, in terms of the wavelength of the sound wave, the first two resonant positions in the resonance tube are one half of the wavelength apart. This is the distance between a tube supporting a standing wave at the fundamental frequency and one supporting a standing wave at the first overtone. Hence, the correct option to the student's question is that the first two resonant positions are one half of the wavelength apart.