The Resonant Seismic Station · AI Generator #GEN-36512D45
Physics
Problem
A seismologist designs an underground detection station to monitor earthquakes. The station consists of a long horizontal tunnel of length L=240m carved into solid bedrock. The tunnel has rigid walls and is sealed at both ends with massive concrete plugs that effectively act as fixed boundaries for acoustic waves.
During a distant earthquake, seismic P-waves traveling through the bedrock generate acoustic waves inside the tunnel. These acoustic waves have a speed of v=340m/s in the air-filled tunnel.
The seismologist notices that the tunnel exhibits particularly strong resonance (maximum amplitude oscillations) when the dominant frequency of the incoming seismic waves is f1=0.708Hz. She also observes that the next higher frequency showing strong resonance is f2=2.125Hz.
However, due to the complex coupling between the bedrock and the air in the tunnel, the effective acoustic length of the tunnel differs from its physical length L. The acoustic waves experience additional phase shifts at each end due to the rock-air interface.
(a) Determine the effective acoustic length Leff of the tunnel.
(b) Calculate the total additional path length ΔL=Leff−L that the acoustic waves effectively travel due to the end effects.