Final answer:
The fundamental frequency of an organ pipe is inversely proportional to its length (1/l). Closed-pipe resonators have a fundamental wavelength of 4L, while open-pipe resonators have a fundamental wavelength of 2L. Both types exhibit a fundamental frequency that is dependent on the speed of sound, which varies with temperature.
Step-by-step explanation:
The fundamental (lowest) frequency of an organ pipe is proportional to 1/l, where 'l' is the length of the pipe. This relationship comes from the fact that, for an organ pipe that is closed at one end (also called a closed-pipe resonator), the fundamental wavelength (λ) is four times the length of the pipe (λ = 4L). The frequency (f) is calculated by dividing the speed of sound (v) by the wavelength (f = v/λ), leading to f = v/(4L), which simplifies to f being proportional to 1/L. For an organ pipe that is open at both ends, the fundamental wavelength is twice the length of the pipe (λ = 2L). Hence, the fundamental frequency is f = v/(2L), again showing that the frequency is inversely proportional to the length of the pipe. Closed-pipe resonator: If an organ pipe closed at one end produces a fundamental frequency of 256 Hz when air temperature is 18.0°C, we can find the length of the pipe using the relationship f = v/(4L). The speed of sound varies slightly with temperature, which would also affect the fundamental frequency. Open-pipe resonator: For a pipe open at both ends, like a flute or an oboe, the fundamental frequency is achieved when the standing wave inside the pipe has a wavelength of twice the length of the pipe.