Simple Nature - Light and Matter

Collapse of the Nimitz Freeway example 47Figure l shows a section of the Nimitz Freeway in Oakland, CA,that collapsed during an earthquake in 1989. An earthquake consistsof many low-frequency vibrations that occur simultaneously,which is why it sounds like a rumble of indeterminate pitch ratherthan a low hum. The frequencies that we can hear are not eventhe strongest ones; most of the energy is in the form of vibrationsin the range of frequencies from about 1 Hz to 10 Hz.l / The collapsed section ofthe Nimitz FreewayAll the structures we build are resting on geological layers of dirt,mud, sand, or rock. When an earthquake wave comes along, thetopmost layer acts like a system with a certain natural frequencyof vibration, sort of like a cube of jello on a plate being shakenfrom side to side. The resonant frequency of the layer dependson how stiff it is and also on how deep it is. The ill-fated sectionof the Nimitz freeway was built on a layer of mud, and analysis bygeologist Susan E. Hough of the U.S. Geological Survey showsthat the mud layer’s resonance was centered on about 2.5 Hz,and had a width covering a range from about 1 Hz to 4 Hz.When the earthquake wave came along with its mixture of frequencies,the mud responded strongly to those that were close toits own natural 2.5 Hz frequency. Unfortunately, an engineeringanalysis after the quake showed that the overpass itself had a resonantfrequency of 2.5 Hz as well! The mud responded strongly tothe earthquake waves with frequencies close to 2.5 Hz, and thebridge responded strongly to the 2.5 Hz vibrations of the mud,causing sections of it to collapse.Physical reason for the relationship between Q and the FWHMWhat is the reason for this surprising relationship between thedamping and the width of the resonance? Fundamentally, it has todo with the fact that friction causes a system to lose its “memory”of its previous state. If the Pioneer 10 space probe, coasting throughthe frictionless vacuum of interplanetary space, is detected by aliensa million years from now, they will be able to trace its trajectorybackwards and infer that it came from our solar system. On theother hand, imagine that I shove a book along a tabletop, it comesto rest, and then someone else walks into the room. There will beno clue as to which direction the book was moving before it stopped— friction has erased its memory of its motion. Now consider theplayground swing driven at twice its natural frequency, figure m,where the undamped case is repeated from figure b on page 171. Inthe undamped case, the first push starts the swing moving with momentump, but when the second push comes, if there is no frictionat all, it now has a momentum of exactly −p, and the momentumtransfer from the second push is exactly enough to stop it dead.With moderate damping, however, the momentum on the reboundis not quite −p, and the second push’s effect isn’t quite as disas-182 Chapter 3 Conservation of Momentum