The Whispering Gallery Interference · AI Generator #GEN-5A8333FD
Physics
Problem
A famous whispering gallery has a circular cross-section with radius R=12.0m. Two musicians stand at opposite ends of a diameter, each playing identical tuning forks at frequency f=440Hz. However, one musician's tuning fork is slightly detuned and produces a frequency that is Δf=2.0Hz higher.
Due to the acoustic properties of the gallery, sound waves travel along the curved walls in both directions around the circular perimeter before reaching the opposite side. A listener standing at the midpoint of the diameter (the center) hears the superposition of four wave components:
- Direct waves from both musicians
- Clockwise circumferential waves from both musicians
- Counterclockwise circumferential waves from both musicians
The direct path length from each musician to the center is R, while the circumferential path length is half the circle's circumference. The speed of sound is v=343m/s.
(a) What is the beat frequency heard by the listener from the direct waves alone?
(b) What is the phase difference between the direct wave and the circumferential wave from the same musician when they arrive at the center?
(c) What is the total amplitude of oscillation at the center as a function of time, expressed in terms of the amplitude A0 of each individual wave component?