Tides

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7. The tidal wave over resonance

Each of the two forced standing waves considered above can be regarded as a steady-state forced oscillation of the ocean under a sinusoidal driving force. Each mode is characterized by a single normal coordinate whose time dependence is found as the periodic solution to the differential equation describing the forced motion of an ordinary damped linear oscillator.

For small damping, the forced oscillation occurs almost in the same phase with the driving force if the driving force varies slowly, that is, if the driving period is longer than the natural period of the mode. Otherwise the forced oscillation has almost the opposite phase with respect to the driving force. To observe such an antiphase oscillation in the simulation, we should enter a value of the natural period greater than the driving period (the driving period is used as the time unit in the simulation). Click here to observe the first standing tidal wave in these conditions. (Click also here to see the applet.)

If the natural period of the oscillatory mode that is excited by the tidal forces is longer than the principal driving period (12 hours for sun-induced tides and 12 hours 25 minutes for moon-induced tides), both forced oscillations (standing waves) occur almost in the opposite phase with respect to the driving forces. This means that the axis of the ellipsoidal circulating forced wave produced by the superposition of these standing waves lags behind the rotating line earth-sun (or earth-moon) by an angle of almost 90 degrees. Click here to observe the circulating tidal wave in these conditions. (Click also here to see the applet.)

For the model of a planet wholly covered by the water envelope of equal depth of about 3 - 4 km, the natural period exceeds the driving period. Therefore the bulges of the tidal wave are aligned almost perpendicularly to the earth-moon (or earth-sun) line, and the flow occurs about the time when the moon (or the sun) is at the horizon.

Conditions of resonance between the tide-generating forces and the natural oscillations of the ocean would be fulfilled if the driving and natural periods were equal. For this hypothetical situation, the phase lag of the steady-state forced oscillations behind the forces equals a quarter of the driving period. In this case the bulges of the circulating tidal wave would lie on the line that lags by 45 degrees behind the earth-moon (or earth-sun) line. Click here to observe the tidal wave in conditions of resonance. (Click also here to see the applet.)

Even though at resonance the tidal wave would have gained much greater amplitude than in the real world situation, the amplitude shown in this simulation is certainly strongly exaggerated.

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The Oceanic Tides Section 7 (of 8)