The Physics of Music - Physics 15 University of California, Irvine ...

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The Physics of Music - Physics 15 University of California, Irvine Instructor: David Kirkby dkirkby@uci.edu Resonance and Damping Why doesn’t the swing keep getting higher and higher until you are doing circles An idealized resonant response builds an unlimited amount of energy. Realistic resonant systems do not do this because of dissipation, i.e., they are damped. Compare the motion of the swing when it is pumped at the right frequency but with different amounts of damping. Resonant Frequencies A physical system may have one or more frequencies at which resonances build up. These are called resonant frequencies (or natural frequencies). The basic requirements for a system to be resonant are that: • It have well-defined and stable boundary conditions, • That it not have excessive damping. This means that most systems have at least one type of resonance! Resonant frequencies are often in the audible range (about 20-20,000 Hz). Try tapping an object to hear its resonant response. Physics of Music, Lecture 5, D. Kirkby 7 Physics of Music, Lecture 5, D. Kirkby 8 A system may have more than one resonant frequency. We call the lowest resonant frequency the fundamental frequency. Any higher frequencies are called overtones. The playground swing has only one resonant frequency. Most of the systems responsible for generating musical sound have many resonances. We will see examples of systems with overtones later in this lecture. A familiar (non-musical) example occurs when different parts of a car rattle at certain speeds. Visualizing Resonance A resonance curve measures how much total energy builds up when a fixed (small) amount of energy is delivered periodically. It is described the the mathematical function: y(x) = 1/(1+x 2 ) Energy Buildup too slow just right log(Driving Frequency) logarithmic axis! too fast http://www.2dcurves.com/cubic/cubicr.html Physics of Music, Lecture 5, D. Kirkby 9 Physics of Music, Lecture 5, D. Kirkby 10 Sidebar on Logarithmic Graph Axes Moving one unit to the right on a normal (linear) graph axis means add a constant amount. Moving one unit to the right on a logarithmic axis means multiply by a constant amount. Example: the exponential decay law (e.g., from damping) results in a decrease by a fixed fraction after each time interval. What would this look like if time is plotted on a logarithmic axis Musical notes (A,B,C,…,G) correspond to logarithmically-spaced frequencies. Therefore a piano keyboard or a musical staff are actually logarithmic axes in disguise! Energy Buildup Physics of Music, Lecture 5, D. Kirkby 11 Physics of Music, Lecture 5, D. Kirkby 12 2
The Physics of Music - Physics 15 University of California, Irvine Instructor: David Kirkby dkirkby@uci.edu Damping and Resonance Quality The amount of damping determines how long a sound takes to die away when you stop adding energy. Resonance Curves of Different Q (curves are rescaled to all go through this point) It also determines how sharply peaked the resonance curve is. We measure this sharpness with a “quality factor” or “Q-value”: Q = resonant frequency / curve width We say that a sharply peaked resonance curve corresponds to a “High-Q” resonator and that a broad resonance curve corresponds to a “Low-Q” resonator. Normalized Energy Buildup log(Driving freq. / Fundamental freq.) Physics of Music, Lecture 5, D. Kirkby 13 Physics of Music, Lecture 5, D. Kirkby 14 Go back to the swing demonstration to see the effect of changing the amount of damping. The famous Tacoma Narrows disaster is an example of a complicated mechanical system that had a resonance (driven by wind) of an unexpectedly high Q. Resonance and Phase Shift If you are pumping a swing below its resonant frequency, the swing responds in synch (in phase) with your pumping. What happens if you pump faster than the swing’s resonant frequency Go back to the swing demonstrations to find out… At frequencies above the resonant frequency, the motion of the swing lags behind. Far above the resonance, the swing motion is the negative of the driving force. In this case, we say that the driving force and the swing motion are 180 o out of phase (or just out of phase). Physics of Music, Lecture 5, D. Kirkby 15 Physics of Music, Lecture 5, D. Kirkby 16 Back to One Dimensional Ropes We have already considered different boundary conditions at one end of a rope. We assumed that the rope was long enough that we could ignore its other end. What if the rope is not so long and we allow reflections from both ends For example, one end might be fixed and the other held (which means fixed + driven). The Rope is a Resonator This is just a combination of boundary conditions that we have seen before, but a fundamentally new feature emerges: resonance! The source of periodic energy is the person wiggling one of the rope at a fixed frequency. The buildup of energy is evident in the amplitude of the rope’s transverse motion. The resonant response is called a standing wave. Try this demo to see for yourself. Physics of Music, Lecture 5, D. Kirkby 17 Physics of Music, Lecture 5, D. Kirkby 18 3
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