Final answer:
To find the path difference between the origin of a 17,000 Hz sound and each ear (20 cm apart), we use the sound's wavelength and the given phase difference. The exact path difference is approximately 2.52 cm, which suggests that the closest answer is 5 cm, although it is not an exact match.
Step-by-step explanation:
The question concerns a wave with a frequency of 17,000 Hz that creates a phase difference of 5π radians between the sound arriving at each ear of an average adult male, where the ears are 20 cm apart. To determine the difference in distance (path difference) between the origin of the sound and each ear, we use the speed of sound in air and the given frequency. Given the speed of sound in air as 343 m/s, we first find the wavelength of the sound:
- Wavelength (λ) = Velocity (v) / Frequency (f)
- λ = 343 m/s / 17,000 Hz
- λ ≈ 0.0201765 m or ≈ 2.01765 cm
The phase difference in terms of wavelength is:
- Phase difference (in radians) = 2π * (Path difference) / Wavelength
- Path difference = (Phase difference / (2π)) * Wavelength
- Path difference = (5π / (2π)) * 2.01765 cm
- Path difference ≈ 5 * 2.01765 cm / 2
- Path difference ≈ 5.044125 cm / 2
- Path difference ≈ 2.5220625 cm
The closest answer from the options is B. 5 cm, but this is rounded since the exact calculated path difference is approximately 2.52 cm.