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    Re: Resonance
    From: Frank Reed CT
    Date: 2003 Dec 29, 18:12 EST
    Fred, you wrote:
    "resonance is not a subject with
    which I am strongly conversant, and I finally came up with an example
    that might be helpful to the similarly challenged.  My example is a
    "Slinky" "

    Ya picked a good one! A driven spring is the example of resonance that's usually taught first to physics students today.

    And you wrote:
    " If you hold one at the end and let it drop, a slight upward
    flick at just the right moment with cause it to spring back up toward
    your hand, whereas a similar flick at another moment has almost the
    opposite effect, dampening the spring action and causing it to dangle
    from your hand, with no satisfying spring back.  I expect the flick
    that causes the Slinky to spring back to your hand is in phase with the
    natural resonance of this system, while the worst example of the other
    flick is one-hundred eighty degrees out of phase."

    You're very close. The phase difference at resonance is 90 degrees. That means that you're at the bottom of your downstroke just as the other end of the spring is at the maximum speed portion of its downward motion.

    In general, any one-dimensional oscillator with a single natural frequency will have exactly the same behavior as a spring. An electrical circuit shows the same behavior, for example. More complicated systems will have different phase relationships. In the case of the tides, the astronomical driving force is known. It can be spectrally analyzed into a series of frequencies. The definitive analysis was published by George Darwin in about 1885 (?).

    The specific response of the ocean to the astronomical driving force has generally been measured rather than calculated. The measurement yields an amplitude and a phase for each frequency in the driving force. With those measured harmonic constants and the predictable changes in the future astronomical driving force, it's possible to predict the tides many years into the future. 

    It's interesting (maybe!) that the harmonic analysis system cannot tell the difference between "real" tides driven fundamentally by the gravity of the Sun and the Moon and other periodic changes in water level that just happen to have the exact same frequency as the astronomical components. For example, if a tidal station has a daily change in water level caused by daily afternoon thunderstorms (e.g.), the signature from runoff cannot be distinguished from the daily solar tide (S1) and it will be folded into that harmonic constituent whether we like it or not.

    Most shallow-water coastal stations have a fairly large "annual" tide (SA). In the astronomical analysis, this annual tide is connected with perihelion, but it should yield only a very small change in the tides. The amplitude of this harmonic should be very small at most tidal stations. Instead, measurements show that the overall water level can be higher by as much as six inches in summer. This ends up in the harmonic analysis as an "annual" tide, but its origin is non-gravitational. Usually, it's caused by summer heating --the water gets warmer and expands. Tide tables almost always include this annual expansion of water on the same footing as the astronomical tides.

    Frank E. Reed
    [X] Mystic, Connecticut
    [ ] Chicago, Illinois
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