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Math Help - exponential decay

  1. #1
    ldenk
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    exponential decay

    Hi!

    I don't understand the following:

    F = 1/tf * V

    F... vector quantity, represents the friction
    tf ... friction timescale
    V ... velocity of an air parcel V= (u,v,w), also called wind velocity

    In the Article "Ray tracing volume densities" Kajiya and Von Herzen (1984) writes:
    "Frictional effects are approximated by a simple relation yielding an exponential decay of wind velocities with time"
    (the wind velocities are described by a simplified equation of motion and also
    change at every time step)

    Why is this a exponential decay ?!

    hope you can help me
    ldenk
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ldenk
    Hi!

    I don't understand the following:

    F = 1/tf * V

    F... vector quantity, represents the friction
    tf ... friction timescale
    V ... velocity of an air parcel V= (u,v,w), also called wind velocity

    In the Article "Ray tracing volume densities" Kajiya and Von Herzen (1984) writes:
    "Frictional effects are approximated by a simple relation yielding an exponential decay of wind velocities with time"
    (the wind velocities are described by a simplified equation of motion and also
    change at every time step)

    Why is this a exponential decay ?!

    hope you can help me
    ldenk
    The friction on the parcel should be in the opposite direction to the
    velocity so you should have:

    F = -V/t_f
    Then Newton tells us that:

    \frac{dV}{dt}\ =\ m.F
    which is:

    \frac{dV}{dt}\ =\ -m.V/t_f
    and as both m and t_f are positive the solution
    of the differential equation has the form of exponential
    decay.

    (I would also assume that at some point things have been normalised
    so that F is the frictional force per unit mass, and so the value of
    m that appears here should be 1.)

    RonL
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